Additional comments related to material from the class. If anyone wants to convert this to a blog, let me know. These additional remarks are for your enjoyment, and will not be on homeworks or exams. These are just meant to suggest additional topics worth considering, and I am happy to discuss any of these further.

• Friday, May 14. We have seen numerous times the importance of optimizing, of finding maxima and minima. For a continuous, differentiable function f of one variable the candidates are the critical points (all x where f'(x) = 0) and the endpoints. To determine if a critical point is a maximum, minimum or neither, we have the First Derivative Test and the Second Derivative Test. The tests become more complicated in several variables. We discussed a continuous, differentiable function f of two variables. At a critical point its gradient vanishes, and thus its second order Taylor series looks like T(x,y) = f(0,0) + (1/2) (x y) (Hf)(0,0) (x y)^T, where (x y)^T means the column vector. If we take the matrix Hf to have first row (2 0) and second row (0 3) we find T(x,y) = (1/2)[2 x^2 + 3 y^2], which is always positive for (x,y) distinct from (0,0). We thus see that we have a minimum. In other words, to generalize the second derivative test to several variables we must figure out what the generalization of g''(0) > 0 is. It becomes (x y) (Hf)(0,0) (x y)^T > 0 for (x y) not equal to zero. This is called positive definiteness; there are similar generalizations for maxima. The easiest way (or at least a common way) to determine if a matrix is positive definiteness is to compute its eigenvalues; if all the eigenvalues are positive it is positive definite, if all are negative its negative definite, and if some are positive and some negative it is indefinite (if some are zero we use the word indefinite). The mathematics behind these classifications and eigenvalues is related to much of our course, especially the Change of Variable formula and finding equations of curves. See the Principal Axis Theorem for more details (which includes how to write down the equation of an ellipse whose axes are not aligned with the standard coordinate axes).

• Wednesday, May 12. Today was a fast introduction to path integrals, line integrals, and Green's Theorem (which is a special case of the Generalized Stokes' Theorem). While our tour of these subjects has to be rushed in a 12 week course, if you are continuing in certain parts of math, physics or engineering you will meet these again and again (for example, see Maxwell equations for electricity and magnetism). In fact, one can view all of classical mechanics as path integrals where the trajectory of the particle (its c(t)) minimizes the action; there is also a path integral approach to quantum mechanics.
  • For those continuing in mathematics or physics, you will see these ideas again if you take complex analysis. In particular, one of the gems of that subject is Cauchy's Integral Theorem, A complex differentiable function satisfies what is called the Cauchy-Riemann equations, and these are essentially the combination of partial derivatives one sees in Green's theorem. In other words, the mathematics used for Green's theorem is crucial in understanding functions of a complex variable.
  • For me, I consider it one of the most beautiful gems in mathematics that we can in some sense move the derivative of the function we're integrating to act on the region of integration! This allows us to exchange a double integral for a single integral for Green's theorem (or a triple integral for a double integral in the divergence theorem). As we've seen constantly throughout the year, often one computation is easier than another, and thus many difficult area or volume integrals are reduced to simpler, lower dimensional integrals.
  • The fact that Int_{t = a to b} grad(f)(c(t)) . c'(t) dt = f(c(b)) - f(c(a)) means that this integral does not depend on the path. If a vector field F = (F1, F2, F3) equals grad(f) for some f, we say F is a conservative force field and f is the potential. The fact that these integrals do not depend on the path has, as you would expect, profound applications.
  • This is a good point to stop and think about the number of spatial dimensions in the universe. Imagine a universe with two point masses under gravity, and assume gravity is proportional to 1/r^{n-1} with r the distance between the masses and n the number of spatial dimensions. If there are three or more dimensions, then the work done in moving a particle from infinity to a fixed, non-zero distance from the other mass is finite, while if there are two dimensions the work is infinite! One should of course ask why the correct generalization to other dimensions is 1/r^{n-1} and not 1/r^2 always. There is a nice geometric justification in terms of flux and surface area; the surface area of a sphere grows like r^2 and thus the only way to have the total flux of force out of it be constant is to assume the force drops like 1/r^2; click here for a bit on the justification of inverse-square laws.
  • Speaking of dimensions, one of my favorite problems from undergraduate days was the Random Walk. In 1-dimension, imagine a person so completely drunk that he/she has a 50% chance at any moment of stepping to the left or the right; what is the probability the drunkard eventually returns home? It turns out that this happens with probability 1. In 2-dimension, we have a 25% chance of moving north, south, east or west, and again the probability of returning is 1. In 3 dimensions, however, the drunkard only returns home with probability about 34%. As my professor Peter Jones said, a three-dimensional universe is the smallest one that could be created that will be interesting for drunks, as they really get to explore! These random walk models are very important, and have been applied to economics (the random walk hypothesis), as well as playing a role in statistical mechanics in physics.

• Monday, May 10. Today we began our drive to the end of the semester, which will culminate in a statement of the Big Three theorems of Vector Calculus (Green's Theorem, Gauss' Divergence Theorem, and Stokes' Theorem). These theorems are massive generalizations of the Fundamental Theorem of Calculus, which can be generalized even more. The idea is to relate the integral of the derivative of something over a region to the integral of the something over the boundary of the region. To state these theorems requires many concepts from vector calculus (parametrizing curves, vectors, ...) as well as the Change of Variable theorem (converting integrals over curves and surfaces to intergrals over simpler curves and surfaces).
  • To truly see and appreciate the richness of the three theorems (which are really three variants of the same theorem), one must be in at least three dimensions. There the Stokes' Theorem states that the integral of a certain function over a surface equals the integral of another over the boundary curve. This means that many integrals turn out to be the same.
  • To see the equivalence of these formulations requires differential forms. Frequently it is not immediately clear how to generalize a concept to higher dimensions or other settings.
  • While we only briefly touched on the subject, conservative forces are extremely important in physics and engineering, primarily because of a wonderful property they have: the work done in moving an object from A to B is independent of the path taken if the exerted force is conservative. Many of the most important forces in classical mechanics are taken to be conservative, such as gravity and electricity. In modern physics, these forces are replaced with more complicated objects. One of the central quests in modern physics is to unify the various fundamental forces (gravity, strong, weak and electricity and magnetism).
  • Click here for more on divergence, and click here for more on curl. Another related object (one we have seen many times) is the gradient. All of these involve the same differential operator, called del (and represented with a nabla). We used our intuition for vectors to define new combiantions involving the del operator (the curl and the divergence). While our intuition comes from vectors, we must be careful as we do not have commutivity. For example, nabla dot F is not the same as F dot nabla; the first is a scalar (number) while the second is an operator. Click here for more on differential operators. For those who want to truly go wild on operators, modern quantum mechanics replaces concepts like position and momentum with differential operators (click here for the momentum operator)! This allows us to rewrite the Heisenberg uncertainty principle in the following strange format.
  • One of the most famous applications of these concepts is the Navier-Stokes equation, which is one of the Millenium Problems (solving one of these is probably the hardest path to one million dollars!). The Navier-Stokes equation describes the motion of fluids, which not surprisingly has numerous practical (as well as theoretical) applications. Click here for a nice derivation, which includes many of the new operators we saw today.
  • Another place where gradients, curls and divergences appear is the Maxwell equations for electricity and magnetism; you can view the equations here.
  • The General Stokes Theorem is a massive generalization of the fundamental theorem of calculus. The idea of formally moving the derivative from the function to the region of integration is meant to be suggestive, but of course is in no ways a proof. Notation should help us see connections and results. The great physicist Richard Feynman showed that all of physics is equivalent to solving the equation U = 0, where U measure the unworldliness of everything. It is made up of squaring the differences between the left and right hand sides of every physical law. Thus it has terms like (F - ma)^2 and (E - mc^2)^2. It is a concise way of encoding information, but it is not useful; everything is hidden. This is very different than the vector calculus formulations of electricity and magnetism, which do aid our understanding. For more information, skim the article here (search for unworldliness if you wish).
  • We saw that we can compute the lengths of curves by evaluating integrals of ||c'(t)||, where c(t) = (x(t), y(t), z(t)) is our curve. While this formulation immediately reduces the problem of finding lengths to a Calc II problem, in general these are very difficult integrals, and frequently cannot be done in closed form even for simple shapes. For example, for extra credit find the length of the ellipse (x/a)^2 + (y/b)^2 = 1. Click here for the solution (the answer involves the elliptic integral of the second kind).
  • Finally, we talked today about generalizing the Fundamental Theorem of Calculus. There are not that many fundamental theorems in mathematics -- we do not use the term lightly! Other ones you may have seen are the Fundamental Theorem of Arithmetic and the Fundamental Theorem of Algebra; click here for a list of more fundamental theorems (including the Fundamental Theorem of Poker!).

• Friday, May 7. Today we saw proofs of the various convergence / divergence tests, as well as more examples.
  • A nice application of a series expansion is Stirling's formula for n!. We can get close to the correct value by the integral test or the Euler-MacLaurin summation formula, This builds on a question asked in each section: we know the integral test tells whether or not a series converges; if it does converge, how close is the sum to the integral? The Euler-MacLaurin formula teaches us how to convert sums to integrals and bound the error.
  • The fact that Sum_{n = 1 to oo} 1/n^2 = pi^2/6 has a lot of applications. It can be used to prove that there are infinitely many primes via the Riemann zeta function. The Riemann zeta function is zeta(s) = Sum_{n = 1 to oo} 1/n^s. By unique factorization (also known as the Fundamental Theorem of Arithmetic), it also equals Prod_{p prime} 1 / (1 - 1/p^s); notice that a generalization of the harmonic sum and the geometric series formula are coming into play. It turns out that zeta(2) =  pi^2/6, as can be seen in many different ways. As pi^2 is irrational, if there were only finitely many primes then the product would be irrational, contradiction! See wikipedia for a proof of this sum.
    • 6/pi^2 is the probability that two randomly chosen integers are relatively prime.
    • It's also the probability a randomly chosen integer is square-free!
    • We can also use the fact that Sum_{n=1 to oo} 1/n diverges to prove there are infinitely many primes: if there were only finitely many primes then as s --> 1 from above the product would be finite, and that contradicts the sum diverging!
    • A large amount of modern mathematics involves taking a series that only converges for certain values and `extending' it. For example, 1 + 2 + 4 + 8 + 16 + ... really does, in some sense, equal -1. This is the principle of meromorphic continuation, one of the central themes of complex analysis. We can similarly continue the zeta function, and we find zeta(-1) = 1 + 2 + 3 + 4 + ... = -1/12.
  • Another interesting application of summing series involving primes is to the Pentium bug (see the links there for more information, as well as Nicely's webpage). The calculation being performed was sum_{p: p prime and either p+2 or p-2 is prime} 1/p; this is known as Brun's constant. If this sum were infinite then there would be infinitely many twin primes, proving one of the most famous conjectures in mathematics; sadly the sum is finite and thus there may or may not be infinitely many twin primes (twin primes are two primes differing by 2).
  • The proof we gave today of the geometric series formula (by shooting baskets) uses many great techniques in mathematics. It is thus well worth it to study and ponder the proof.
    • Memoryless process: once both people miss, it is as if we've just started the game fresh.
    • Calculating something two different ways: a good part of combinatorics is to note that there are two ways to compute something, one of which is easy and one of which is not. We then use our knowledge of the easy calculation to deduce the hard. For example, Sum_{k = 0 to n} (n choose k)^2 = (2n choose n); the right side is easy to compute, the left side not so clear.Why are the two equal? It involves finding a story, which we leave to the reader.
  • For another example of applications of harmonic numbers, see the coin collector problem (if you want more info on this problem, let me know -- I have lots of notes on it from teaching it in probability).
  • In Section 2 today a basketball shot basically went in and out; see the following article on golf for some info on related problems in golf.
  • Finally, we talked about how the hardest part of the integral test is finding the appropriate function to use. Typically you try to replace all n's with x's. Thus if a_n = 1/n try f(x) = 1/x. If we have a_n = 1/n!, it's not so clear. What does x! mean if x is not an integer? This can be done through an extension of the factorial function to what is called the Gamma function. We have Gamma(n+1) = n! if n is an integer, and use this to generalize the factorial function.
    • The fact that Gamma(1/2) = (-1/2)! is one of the most important facts in probability. It's used in finding the normalization constants for the normal distribution and for the Student t-distribution. I give one of the standard proofs of this fact in my paper proving Wallis' formula for pi; my paper is available here (see page 3). Amazingly, this is related to computing zeta(2).

• Wednesday, May 5. We discussed the various tests for whether or not a series converges. We briefly discuss the tests below, and then we say a few words about the rate of convergence. I see a lot of similarities between these tests and finding roots of quadratic polynomials. The 'fastest' way to find the roots is to factor, but of course that only works if you can 'see' the roots. If you can't see them, you can use the mechanical grind of the quadratic formula. It's similar with these tests. The 'easiest / fastest' to use is the comparison test, but its effectiveness is tied to how many series you know that converge or diverge. If you can't see a good series to compare it with, you then move to the ratio and root tests, and then to the integral test.
  • The big tests are:
    • Comparison test: This is one of my favorites, though to be effective you must know how a lot of examples of convergent and divergent series.
    • Ratio Test: Remember that the ratio test provides no information if the value is 1; thus it says nothing about the convergence or divergence of 1/n^p for any fixed p > 0.
    • Root Test: Remember that the root test provides no information if the value is 1; thus it says nothing about the convergence or divergence of 1/n^p for any fixed p > 0.
    • Integral Test: The most common example is the harmonic sum, 1/n. The integral test not only gives the divergence, but with a bit more work shows that the sum of the first n reciprocals of integers is about ln(n).
  • We can use the tests above to show that the Taylor series expansions for exp(x), cos(x), sin(x), ln(1-x) et cetera converge in various neighborhoods (the first three for all x, the last for |x| < 1). Thus our results on convergence of series lets us know when we may replace a function with its Taylor series expansion. Amazingly, in complex analysis we have the following wondrous result: if a function of a complex variable z = x + i y is differentiable once with respect to z then it is infinitely differentiable and it equals its Taylor series expansion! This is violently false for functions of a real variable. What's going on? The difference is that to be differentiable once requires us to allow x + i y to tend to a prescribed value along any path, and this freedom of travel imposes sharp constraints as to what functions will be differentiable. The end result is that complex differentiable functions are a lot nicer than functions of one variable that are differentiable. Lurking in the background are the Cauchy-Riemann equations, implying that a form of Stokes theorem holds.
  • We wanted to show the series Sum_{n = 1 to oo} 1 / (4^n - 3^n) converges. I claimed that for n large, we have 4^n > 2 * 3^n. We can prove this by using logarithms. Say we have 4^n ? 2 * 3^n; we want to figure out is ? equal to > or to <.Taking logarithms of both sides gives n ln(4) ? n ln(3) + ln(2), where we used several log rules to simplify ln(2*3^n). Subtracting gives n( ln(4) - ln(3) ) ? ln(2), or n ? ln(2) / ln(4/3) (where we again used log rules, and noted that ln(4) - ln(3) = ln(4/3) > 0, so dividing by it does not change the relation). As ln(2) / ln(4/3) is about 2.41, we see that for n > 2 that 4^n > 2*3^n,which is what we needed to show. This type of argument is used all the time.
  • Finally, last class we mentioned that certain notations become standardized, no matter how much we may desire to change them. A great example of how things have become fixed is the qwerty keyboard -- if we were to start from scratch, we would not choose this! Another favorite of mine is the use of cc in an email -- anyone remember what cc standa for?

• Minday, May 3. We finished the last of the Big Three coordinate changes, spherical coordinates (be aware that physicists and mathematicians have different definitions of the angles!) and started sequences and series.
  • One can generalize spherical coordinates to hyperspheres in n-dimensional space. These lead to wonderful applications of special functions, such as the Gamma function, in writing down formulas for the `areas' and `volumes'.
  • There are many fascinating question involving spheres:
    • The kissing number, which is related to spherical codes.
    • Sphere packing (the special case in 3-dimensional space is known as the Kepler Conjecture, and a proof was presented by Hales in 1998).
  • One of the most important applications of spherical coordinates is to planetary motion, specifically, proving that the force one sphere exerts on another is equivalent to all of the mass being located at the center of the sphere. This is the most important integral in Newton's great work, Principia (we have a first edition at the library here). I strongly urge everyone to look at this problem. Proving that one can take all of the mass to be at the center enormously simplifies the calculations of planetary motion. See the Wikipedia article on the Shell Theorem for the computation. As this is so important, here is another link to a proof. Oh, let's do another proof here as well as another proof here. For an example of a non-proof, read the following and teh comments.
  • Instead of the standard examples of sequences and series, it is fun to explore some of the more exotic possibilities:
    • The look and say (or the see and say) sequence of John Conway
    • The 3x+1 problem (I have a paper on the 3x+1 problem, concerning the distribution of leading digits of the iterates and applications to fighting tax fraud; this is related to Benford's law, and is a very important area of research that is accessible to undergraduates).
  • Standard examples of sequences and series include
    • The harmonic series (see also In Perfect Harmony by John Webb for examples of where it arises). The standard example given is that you can use this to place dominoes on top of each other hanging out infinitely far without having them fall or be supported other than their weight! See http://www.ken.duisenberg.com/potw/archive/arch03/030728sol.html as well as http://www.cs.cmu.edu/afs/cs/academic/class/16741-s07/www/projects06/chechetka_16-741_project_report.pdf.
    • The geometric series
    • The Fibonacci series (an example of a difference or recurrence equation)
  • An infinite series of surprises: a nice article going from the geometric series to the harmonic series to other important examples.
  • We mentioned that sequences and series are very important; two of the most powerful applications are Taylor series (approximating complicated functions with simpler ones) and Riemann sums (allowing us to calculate areas with integrals).
  • L'Hopital's rule is frequently used to analyze the behavior of sequences. Remember that you can only use it if you have 0 over 0 or infinity over infinity.
    • Extra credit: What is wrong with the following argument: Let's say we want to compute lim_{h --> 0} sin(h) / h; this is the most important trig limit. We use L'Hopital's rule and note that it is the same as lim_{h --> 0} cos(h) / 1; as cos(h) tends to 1, the limit is just 1. Why is this argument not valid? The answer is one of the most important principles in mathemtatics!

• Friday, April 30. We sketched the proof of the Change of Variable formula in two-dimensions. The main idea is to keep track of how the area changes under the mapping. The proof used many of the techniques and concepts we've studied all semester, including the cross product giving the area of a parallelogram and the derivative is defined as the tangent plane giving an excellent approximation. We then applied the change of variables formulas to polar coordinates and cylidrical coordinates. We saw how much easier problems with angular symmetry become.
  • Frequently we are confronted with the need to find the integral of a function that we have never seen. One approach is to consult a table of integrals (here is one at wikipedia; see also the table here). Times have changed from when I was in college. Gone are the days of carrying around these tables; you can access Mathematica's Integrator on-line, and it will evaluate many of these. One caveat: sometimes these integrals are doable but do not appear in the table in the form you have, and some work is required to show that they equal what is tabulated.
  • Probably my favorite example (and one of the most important!) of using polar coordinates to evaluate an integral is to find the value of the Gaussian integral Int_{x = -oo to oo} exp(-x^2)dx. Of course, it seems absurd to use polar coordinates for this as we are in one-dimension! Our book has a good discussion of this problem, as does the wikipedia page. This is one of the most important integrals in the world, and leads to the normalization constant for the normal distribution (also known as the bell curve or the Gaussian distribution), which may be interpreted as saying the factorial of -1/2 is the square-root of π!
  • Another good example, of course, is just computing the area of a circle! In Cartesian coordinates we quickly see we need the anti-derivative of sqrt(1 - x^2), which involves inverse trigonometric functions; it is very straightforward in polar! In fact, we can easily get the volume of a sphere by integrating the function sqrt(1 - x^2 - y^2) over the unit disk!
  • Famous tables are Abramowitz and Stegun and Gradshteyn and Ryzhik.
  • For those interested in some of the history of special functions and integrals, see the nice article here by Stephen Wolfram. There's a lot of nice bits in this article.
    • One of my favorites is the throw-away comment in the beginning on how the Babylonians reduced multiplication to squaring. Here's the full story. The Babylonians worked base 60; if you think memorizing our multiplication table is bad, consider their problem: 3600 items! Of course, you lose almost 1800 as xy = yx, but still, that's a lot of tablets to lug. To compute xy, the Babylonians noted that xy = ((x+y)^2 - x^2 - y^2) / 2, which reduces the problem to just squaring, subtracting and division by 2. There are more steps, but they are easier steps, and now we essentially just need a table of squares. This concept is still with us today: it's the idea of a look-up table, computing new values (or close approximations) from a small list. The idea is that it is very fast for computers to look things up and interpolate, and time consuming to compute from scratch.

• Wednesday, April 28. The Change of Variable formula ties together many of the topics of the semester and generalizes a similar result from one-variable calculus. With complicated formulas such as this, it is quite useful to look at special cases first to get a sense of what is going on and then to try and generalize, being aware of course that sometimes there are features that are missed in the special cases. I like to look at this formula as giving the exchange rate from measuring in one coordinate system to another. For example, going from Cartesian to Polar coordinates we cannot have dx dy go to dr dtheta, as dx dy would have units of meters-squared while dr dtheta has units of meters-radians and radians are essentially unitless. (As a side note, the most important unitless number in physics is the fine structure constant.) We will see later that dx dy transforms to r dr dtheta.
  • Our analysis shows that when we have a linear rescaling, say u = 2x and v = 3y, then if T(x,y) = (u,v) and T^{-1}(u,v) = (x,y), then dx dy transforms to |det(D T^{-1})|. Note how many concepts are being applied here. We have the derivative of a vector valued function and we have determinants. The reason for the absolute value is a bit tricky, but comes from the danger of having signed areas. Remember in Calc I that Int_{x = a to b} f(x) dx = - Int_{x = b to a} f(x) dx. We are looking at how the area elements transform. In order to make sure the areas are positive, we need to insert absolute values here.
  • Another caveat is where to evaluate our function when we integrate it over the transformed region. Assume we have a map T from xy-space to uv-space. Let R = T(S). What should Int Int_S f(x,y) dx dy equal in uv-space? It becomes Int Int_T(S) f(T^{-1}(u,v)) |det DT^{-1}(u,v)| du dv. This is similar to the chain rule. If A(x) = f(g(x)) remember A'(x) = f'(g(x)) g'(x) and not f'(x) g'(x). This is one of the most common mistakes, namely evaluating f at the wrong point. Similarly here we need to make sure we evaluate f at the right place. In uv-space, our inputs are u and v, but f is expecting as inputs x and y. As T sends x and y to u and v, T^{-1} sends u and v to x and y, and thus the new function say g(u,v) = f(T^{-1}(u,v)) is what we should integrate over T(S).
  • There are many applications of the Change of Variables formula, especially in probability theory; see here for a one-dimensional example (if you have access to JStor, here is one in economics).

• Friday, April 23.  In the first part of class we discussed Monte Carlo Integration. Two good papers on the subject are available here: Metropolis (beginnings of method) and Metropolis-Ulam (the Monte Carlo Method). The basic idea is that if you have a region inside a big rectangle (or box or hyperbox in higher dimensions) where it is easy to tell if a point is in or not, then by choosing numbers randomly you can approximate the area very well and very quickly. To quantify how fast and how accurate you are, you can use various results from probability theory, such as Chebyshev's Theorem or even better the Central Limit Theorem; the arguments rely on knowledge of the concepts of mean (or expected value) and variance (or standard deviation).
  • There are a multitude of applications of these techniques; see for example the wikipedia article on Monte Carlo Methods in Finance, or the book http://www.amazon.com/Financial-Engineering-Stochastic-Modelling-Probability/dp/0387004513 (a book on applying Monte Carlo to financial engineering).
  • We also talked about how to evaluate certain integrals quickly by noting that we're integrating an odd function over a symmetric region.
  • Certain definite integrals can be evaluated easily, even though the anti-derivatives are hard to find. We saw this for integrating cos^2(x) from 0 to π/4; there are many others. After doing the change of variable formula, we'll see this for integration exp(-x^2/2) from minus infinity to infinity (unlike the cosine example, there is NO closed form expression for the anti-derivative; it's not just that it is messy, it's that it doesn't exist!). Personally, I find it a bit miraculous at times that we can exploit identities and get clean expressions for quantities such as these.
  • In the sabermetrics lunch today we digressed a bit, and eventually started talking about Erdos number, Bacon numbers, and my favorite: Erdos-Bacon numbers. You can use MathSciNet (click on collaborative distance) to search for Erdos numbers, and the Oracle of Bacon for Bacon numbers. It is a very non-trivial problem to take a large network and find the shortest route between two points. Another good example of the mathematics of efficient searching is Google's PageRank.

• Wednesday, April 21.  Fubini's Theorem (changing the order of integrations) is one of the most important observations in multivariable calculus. For us, we assume our function f(x,y) is either continuous or bounded, and that it is defined on a simple region D contained in a finite rectangle. If D is an unbounded region, say D = {(x,y): x, y >= 0} then Fubini's theorem can fail for continuous, bounded functions. In class we did an example involving a double sum, where a_{0,0} = 1, a_{0,1} = -1, a_{0,n} = 0 for all n >= 2, then a_{1,0} = 0, a_{1,1} = 1, a_{1,2} = -1, and then a_{1,n} = 0 for all n >= 3, and so on. If we want to have a continuous function, we can tweak it as follows. Consider the indices {m,n}. Draw a circle of radius 1/2 with center {m,n} (note no two points will have circles that intersect or overlap). If a_{m,n} is positive, draw a cone with base a circle of radius 1/2 centered at {m,n} and height 12/π. As the area of a cone is (1/3) (area base) (height), this cone will have volume 1; if a_{m,n} was positive we draw a similar cone but instead of going up we go down, so now the volume is -1. What is going wrong? The problem is that Sum_m Sum_n |a_{m,n}| = ∞ (the sum of the absolute values diverges), and when infinities enter strange things can occur. Recall we are not allowed to talk about ∞ - ∞; the contribution from where our function or sequence is positive is +∞, the contribution where it is negative is -∞, and we are not allowed to subtract infinities.
  • To motivate the Change of Variable Formula, we discussed trying to find the area of a circle by doing the integration directly. While there are many ways to justify learning the Change of Variable Formula (it's one of the key tools in probability), I wanted to take the path of looking at what should be a simple integral and seeing how hard it can be to evaluate in the given coordinate system. Much of modern physics is related to changing coordinate systems to where the problem is simpler to study (see the Lagrangian or Hamiltonian formulations of physics); these are equivalent to F = ma, but lead to much simpler algebra. The problem we considered was using one-variable calculus to find the area under a circle. This requires us to integrate sqrt(1 - x²) from x=0 to x=1. This is one of the most important shapes in mathematics -- if calculus is such a great and important subject, it should be able to handle this!
  • To attack this problem, we recalled a powerful technique from Calc I: if f(g(x)) = x (so f and g are inverse functions, such as sqrt(x^2)), then g'(x) = 1 / f'(g(x)); in other words, knowing the derivative of f we know the derivative of its inverse function. This was used in Calc I to pass from knowing the derivative of exp(x) to the derivative of ln(x). We tried various inverse trig functions; while many were close to sqrt(1-x^2), none of them were exactly that (a list of the derivatives of these are here). This highlights one of the most painful parts of integration theory -- just because we are close to finding an anti-derivative does not mean we can actually find it! While there is a nice anti-derivative of sqrt(1 - x^2), it is not a pure derivative of an inverse trig function. There are many tables of anti-derivatives (or integrals) (a fun example on that page is the Sophomore's Dream). Unfortunately it is not always apparent how to find these anti-derivatives, though of course if you are given one you can check by differentiating (though sometimes you have to do some non-trivial algebra to see that they match). In fact, there are some tables of integrals of important but hard functions where most practitioners have no idea how these results are computed (and occasionally there are errors!). We will see later how much simpler these problems become if we change variables; to me, this is one of the most important lessons you can take from the course:  Many problems have a natural point of view where the algebra is simpler, and it is worth the time to try to find that point of view!
  • For another example of changing your viewpoint, think of trying to write down an ellipse aligned with the coordinate axes, and one rotated at an angle. Linear algebra provides a nice framework for doing these coordinate transformations, changing hard problems to simpler ones already understood.

• Monday, April 19.  Today we discussed modeling, in particular, the interplay between finding a model that captures the key features and one that is mathematically tractable. While we used a problem from baseball as an example, the general situation is frequently quite similar. Often one makes simplifying assumptions in a model that we know are wrong, but lead to doable math (for us, it was using continuous probability distributions in general, and in particular the three parameter Weibull). For more on these and related models, my baseball paper is available here; another interesting read might be my marketing paper for the movie industry (which is a nice mix of modeling and linear programming, which is the linear algebra generalization of Lagrange multipliers).
  • One of the most important applications of finding areas under curves is in probability, where we may interpret these areas as the probability that certain events happen. Key concepts are:
    • Probability distribution
    • Mean or Expected Value
    • Standard Deviation
    • Independence
    • Skewness and kurtosis (for the hypercompetitive students who really want to compare themselves to the class)
  • The more distributions you know, the better chance you have of finding one that models your system of interest. Weibulls are frequently used in survival analysis. The exponential distribution occurs in waiting times in lines as well as prime numbers.
  • In seeing whether or not data supports a theoretical contention, one needs a way to check and see how good of a fit we have. Chi-square tests are one of many methods.
  • Much of the theory of probability was derived from people interested in games of chance and gambling. Remember that when the house sets the odds, the goal is to try and get half the money bet on one team and half the money on the other. Not surprisingly, certain organizations are very interested in these computations. Click here for some of the details on the Bulger case (the bookie I mentioned in class is Chico Krantz, and is referenced briefly).
  • Any lecture on multivariable calculus and probabilities would be remiss if it did not mention how unlikely it is to be able to derive closed form expressions; this is why we will study Monte Carlo integration later. For example, the normal distribution is one of the most important in probability, but there is no nice anti-derivative. We must resort to series expansions; that expansion is so important it is given a name: the error function.
  • I strongly urge you to read the pages where we evaluate the integrals in closed form. The methods to get these closed form expressions occur frequently in applications. I particularly love seeing relations such as 1/c = 1/a + 1/c; you may have seen this in resistors in parallel or perhaps the reduced mass from the two body problem (masses under gravity). Extra credit to anyone who can give me another example of quantities with a relation such as this.
  • The probability distribution 6 x (1-x) for 0 <= x <= 1 is an example of a Beta distribution, which is very useful in modeling a wide variety of phenomena (for example, see the Laffer curve).
  • Click here for a clip of Plinko on the Price I$ Right, or here for a showcase showdown.

   

• Friday, April 16.  The main result today was a method for integrating over regions other than a rectangle. We discussed a theoretical way to do it by replacing our initial function f on a rectangle including D with a new function f^*, with f^*(x,y) = f(x,y) if (x,y) is in our domain D and 0 otherwise. To make this rigorous we need to argue and show that we may cover any curve with a union of rectangles with arbitrarily small area. This leads to some natural, interestin gquestions.
  • The first, and most important, involves what happens to a function when we force it to be 0 from some point onward (say outside D). The function may be discontinuous at the boundary, but then again it may not. There are many interesting and important examples from mathematical physics where we are attempting to solve some equation that governs how that system evolves. One of the most studied are the vibrations of a drum, where the drumhead is connected and stationary. We can thus view the vibrating drumhead as giving the values of our function on some region D, with value 0 along the boundary. This leads to the fascinating question of whether or not you can hear the shape of a drum. This means that if you hear all the different harmonics of the drum, does that uniquely determine a shape? Sadly, the answer is no -- different drums can have the same sounds. An excellent article on this is due to Kac, and can be read here.
  • We discussed y-simple, x-simple an simple regions (note that a region is said to be elementary if it is either x-simple or y-simple). The point of our analysis here is to avoid having to go back to the definition of the integral (ie, the Riemann sum). While not every region is elementary, many are either elementary or the union of elementary regions. Below are two interesting tidbits about how strange things can be:
    • Space filing curves: click here for just how strange a curve can be!
    • Koch snowflake: This is an example of a fractal set; the boundary has dimension greater than 1 but less than 2! It's fractal dimension is log 4 / log 3.
    • Jordan curve theorem: It turns out to be surprisingly difficult to prove that every non-intersecting curve in the plane divides the plane into an inside and outsider region. It's not too bad for polygons, but for more general curves (such as the non-differentiable boundary of the Kock snowflake), it's harder.

• Monday, April 12.  Today we proved the Fundamental Theorem of Calculus in one variable. To simplify the prove, we made the additional assumptions that our function was continuously differentiable and the derivative was bounded. These assumptions can all be removed; it suffices for the function to be continuous on a finite interval (in such a setting, a continuous function is actually uniformly continuous; informally, this means in the epsilon-delta formulation of continuity that delta is independent of the point. Such a result is typically proved in an analysis class. What I find particularly interesting about the proof is that the actual value that bounds the function is irrelevant; all that matters is that our function is bounded. Theoretical math constantly uses such tricks; this is somewhat reminiscent of some of the Lagrange Multiplier problems, where we needed to use the existence of lambda to solve the problem, but frequently we never had to compute the value of lambda.
  • The key ingredients in the proof are using the Mean Value Theorem and observing that we have a telescoping sum. One has to be a little careful with telescoping sums with infinitely many terms. The wikipedia article has some nice examples of telescoping sums and warnings of the dangers if there are infinitely many summands.
  • Whenever you are given a new theorem (such as the Fundamental Theorem of Calculus), you should always check its predictions against some cases that you an readily calculate without using the new machinery. For example, if we want to find the area under f(x) from x=0 to x=1, obviously the answer will depend on f. If f is constant it is trivial; if f is a linear relation then the answer is still readily calculated. For more general polynomial n, one can compute the Riemann sums (the upper and lower sums) by Mathematical Induction. For example, using induction one can show that the sum from n=1 to n=N of n is simply n(n+1)(2n+1)/6, and this result can then be used to find the area under the parabola y = x².
  • The integration covered through Calc III is known as Riemann sums / Riemann integrals. In more advanced math classes you'll meet the successor, Lebesgue integrals. Informally, the difference between the two is as follows. Imagine you have a large number of coins of varying denominators to assist; your job is to count the amount of money. Riemann sums work by breaking up the domain of the function; Lebesgue integration works by breaking up the domain.
  • In the review session on Sunday we talked about paths on surfaces; in particular, we talked about how Pac-man is really played on a torus / donut. The mathematics of tori and shapes in general is quite fascinating; if you want, click here and download games for tori (including tic-tac-toe for the torus!).
  • For those looking for a challenge: Let f satisfy the conditions of the Fundamental Theorem of Calculus. Let L(n) denote the corresponding lower sum when we partition the interval [0,1] into n equal pieces, and similarly let U(n) denote the upper sum. We know U(n) - L(n) tends to zero and L(n) <= True Area <= U(n); as U(n) - L(n) --> 0 as n --> oo, both U(n) and L(n) tend to the true area. Must we have L(n) <= L(n+1), or is it possible that L(n+1) might be less than L(n)?

• Friday, April 9.  In one dimension, there is not much choice in how we integrate; however, if we are trying to integrate a function of several variables over a rectangle (or other such region), not surprisingly the situation is markedly different. Similar to the freedom we have with limits in several variables (where we have to consider all possible paths), there are many ways to integrate. Imagine we have a function of two variables and we want to integrate it over the rectangle [a, b] x [c, d], with x in [a, b] and y in [c, d]. One possibility is we can fix x and let y vary, computing the integral over y for the fixed x, and then let x vary, computing the integral over x. Of course, we could also do it the other way. As we are integrating the same function over the same region (just in a different order), we hope that the answers are the same! So long as everything is nice, this is the case. There are many formulations as to exactly what is needed to make the situation nice; if our function is continuous and bounded and we are integrating over a finite rectangle, then we can interchange the order of integration without changing the answer. This is called Fubini's theorem, and is one of the most important results in integration theory in several variables. There really isn't an analogue in one dimension, as there we have no choice in how to integrate!
  • Whenever you are given a theorem, it is worthwhile to remove a condition and see if it is still true. Typically the answer is no (or if it is still true, the proof is frequently much harder). There are many functions and regions where the order of integration matters. The simplest example is looking at double sums rather than double integrals, though with a little work we can convert this example to a double integral. We give a sequence a_{mn} such that Sum_{m = 0 to oo} Sum_{n = 0 to oo} a_{m,n}) is not equal to Sum_{n = 0 to oo} Sum_{m = 0 to oo} a_{m,n}). For m, n >= 0 let a_{m,n} = 1 if m = n, -1 if n=m+1 and 0 otherwise. Show that the two different orders of summation yield different answers. The reason for this is that the sum of the
    absolute value of the terms diverges.
  • Click here for another example where we cannot interchange the order of integration; a more involved example is available here.
  • Click here for a video by Cameron on how he applies Fubini's theorem to change the order of operations (he does a double sum instead of a double integral, but the principle is the same).

• Wednesday, April 7.  Though today's lecture covered two themes (Lagrange Multipliers and the Method of Least Squares), there are similarities between applications of each. Both highlight how our choice of measuring can affect the answer. For the Lagrange problem, where to build the base or distribution center varies greatly depending on how we weight differences. Similarly the best fit value of the parameters depends on how we choose to measure errors. It is very important to think about how you are going to measure / model, as frequently people reach very different conclusions because they have different starting points / different metrics.
  • A complete write-up of the five different models for measuring distance between our base and the trouble points is available here. For three of the five models one can use Lagrange Multipliers and obtain a system of equations that is solvable with straightforward algebra; for two of the methods the gradients end up involving very nasty functions (with square-roots of polynomials in the denominator), and we need to resort to numerical techniques. The notes referred to above describe the different approaches, especially the merits and disadvantages of each model. These notes are a good introduction to how sensitive an answer can be on how you measure success.
  • The Method of Least Squares is one of my favorites in statistics (click here for the Wikipedia page, and click here for my notes). The Method of Least Squares is a great way to find best fit parameters. Given a hypothetical relationship y = a x + b, we observe values of y for different choices of x, say (x1, y1), (x2, y2), (x3, y3) and so on. We then need to find a way to quantify the error. It's natural to look at the observed value of y minus the predicted value of y; thus it is natural that the error should be Sum_{i=1 to n} h(yi - (a x_i + b)) for some function h. What is a good choice? We could try h(u) = u, but this leads to sums of signed errors (positive and negative), and thus we could have many errors that are large in magnitude canceling out. The next choice is h(u) = |u|; while this is a good choice, it is not analytically tractable as the absolute value function is not differentiable. We thus use h(u) = u²; though this assigns more weight to large errors, it does lead to a differentiable function, and thus the techniques of calculus are applicable. We end up with a very nice, closed form expression for the best fit values of the parameters.
  • Unfortunately, the Method of Least Squares only works for linear relations in the unknown parameters. As a great exercise, try to find the best fit values of a and b to y = c/x^a (for definiteness you can think of this as the force due to two unit masses that are x units apart). When you take the derivative with respect to a and set that equal to zero, you won't get a tractable equation that is linear in a to solve. Fortunately there is a work-around. If we change variables by taking logarithms, we find ln(y) = ln(c/x^a); using logarithm laws this is equivalent to ln(y) = a ln(x) + ln(c); setting Y = ln(y), X = ln(X) and b = ln(c) this is equivalent to Y = a X + b, which is exactly the formulation we need! This example illustrates the power of logarithms; it allows us to transform our data and apply the Method of Least Squares.
  • There are many examples of power laws in the world. Many of my favorite are related to Zipf's law. In class we discussed the frequencies of the most common words in English (click here for the data; see also this site); this works for other languages as well, for the size of the most populous cities, ...; if you consider more general power laws, you also get Benford's law of digit bias, which is used by the IRS to detect tax fraud (the link is to an article by a colleague of mine on using Benford's law to detect fraud). The power law relation is quite nice, and initially surprising to many. My Mathematica programming analyzing this is available here. See also this paper by Gabaix for Zipf's law and the growth of cities. As a nice exercise, you should analyze the growth of city populations (you can get data on both US and the world from Wikipedia).
  • We discussed Kepler's Three Laws of Planetary Motion (the Wikipedia article is very nice). Kepler was proudest (at least for a longtime) of Mysterium Cosmographicum (I strongly urge you to read this; yes, the same Kepler whom we revere today for his understanding of the cosmos also advanced this as a scientific theory -- times were different!).
  • Finally, a theme of the past two days is the importance of how we choose to measure things; how we model and how we judge the model's prediction will greatly affect the answer. In a similar spirit, I thought I would post a brief note about Oulipo, a type of mathematical poetry (this is a link to the Wikipedia page, which has links to examples). There was a nice article about this recently in Math Horizons (you can view the article here). This is a nice example of the intersection of math and the arts, and discusses how the structure of a poem affects the output, and what structures might lead to interesting works.

• Monday, April 5.  Welcome back! Today's class and Wednesday's are about applications of differentiating functions of several variables. Specifically, we first generalized the methods from one variable calculus on how to find maxima and minima of functions. Recall that if f is a differentiable real-valued function on an interval [a,b], then the candidates for maxima / minima are (1) the critical points, namely those x in [a,b] where f'(x) = 0, and (2) the endpoints. How does this generalize to several variables? In one-dimension the boundary of an interval is `boring'; it's just the two endpoints, and thus it isn't that painful to have to check the value of the function there as well as at the critical point. What about several variables? The situation is quite different. For example, the interval [-1,1] might become a sphere x^2 + y^2 + z^2 <= 1; the interior is all points (x,y,z) such that x^2 + y^2 + z^2 < 1, while the boundary is now the set of points with x^2 + y^2 + z^2 = 1. Unfortunately this leads to infinitely many points to check; while we could afford to just check the endpoints by brute force in one-dimension, that won't be possible now. The solution is the Method of Lagrange Multipliers.
  • Two good links: An introduction to Lagrange Multipliers    and    Lagrange Multipliers.
  • The Method of Lagrange Multipliers is one of the most frequently used results in multivariable calculus. It arises in physics (Hamiltonians and Lagrangian, Calculus of Variations), information theory, economics, linear and non-linear programming, .... You name it, it's there. The two webpages referenced above have several examples in these and other subjects; there are of course many other sources and problems (click here for a nice post on gasoline taxes, pollution and Lagrange multipliers). For more on the economics impact, click here, as well as see the following papers:
    • Lagrange Multipliers and the Fundamental Theorem of Algebra (Theo de Jong), American Mathematical Monthly, Volume 116, Number 9, November 2009, pp. 828-830(3). The Fundamental Theorem of Algebra is one of the gems of mathematics (we don't have that many fundamental theorems). This is a beautiful proof mostly using what we've done in class (gradients, Lagrange Multipliers, Level Sets). There is a bit of real analysis terminology and results, but not too much, and I think confined enough so that you can get the general flavor.
    • Lagrange Multipliers and Optimality (R. Tyrrell Rockafellar)
    • Arbitrage Theory: A Mathematical Introduction (David P. Ellerman)
  • The Method of Lagrange Multipliers ties together many of the concepts we've studied this semester, as well as some from Calc I and Calc II (vectors, directional derivatives and gradients, and level sets, to name a few). The goal is to show you how the theoretical framework we developed can be used to solve problems of interest. The military example we discussed is just one of many possible applications. We were concerned with how to deploy a fleet to minimize average deployment time to trouble spots (for more information, see my notes on the problem and the Mathematica code); of course, instead of considering each place equally important we could easily add weights. One consequence of war is that it does strongly encourage efficiency and optimization; in fact, many optimization algorithms and techniques were developed because of the problems encountered. The subject of Operations Research took off during WWII; see the excellent wikipedia article on Operations Research, especially the subsection on the problems OR attempts to solve. Not surprisingly, there are also numerous applications in business. Feel free to talk either to my wife (who is a Professor of Marketing) or myself (I've written several papers with marketing professors, applying such techniques to many companies, my favorite being movie theaters). As mentioned, we can reinterpret our problem as minimizing shipping costs from a central distributor to various markets (where some markets may be more valuable than others, leading to a weighted function).
  • One of the most important takeaways of the deployment problem is that the answer you get, as well as the difficulty of the math needed to arrive at the answer, depends on how you choose to model the world. For us, it depends on how we choose to measure 'distance'. My notes on the problem give four different methods yielding three different solutions, all of which differ from what you get if you use the 'correct' measure of distance. This is an extremely common outcome -- your answer depends on how you choose to model / measure! You need to be very aware of this when you compare different people's answers to the same problem. For a nice example of how the answer can depend on your point of view, consider the riddle below (passed on by G. Mejia). What's the right answer?
    • The police rounded up Jim, Bud and Sam yesterday, because one of them was suspected of having robbed the local bank. The three suspects made the following statements under
      intensive questioning (see below). If only one of therse statements turns out to be true, who robbed the bank?
      • Jim: I'm innocent. 
      • Bud: I'm innocent.
      • Sam: Bud is the guilty one.
  • For more on the problem of building an efficient computer in terms of retrieval of information, see the solution to the related extra credit problem from earlier in the semester. Note the problem is harder without the tools of multivariable calculus. See also the article by Hayes in the American Scientist, Third Base.
  • I've scanned in a chapter by Lanchester on The Mathematics of Warfare; you can also view it through GoogleBooks here. This article is from a four volume series, The World of Mathematics. (I am fortunate enough to own two sets; one originally belonged to a great uncle of mine, another to a grandfather-in-law of my wife). I've written some Mathematica code to analyze the Battle of Trafalgar, which is described in the Lanchester article; the Mathematica code is here (though it might not make sense without comments from me). (The file name is boring because, during the 200th anniversary re-enactment, in order to avoid hurting anyone's feelings they refused to call the two sides 'English' and 'French/Spanish'). This is a terrific problem to illustrate applying mathematics to the real world. One has a very complicated situation, and you must decide what are the key features. The more features you include the better your model will be, but the less likely you'll be able to solve it! It's a bit of an art figuring out exactly how much to include to capture what truly matters and still be able to solve your model. We'll discuss this in greater detail when we do the Pythagorean Won-Loss theorem from baseball, which is a nice application of probability and multiple integrations.
  • Finally, a common theme that surfaces as we do more and more mathematical modeling is that simple models very quickly lead to very hard equations to solve. The drowning swimmer problem is actually the same as Snell's law, for how light travels / bends in going from one medium to another. If you write down the equations for the drowning swimmer, you quickly find a quartic to solve. For interesting articles related to this, see the two papers below by Pennings on whether or not dogs know calculus. Click here for a picture of his dog, Elvis, who does know calculus.
    • Do dogs know calculus?  
    • Do dogs know bifurcations?

• Friday, March 19. 
  • We began today by seeing how well Taylor series approximate functions. The Mathematica program here is (hopefully) easy to use. You can specify the point and number of terms of the Taylor series of cos(x) to do. At first it might seem surprising that there is no improvement in fit when we go from a second order to a third order Taylor series approximation; however, we have cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + .... In other words, all the odd derivatives vanish at the origin, and thus there is no improvement at the origin in adding a cubic term (ie, the best cubic coefficient at the origin is 0). If we go to a fourth order, we do see improvement. By n=10 or 12 we are already getting essentially an entire period correct; by n=40 we have several cycles.
  • We then compared two methods to find roots of polynomials. In some special cases we can find closed form expressions for roots in terms of the coefficients. For example, any linear equation (ax+b=0), quadratic (ax^2+bx+c=0), cubic (ax^3+bx^2+cx+d=0) or quartic (ax^4+bx^3+cx^2+dx+e=0) has a formula for the roots in terms of the coefficients of the polynomials; this fails for polynomials of degree 5 and higher (the Abel-Ruffini Theorem; see also Galois). It is very convenient when we have a solution that is a function of the parameters; we can then use our methods to find the optimal values of the parameters. Sadly in industry it is often difficult to get closed form expressions; if you are looking for the most potent compound, for example, you might be required to do numerous different trial runs and just observe which is best. We thus need a way to find optimal values / solve equations. We describe two below.
    • Newton's method is significantly more powerful than divide and conquer (also called the bisecting algorithm); this is not surprising as it assumes more information about the function of interest (namely, differentiability). The numerical stability of Newton's method leads to many fascinating problems. One terrific example is looking at roots in the complex plane of a polynomial. We assign each root a different color (other than purple), and then given any point in the complex plane, we apply Newton's method to that point repeatedly until one of two things happen: it converges to a root or it diverges. If the iterates of our point converges to a root, we color our point the same color as that root, else we color it purple. This leads to Newton fractals, where two points extremely close to each other can be colored differently, with remarkable behavior as you zoom in. If you're interested in more information, let me know; a good chaos program is xaos (I have other links to such programs for those interested). One final aside: it is often important to evaluate these polynomials rapidly; naive substitution is often too slow, and Horner's algorithm is frequently used. We also talked about the dangers of interchanging operations (such as interchanging the order of summation or a limit and an integral). For limit-integral problems, frequently one appeals to Lebesgue's Dominated Convergence Theorem to justify these interchanges. Measure theory is a generalization of integration, and allows us to handle more general sets. One example is the characteristic function of the rationals on [0,1], ie, the function that is 1 if x is a rational in [0,1] and 0 otherwise. This function is not Riemann integrable, as the upper sums are always 1 and the lower sums are always 0. It can be shown in a 'natural' generalization of integration that this function integrates to 0 (which agrees with our intuition that there are a lot more irrationals than rationals).
    • The fractal behavior exhibited by Newton's method applied to finding roots of polynomials is one of many examples of Chaos Theory, or extreme sensitivity to initial conditions. While one of the earliest examples was the work of Poincare on the motion of three planetary bodies, the subject really took off with Lorenz work on weather (the Butterfly Effect). Another nice example is the orbit of Pluto; while we know it will orbit the sun, its orbit is chaotic and we cannot say where exactly in the orbit it will be millions of years from now.
  • We ended today by discussing the Birthday Paradox: assuming each day of the year is equally likely (not true for hockey players!) to be someone's birthday, and no one is born on Feb 29, how many people do you need in a room before there is a 50% chance of two sharing a birthday? The answer is surprisingly low, only about 23. To compute this, if there are n people then the probability no one shares a birthday with anyone else is (1 - 0/365) (1 - 1/365) (1 - 2/365) * ... * (1 - (n-1)/365), or Prod_{k=0 to n-1}(1 - k/365). If we set this equal to 1/2, we just need to keep multiplying. But this is inelegant, and it's not at all clear how the answer depends on the number of days in the year (ie, we'd have to do an entirely new calculation if we move to Pluto). We can solve this by using logarithms to summify the expression. In other words, those log laws we make you learn in junior high / high school can be used for problems you're interested in! Taking logarithms gives log(1/2) = Sum_{k=0 to n-1} log(1 - k/365), as the log of a product is the sum of the logs. We then use Taylor series to expand log(1-x), noting that log(1-x) is approximately -x. This gives us log(1/2) =approx= -(1/365) Sum_{k=0 to n-1} k; we approximate the sum with an integral (Int_{x=0 to n-1} xdx, which is (n-1)^2/2), and find that (n-1)^2/2 =approx= 365 log 2, or n =approx= 1 + Sqrt(365 * 2 log 2), which allows us to see how things would change if we move to Pluto, where there are about 90,000 days in a year!
    • I forgot to mention applications of this in Section 2. One application is the following: say you have a steel girder, and acid rain is equally likely to hit it anywhere. It is safe until rain hits the same spot twice -- how long must you wait until you have a 50% chance of seeing it crack? You can of course generalize this to you need 5 hits for it to crack.
    • Another application (not mentioned in either section due to time) is to birthday attacks in cryptography.
  • In the baseball lecture today we discussed one of my favorite riddles: Find a way to place 5 queens on a 5x5 chess board such that there are three squares where we may safely place pawns so that no queen attacks any pawn. This is one of my favorite riddles (if you enjoy problems like this, check out my riddles homepage; while the site is poorly designed, it receives a lot of hits, and is usually in the top 10 in the world when you google math riddles). The solution of this is related to linear programming (click here for my notes on the subject). This is a phenomenally useful field, and is a natural outgrowth of linear algebra and the optimization methods we've been discussing. Linear programming can be used to find solutions to problems ranging from determining cost-effective diets for poor nations to scheduling baseball seasons, airlines, or movies (click here for a paper of mine helping theaters optimizing scheduling). If anyone is interested in discussing these topics further, let me know.

• Wednesday, March 17.  The search for extrema is a central pursuit in modern science and engineering. It is important to have techniques to winnow the list of candidate points. The methods discussed in class are the natural generalizations from one-variable calculus. While one must prove that the function under consideration does have a max/min, typically this is clear from physical reasons (for example, there should be a pen of maximal area for given perimeter; there should be a path of least time).
  • In one-dimension, boundaries of sets aren't too bad; for example, the boundary of [a, b] is just two points, a and b. The situation is violently different in several variables. There the boundary can have infinitely many points, and reducing a problem to interior critical points and checking the function on the boundary is not enough; we must have a way to evaluate all these points on the boundary.
  • The generalization of the second derivative tests involves determinants and whether or not the Hessian is a positive definite matrix, a negative definite matrix, et cetera. What is really going on is that we want to use the Principal Axis Theorem and change to a coordinate system where the Hessian is easier to understand because, in this new coordinate system, it is a diagonal matrix!
  • In the Friday sabermetrics lecture, we'll discuss linear programming. This is a wonderful topic, and it allows us to solve (or approximate the solutions) to a wealth of problems very quickly. My lecture notes are online here. One of my favorite applications of linear programming is to determining when teams are eliminated from playoff contention; MLB and ESPN frequently do the analysis incorrectly by not taking into account secondary effects of teams playing teams playing teams. For example, ESPN or MLB back in '04 had the wild-card unclinched for one extra day. (The Sox had a big lead over the Angels and a slightly smaller lead over the As; however, the As and the Angels were playing each other and thus at least one team would get 2 losses, and one had to win the ALWest. Thus the Sox had clinched the wildcard a day earlier than thought.) If you're interested, click here for a paper I wrote with colleagues applying linear programming to helping a movie theatre determine optimal schedules.
  • Finally, it is worth remarking that for many applications in the real world, we do not need to find the true extremum, but rather just something very close. For example, say we are trying to determine the optimal schedule for an airline for a given day. We can write the linear programming problem down, but it might take days to years to run; however, frequently we can obtain bounds showing how close our answer is to the theoretical best (ie, we can show we are no more than X away from optimal). It often happens that X is small, and thus with a small run-time we can be close enough. (It isn't worth it to ground our airfleet for a few years to find the optimal schedule!)
  • In honor of Kayla's guest appearance today, I think it's fitting to end with some comments about children's books. Click, Clack Moo. Cows that type is the first of a series of well-illustrated and entertaining adventures of Duck and his compatriots. It's a Caldecott Honor award winner, which after winning the Caldecott Medal is, I believe, considered the top honor for a children's story. One of my favorite childhood books is Make Way For Ducklings, which has now been read to three generations of Millers. This is extremely popular in the eastern part of the state (ie, Boston). We've taken Cam to the statues; it's a fun area. For the western part of the state, the big children's attractions are Dr Seuss (from Springfield) and the Eric Carle Museum (in Amherst -- okay, there is something nice there!). Finally, no `cultural' introduction to the Commonwealth of Massachusetts would be complete without providing a link to Norman Rockwell.

• Monday, March 15.  Today we discussed Taylor's theorem (in one and several variables). This is one of the most important applications of calculus. It allows us to replace complicated functions with simpler ones. There are numerous questions to ask.
  • Are Taylor series unique? Yes. The definition just involves taking sums of derivatives; the process is well-defined.
  • Does every infinitely differentiable function equal its Taylor series expansion? Sadly, no; the function f(x) = exp(1/x^2) if |x| > 0 and 0 if x=0 is the standard example. This function causes enormous problems in probability. There are many functions which do equal their own Taylor series expansion, such as exp(x), cos(x) and sin(x). It's not surprising that these three are listed together, as we have the wonderful Euler-Cotes formula: exp(i x) = cos(x) + i sin(x), with i = sqrt(-1). At first this formula doesn't seem that important; after all, we mostly care about real quantities, so why complexify our life by adding complex (i.e., imaginary) numbers? Amazingly, even for real applications (applications where everything is real), complex numbers play a pivotal role. For example, note that a little algebra gives cos(x) = (exp(i x) + exp(-i x)) / 2 and sin(x) = (exp(i x) - exp(-i x)) / 2i. Thus properties of the exponential function transfer to our trig functions. The hyperbolic cosine and sine functions are similarly defined; cosh(x) = cos(i x) = (exp(-i x) + exp(x)) / 2. The Foxtrot strip below (many thanks to the author, Bill Amend, for permission to post) illustrates the confusions that can happen between hyperbolic and regular trig functions (as a note, why does Eugene know that the calculator cannot be giving the right answer?). It's worth noting that the formula exp(i x) = cos(x) + i sin(x) allows us to derive ALL trig identities painlessly! See the comments from Friday, February 12.
  • 
  • FoxTrot (c) Bill Amend. Used by permission of Universal Uclick. All rights reserved.
  • How easy are Taylor series to use? If we keep just a few terms, it's not too bad; however, as the great Foxtrot strip below shows, it's not always clear how nicely something simplifies.
  • 
  • FoxTrot (c) Bill Amend. Used by permission of Universal Uclick. All rights reserved.
  • In the strip above, notice the large factorials in the denominator. Note 52! is about 10⁶⁸; in other words, these terms are small! For interest, 52! is the number of ways (with order mattering) of arranging a standard deck of cards. There are about 10⁸⁵ or so subatomic thingamabobs in the universe; we see quite quickly reach numbers this high (a deck with 70 cards more than sufices; in other words, we could not have each subatomic object in the universe represent a different shuffling of a deck of 70 cards). In a related note, it's important to think a bit and decide what 0! should be. It simplifies many formulas to have 0! = 1, and we can make this somewhat natural by saying there is only 1 way to do nothing (mathematically, of course). The definition of the factorial function on Wikipedia talks a little bit about this.
  • Unlike 0!, 0^0 is a bit more controversial as to what the definition should be. As I don't want to pressure anyone, I will not publically disclose where I stand in the great debate, though I'm happy to tell you privately / through email.
  • It's worth remarking on why we have n! in the denominators. This is to ensure that the nth derivative of our function equals the nth derivative of the Taylor expansion at the point we're expanding. In other words, we're matching up to the first 2 derivatives for the 2nd order Taylor expansion, up to the first 3 for the 3rd order Taylor expansion, and so on. It isn't surprising that we should be able to do a good job; the more derivatives we use, the more information we have on how the function is changing near the key point.
  • For many purposes, we just need a first order or second order Taylor series; one of my favorites is the proof of the Central Limit Theorem in probability. One of my favorite proofs involves second order Taylor expansions of the Fourier Transforms (these were mentioned in the additional comments on Friday, March 12).
  • If f(x) equals its infinite Taylor series expansion, can we differentiate term by term? This needs to be proved, and is generally done in a real analysis course. For some functions such as exp(x) we can justify the term by term differentiation, but note that this is something which must be justified.
  • A terrific application of just doing a first order Taylor expansion is Newton's Method, which we'll discuss in great detail on Friday.
  • For some reason, most books don't mention the trick on how to quickly compute higher order Taylor expansions in several variables. The idea is to 'bundle' variables together and use one-dimensional expansions. For example, consider f(x,y) = exp(-(x^2 + y^2)) cos(xy). We saw in class how painful it is to compute the Hessian, the matrix of second partial derivatives. That involved either two product rules or knowing the triple product formula. If we use our trick, it's much easier. Note exp(u) = (1 + u + u^2/2! + ...) and cos(v) = (1 - v^2/2! + ...). A second order Taylor expansion means keep only terms with no x's and y's, with just x or y, or with just x^2, xy or y^2 (a third order would allow terms such as x^3, x^2 y, x y^2, y^3, and so on). Thus we expand exp(u) cos(v) and then set u = -(x^2+y^2) and v = xy. For exp(u), we just need 1 + u, as already the u^2/2 term will be order 4 when we substitute -(x^2+y^2). For cos(v), we only keep the 1 as v^2/2 is order 4. Thus the Taylor expansion of order 2 is just (1 -(x^2+y^2)) (1) = 1 - x^2 - y^2; this is a lot faster than the standard method! That method works in general, but there are so many cases where this is faster that it's worth knowing.

• Friday, March 12.  The equality of mixed partial derivatives is one of the most important results in multivariable calculus. It is unusual to have two operations whose order commutes; the square root of a sum is not the sum of the square-roots. Higher derivatives arise in partial differential equations. Fortunately, most of these equations involve at most second order derivatives, and most don't have mixed partials.
  • Famous examples: wave equation, heat equation, Navier-Stokes equation.
    • A particularly nice example is writing some fluid flow problems in cylindrical coordinates. Here the differential equations do involve explicitly mixed partials.
  • There is another, higher-level way of viewing the equality of mixed partial derivatives. It involves the Fourier Transform. The Fourier transform converts differentiation to multiplication. Noting this, the equality of mixed partials is then equivalent to multiplication being commutative! It's tough to read about this without a solid background in math and physics, so please contact me if you want to know more.
    • One very fun and strange application of Fourier analysis is to the Heisenberg uncertainty principle, which may be interpreted as a statement about a function and its Fourier transform (one can not localize both well).
  • Note that, as usual, our proof involves multiplying by 1 and the Mean Value Theorem. The MVT is one of my favorite tools in mathematics. It is, in some sense, a variant of Taylor series, which we will meet very soon. It allows us to work with differences between a function at two close points.
    • The way we showed the two mixed partials are equal (for C² functions) was to show that both of them equal Lim_{Δx,Δy --> 0} S(Δx,Δy) / ΔxΔy, and then two quantities equal to a third are equal to each other. This is one of the most powerful ideas in mathematics, going at least as far back as being one of Euclid's 5 common notions (there are 10 axioms in Euclid's Elements, 5 about constructions and 5 about common notions). This principle of things equal to a common object must be equal is one of the major tools in combinatorics. The way we compute many sums is to give two interperations, one of which is doable and the other what we want to know.
      • Here is one of my favorite combinatorial problems. Let (n C r) denote the number of ways of choosing r people from n when order does not matter; thus (n C r) = n! / r!(n-r)!. One can show that the number of ways of dividing n identical cookies among p people is (n + p - 1 C p - 1). Find a nice formula for Sum_{n = 0 to N} (n + p - 1 C p - 1). What makes this a particularly challenging combinatorial sum is that the numerator of the binomial coefficient is varying! If you look at this problem the right way (all the info you need is in the previous lines!), you can evaluate this immediately! In other words, what is Sum_{n = 0 to 2010} (n+76)! / 76! n!? 
  • Today we talked about how to use Mathematica to attack many problems (plotting, integrating, finding derivatives). My template (and more) is available here. While these programs are great, they do not replace knowing how to do the calculations yourself. There are many reasons for this. One, of course, is that these programs can make mistakes, and it is useful to have a sense of what the right answer should be. This is why the extra credit problem from the first day is one of my favorite problems, namely trying to see which of 4 possible formulas is reasonable. I can actually give an example of how this has impacted my professional research today. I'm working on a paper with a colleague, and I conjectured that the following combinatorial identity holds: Sum_{r = 0 to m} (-1)^r (m C r) (m+r C r) / (r+1)(m+r) = 0 for all integers m > 1. There are computer programs to PROVE conjectures like this. I ran it adn it generated the following: The equation is routinely verifyable by dividing the right-hand side by the left-hand side and simplifying the resulting rational function to 1. Not very illuminating, and I couldn't get it to output more. I was at least convinced that there was a proof, but I felt bad not being able to see it. After some work, I was able to prove it myself. Just yesterday (ok, technically today as it was 3am) I was finishing up a paper where I needed to prove a formula for a much simpler quantity, namely what is Sum_{k = 0 to n/2} k (n-k C k), which the computer program could not do. Fortunately my experience on similar problems left me trained to handle stuff such as this.
  • I think I only discussed the following example in one of the sections today. It relates to why I find it so important to know proofs and definitions. It's about the fall of Western Civilization (orif you're not quite as pessimistic, the financial mortgage meltdown). While there are many reasons behind the collapse (I have close family that has worked in the upper levels of many of the top financial firms; if you are interested in stories of what isn't reported in the news, let me know), one large component was an incorrect use of Gaussian copulas. It's similar to looking at low-velocity data and extrapolating to relativistic speeds -- there is an enormous danger when you apply results from one region in another with no direct data in that second realm. A great article on this is from Wired Magazine (The Formula That Killed Wall Street). It's worth reading this. Some particularly noteworthy passages:
    • Bankers should have noted that very small changes in their underlying assumptions could result in very large changes in the correlation number. They also should have noticed that the results they were seeing were much less volatile than they should have been which implied that the risk was being moved elsewhere. Where had the risk gone? They didn't know, or didn't ask. One reason was that the outputs came from "black box" computer models and were hard to subject to a commonsense smell test. Another was that the quants, who should have been more aware of the copula's weaknesses, weren't the ones making the big asset-allocation decisions. Their managers, who made the actual calls, lacked the math skills to understand what the models were doing or how they worked. They could, however, understand something as simple as a single correlation number. That was the problem.
    • No one knew all of this better than David X. Li: "Very few people understand the essence of the model," he told The Wall Street Journal way back in fall 2005. "Li can't be blamed," says Gilkes of CreditSights. After all, he just invented the model. Instead, we should blame the bankers who misinterpreted it. And even then, the real danger was created not because any given trader adopted it but because every trader did. In financial markets, everybody doing the same thing is the classic recipe for a bubble and inevitable bust.
  • Application of the day: I was asked to incorporate bicycle math into the course. After some web-browsing, I came across powerpoint slides by Jason Achilich. Slides 11 and 12 discuss wheel size, slide 13 discusses gears, and slides 16 - 18 discuss banking (which is essentially determining an optimal choice for your c(t) given your opponents choices!).

• Monday, March 8.  TBD, including
  • One of the requests was to talk about applications of multivariable calculus to molecular gastronomy. After some web browsing, I eventually beecame interested in how bees communicate amongst themselves as to where food is. There appear to bee two schools; one is the waggle dance / language school, and the other is the odor plume theory. In addition to controversies on how bees learn, there are lots of nice applications to gradients and (I believe) directional derivatives. The goal is to convey information about a specific path through a very complex space.
    • See also the paper: Odor landscapes and animal behavior: tracking odor plumes in different physical worlds (Paul Moorea, John Crimaldib). Abstract: The acquisition of information from sensory systems is critical in mediating many ecological interactions. Chemosensory signals are predominantly used as sources of information about habitats and other organisms in aquatic environments. The movement and distribution of chemical signals within an environment is heavily dependent upon the physics that dominate at different size scales. In this paper, we review the physical constraints on the dispersion of chemical signals and show how those constraints are size-dependent phenomenon. In addition, we review some of the morphological and behavioral adaptations that aquatic animals possess which allow them to effectively extract ecological information from chemical signals.
  • We discussed directional derivatives. It is natural that we develop such a concept, as up until now we have only considered derivatives in directions parallel to the various coordinate axes. A central theme of multivariable calculus is the need to be able to approach a point along any path, and that in several dimensions numerous paths are available (unlike the 1-dimensional case, where essentially we just have two paths). Directional derivatives will play a key role in optimization problems.
     

• Friday, March 5.  Today we discussed the Chain Rule. We essentially proved it in the special case of Calc I (i.e., the one variable chain rule). The key step in the proof is a clever multiplication by 1. The reason we need to do this is, as always, to be able to isolate out the definition of the derivative.
  • One item we must deal with carefully is that we had g(x+h) - g(x) divided by itself; what if g(x+h) = g(x) for infinitely many arbitrarily small h? Then we are dividing by zero. Can you prove that this cannot happen if g is differentable? If that is not a strong enough condition, what if we assume the derivative g' does not vanish at x -- does that suffice to prove that g(x+h) cannot equal g(x) infinitely often for arbitrarily small h?
  • One way to view the Chain Rule is that it is all about giving you the freedom to choose. You can either plug everything in and differentiate directly by brute force, or you can use the Chain Rule to find the derivative of the composition in terms of the derivatives of the constituent pieces. Depending on the problem, one way could be easier than the other; there are examples of situations where direct substitution is best, and examples where it is better to use the Chain Rule. With experience it becomes clear which way is better. In Section 2.6, when we discuss gradients and directional derivatives, we'll see a theoretical interpretation of the Chain Rule. This will play a fundamental role when we turn to optimization problems in Chapter 3. Finally, of course, it is useful to be able to compute an answer two different ways, as this provides a nice check of your work.
  • To use the Chain Rule in full glory, we needed to understand how to multiply matrices, as h(x) = f(g(x)) implies (Dh)(x) = (Df)(g(x)) (Dg)(x), where x = (x₁, ..., x_n). I chose to motivate matrix multiplication through the dot product, as we know how to take the dot product of two vectors of the same number of coordinates. Matrix multiplication looks quite mysterious at first.Wikipedia has a nice article (with color) on multiplying matrices, though it is a bit short on motivation. The advanced reason as to why we do this comes from also viewing matrices as linear transformations, and we want the product of two matrices to represent the composition of the transformations. This is an advanced topic, and sadly is frequently mangled in a linear algebra course. I've posted a little bit about this in the advanced notes from Thursday's lecture. The best motivation I know is to consider 2 x 2 rotation matrices. If R(a) corresponds to rotating by a radians, and R(b) to rotating by b radians, then R(b) R(a) should equal R(b+a); this does happen if we use the matrix multiplication method we discussed.

• Wednesday, March 3. 
  • General comment: frequently it is easier to attack a problem directly than to use the advanced theorems. For example, we saw today that if f(x,y) = x²y and g(x,y) = x² + y², it is easier to take the partials of h(x,y) = f(x,y)g(x,y) directly. In other words, just note that h(x,y) = x⁴y + x²y³. One place you may have seen this is in the Farmer Brown optimization problem. Recall Farmer Brown has 40 meters of fence and wants to enclose as large of an area as possible for his cows, but he is particular and won't consider any shape other than a rectangle. The enclosed area equals xy, but since 2x+2y=40 we have y=20-x, which implies that the area is A(x) = x(20-x). We could differentiate this with the product rule, or just note that the answer is A(x) = 20x-x², which is simpler to differentiate.
  • Art fraud: See the wikipedia article for information about the history of art forgeries, both creation and detection. There are some very entertaining bits, especially about how some forgers became so famous that people forged the forgeries! There's another bit about a son forging his dad's work and providing statements that it was done by his father, and another about the forger Meegeren, who had to create a forgery under police supervision to prove himself innocent of treason charges! There's also the Getty Kouros, which is displayed with the note (according to the Wikipedia article) Greek, 530 B. C. or modern forgery.
  • Below are some articles to read about some of the mathematics of detecting art forgeries:
    • A digital technique for art authentication (Lyu, Rochmore and Farid, 2004).
    • Detecting forged handwriting samples with wavelets and statistics (Lytle and Yang, Research Experience for Undergraduates project at Grand Valley State University, 2006).
    • Quantification of artistic style through sparse coding analysis in the drawings of Pieter Bruegel the Elder (Hughes, Graham and Rockmore, 2010).
    • Teaching a computer to spot a bogus Bruegel (Julie Rehmeyer, 2010).
  • The Chain Rule is one of the most important results in multivariable calculus, as it allows us to build complicated functions depending on functions of many inputs. To state it properly requires some linear algebra, especially matrix multiplication. The proof uses multiple applications of adding zero. This is a essential skill to master if you wish to continue in mathematics. It is somewhat similar to adding auxiliary lines in geometry. With experience, it becomes easier to `see' where and how to add zero. The idea is we want to add zero in such a way that we convert one expression to several, where the resulting expressions are easier to analyze because we are subtracting two quantities that are quite close. For the chain rule, we did this by adding numerous intermediary points.
  • For a baseball / sabermetrics application of these rules, consider estimating a team's winning percentage. The Pythagorean Won-Loss Theorem (I've written a paper providing a theoretical justification for this observation) says that if RS (resp. RA) represents the average number of runs a team scores (resp. allows), #Wins/#Games is about RS^γ / (RS^γ + RA^γ); originally γ was taken to be 2, but now the best fit value is believed to be about 1.8. If we want to figure out how many wins a team will have, we can estimate RS and RA by a micro analysis of players, using the Runs Created Formulas (and similar examples). We can then investigate the changes in winning percentage as we vary parameters used to model player productivity.

• Monday, March 1.  Note: for a general, all purpose Mathematica template (as well as a LaTeX one) that I've written, click here.
  • Today we talked about parametrized curves. Frequently a curve or a surface may be regarded as the level set of a given function. For example, a sphere of radius 2 is the level set with value 4 of the function f(x,y,z) = x² + y² + z². A circle of radius 5 may be regarded as the level set of value 25 of the function f(x,y) =  x² + y².
  • As I had to prepare much of the lecture without hearing what I should try to incorporate into the class, I decided to do another Isaac Asimov reference. Today's choice is the short story Runaround (you can read the story here). If you do choose to read this story, remember it was written in 1941 -- try and remember what technology was around back then (of course, that doesn't explain the writing style...). One may interpret the entire story as a study of level sets and parametrized curves; though this is probably not what Asimov intended, it is a nice way to view it. (On another note, it seems everything these days has a wikipedia page!)
  • In discussing paths of objects, the standard examples are planetary motion (either orbits of planets or rocket ships and probes) and cannonballs / bullets.
    • Planetary motion: Kepler's laws describe the orbits of planets, but give no reason as to why the planets follow these paths. These were deduced from observational data, and were crucial in leading Newton to the inverse-square law of gravity.
    • Probes: Approximately every 176 years, the outer gas giants align and one probe launched from Earth can visit them all; this is called the (planetary) Grand Tour (in analogy with the Grand Tour of Europe). NASA didn't have the technology to prepare everything for the mission at the needed launch date, and gambled that they could successfully reprogram the probe billions of miles later. It's a phenomenal success story. See in particular the details of the Voyager 2 mission; Voyager 1 is currently the farthest man-made object, and the fictional Voyager 6 sadly became the basis for a really bad Star Trek movie, Star Trek: The Motion Picture.
    • Ballistics: Ballistics deals with the trajectories of objects such as bullets and cannonballs. This is an extremely important application of mathematics; for years people were employed in creating firing tables for artillery. One interesting application is in the Falklands War between Great Britain and Argentina. The Earth's rotation causes the trajectory of objects fired in the northern and southern hemispheres to be different; it is claimed the British missed the Argentinians in their first volley, but quickly corrected. The explanation is the Coriolis Effect.
  • We ended the day with a discussion of Greek Astronomy. Particularly fascinating (to me) is how well they can do with circles on circles; click here for some more information, and click here for the Mathematica file from class today. In some sense, you can view the circles on circles as a Taylor series expansion of planetary motion! The idea that planets must move in circles seriously slowed down scientific advancement. That said, it is truly impressive how well one can do with all these circles, but the theory is not elegant (and that bothers a mathematician!). The world need not be elegant, of course, but so much is that we tend to seek out elegance. If you are given enough free parameters, you can fit almost any data set; thus we tend to prefer theories with just a few input parameters but sweeping predictions. The motions we saw are very similar to what you see for orbits of atoms in electrons in the Bohr model when we represent the electron's path by wiggles.

• Friday, February 26.  Today was hopefully the first of many days when you'll challenge me to work specific items in a natural way into the lecture. Today's challenge was to use Isaac Asimov and Star Trek: The Next Generation. I did have a way to do ST:TNG (as we geeks write it), but forgot.
  • One of my favorite science articles by Isaac Asimov is the Relativity of Wrong. I strongly urge you to read it. The article can really be read as a homage to the tangent plane. The Earth (locally) is so close to being flat that it really isn't that bad of an approximation to say the tangent plane is perfect, and not just an approximation. Basically, the Earth's curvature is 8 inches per mile, which again is so slight that it's not surprising the ancients (but not The Ancients) thought the Earth was flat. Of course, eventually people realized the Earth is not flat (ships on the horizon and shadows were two key inputs). Later we realized that, since the Earth is rotating, there will be an equatorial bulge. It's about 8.027 inches / mile near the equator, and about 7.973 near the poles. (Of course, this isn't quite right, see the article for more details). The point of all this is that tangent planes can do a great job, and small differences (which are hard to see) can be important and add up!
    • I believe the physical quantity that has been measured so accurately as to cause Feynman to remark it is equivalent to measuring the distance from LA to NY to within the thickness of a human hair is the Gyromagnetic ratio.
  • The ST:TNG reference was going to be the episode "The Loss". The premise is that the Enterprise runs into two-dimensional beings, and there are singularity issues. There's also a cosmic string fragment (or some such technobabble, I forget exactly what). The reason I was going to mention this is to talk about singularities and domains of definitions of the function. Basically, things on the Enterprise go haywire b/c part of it is now being intersected by the plane of two-dimensional beings. There are some animations of the Enterprise being intersected by the plane (i.e., a level set!). You can watch the episode on YouTube; the key clip is part 2 at around 8:11:
    • Part 1        Part 2 (start at 8:11)        Part 3        Part 4        Part 5
  • We talked a lot about different notations for the derivative. It is very convenient to be able to refer to all the different derivatives (functions with one input, several inputs and one output, several inputs and several outputs) with just one notation. Our definition is that a function is differentiable if the error in the tangent plane approximation tends to zero faster than the distance of where we are to where we start tends to zero. It is sadly possible for the partial derivatives to exist without the function being differentiable. We showed how it is not sufficient for the partial derivatives to exist; that is not enough to imply our function is differentiable. The example was f(x,y) = (xy)^1/3. What must we assume in order for the partial derivatives to imply our function is differentiable? It turns out it suffices to assume the partial derivatives are continuous. This is the major theorem in the subject, and provides a nice way to check for when a function is differentiable.
    • The proof of the alluded to theorem above uses two of my favorite techniques. While sadly we do not multiply by 1, we do get to add 0 and we do use the Mean Value Theorem. One of my goals in the class is to illustrate how to think about these problems, why we try certain approaches for our proofs. We want to study how well the tangent plane approximates our function, thus we need to study f(x,y) - f(0,0) - (δf/δx)(0,0) x - (δf/δy)(0,0) y. Our theorem assumes the partial derivatives are continuous, thus it stands to reason that at some point in the proof we should use the partial derivatives are continuous! The trick is to try and see how we can get another δf/δx and another δf/δy to appear. The key is to recall the MVT. If we add 0 in a clever way, we can do this. Our expression equals f(x,y) - f(0,y) + f(0,y) - f(0,0) - (δf/δx)(0,0) x - (δf/δy)(0,0) y. We now use the MVT on f(x,y) - f(0,y) and on f(0,y) - f(0,0). In each of these two expressions, only one variable changes. Thus the first is (δf/δx)(c,y) x and the second is (δf/δy)(0,ĉ). Thus the error in using the tangent plane is [(δf/δx)(c,y) - (δf/δx)(0,y)] x + [(δf/δy)(0,ĉ) - (δf/δx)(0,o)] y. We now see how the continuity of the partials enters -- it ensures that these differences are small, even when we divide by ||(x,y)-(0,0)||.

   

• Wednesday, February 24. 
  • Continuing our economics theme, one powerful application of the concept of the derivative is that if f'(x) is positive then f is increasing to the right and decreasing to the left, while if f'(x) < 0 then f is decreasing to the right and increasing to the left. This has numerous applications in optimization problems, and explains the shape of objects such as the Laffer curve.
  • Unfortunately, the composition of differentiable functions need not be differentiable. The example we discussed, f(x,y) = (xy)^1/3 and g(x) = (x,x) illustrates a major difference between one and several variables. The real issue is that when we compute the partials, we are only moving in a few directions and not exploring all directions. We will remedy this later.
  • Instead of approximating a function locally by a line, we now use a plane (in 2-dimensions) or hyperplane (in general). We can use the Mean Value Theorem to get some information on how close the estimation is, and then use these estimations to approximate our function. One of the most important applications of the tangent line approximation is Newton's Method, which we'll discuss in much greater detail later. It's a phenomenal way of finding roots of polynomials, and also leads to chaotic / fractal behavior when applied to finding roots of cubics and other functions; click here for the wikipedia article (which has many fascinating pictures). The Mathematica file with the tangent line and tangent plane approximations is here.
  • We discussed how important it is to quantify how close our approximated value is to the function's value. We will expand on this later in the semester; for an entertaining clip on a miscalculation, see the footage here of the Tacoma Narrows Bridge collapse; warning: this video is from the 1940s, and is presented in a very different style than you may be used to!
  • This is the second day in a row where the additional comments have involved fractals; this is not a coincidence; these objects appear throughout modern math / physics / econ / .... The most famous example is the Mandelbrot set. A very nice popular book on the subject is James Gleik's Chaos: Making a New Science (the webpage has links to excerpts from the book). Another wonderful read of his is Genius, which is about the great physicist Richard Feynman (who's work is mentioned in our book).

• Monday, February 22. 
  • We talked about limits and continuity. Informally, a continuous function is one where we can draw its graph without lifting our pen/pencil from the paper. If we take this as our working definition, however, we can easily be misled in terms of properties of continuous functions. For example, are all continuous functions differentiable? Clearly not, as we can take f(x) = |x|. While this function is not differentiable at x=0, it is differentiable everywhere else. Thus we might be led to believe that all continuous functions are differentiable at most points. This sadly is not true.
    • Weierstrass showed (the first text where I read this used the phrase 'Weierstrass distressed 19th century mathematicians) that it is possible to have a function which is continuous and differentiable nowhere! The wikipedia article is a good starting point. In addition to explicitly stating what the function is, it has a nice plot and good comments. The function exhibits fractal behavior, though the term fractal wasn't used until many years later by Mandelbrot.
      • In higher mathematics we learn to quantify orders of infinity. We see that there are more real numbers than rational numbers (see Cantor's diagonalization argument); the comments from Wednesday, February 19th discuss whether or not there is a set whose size is strictly between the rationals and the real. Amazingly, if you count functions properly, it turns out almost every continuous function is differentiable nowhere! See here for some comments on this strange state of facts. The key ingredient is an advanced result from Functional Analysis, the Baire Category Theorem.
    • Fractals have a rich history and numerous applications, ranging from economics to Star Trek II: The Wrath of Khan (where they were used to generate the simulated landscapes of the Genesis Torpedo; see the wikipedia article on fractals in film). The economics applications are quite important. One of the most influential papers is due to Mandelbrot (The Variation of Certain Speculative Prices). This is one of the most important papers in all of economics, and argues that the standard Brownian motion / random walk model of wall street is wrong. The crux of the argument is that these standard theories do not allow enough large deviation days. For more on this, see Mandelbroit-Hudson: The fractal (mis)behavior of markets (I have a copy of this book and can lend you part of it if you are interested).
  • We saw that for many limits of the form 0/0, a good way to attack the problem is to switch to polar coordinates. We then replace (x,y) goes to (0,0) through an arbitrary path with r tends to 0 and θ does whatever it wants. This works for many problems, and is a good thing to try.

   

• Wednesday, February 19. 
  • Russell's paradox is one of the most famous in all of mathematics; it showed that we didn't even understand what it meant to be a set or an element of a set! Another famous one is the Banach - Tarski paradox, which tells us that we don't understand volumes! It basically says if you assume the Axion of Choice, you can cut solid sphere into 5 pieces, and reassemble the five pieces to get two completely solid spheres of the same size as the original! While it is rare to find these paradoxes in mathematics, understanding them is essential. It is in these counter-examples that we find out what is really going on. It is these examples that truly illuminate how the world is (or at least what our axioms, imply). Most people use the Zermelo-Fraenkel  axioms, abbreviated ZF. If you additionally assume the Axiom of Choice, it's called ZFC or ZF+C. Not all problems in mathematics can be answered yea or nay within this structure. For example, we can quantify sizes of infinity; the natural numbers are much smaller than the reals; is there any set of size strictly between? This is called the Continuum Hypothesis, and my mathematical grandfather (my thesis advisor's advisor), Paul Cohen, proved it is independent (ie, you may either add it to your axiom system or not; if your axioms were consistent before, they are still consistent).
  • In a real analysis course, one develops the notation and machinery to put calculus on a rigorous footing. In fact, several prominent people criticized the foundations of calculus, such as Lord Berkeley; his famous attack, The Analyst, is available here. It wasn't until decades later that a good notion of limit, integral and derivative were developed. Most people are content to stop here; however, see also Abraham Robinson's work in Non-standard Analysis. He is one of several mathematicians we'll encounter this semester who have been affiliated with my Alma Mater, Yale. Another is the great Josiah Willard Gibbs.
  • One of my favorite applications of open and closed sets is Furstenberg's proof of the infinitude of primes; one night while a postdoc at Ohio State I had drinks with Hillel Furstenberg and one of his students, Vitaly Bergelson. This is considered by many to be one of the best proofs of the infinitude of primes; it is so good it is one of six proofs given in THE Book. Unlike most proofs of the infinitude of primes, this gives no bounds on how many primes there are at most x; even Euclid's proof (if there are only finitely many primes, say p₁, ..., p_n, then consider (p₁*...*p_n)+1; either this is new prime or is divisible by a prime not in our list, since each prime in our list has remainder 1 when we divide by it) gives a lower bound, namely log log x (the true answer is that there are about x / log x primes at most x). As a nice exercise (for fun), prove this fact. This leads to an interesting sequence: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357....   This sequence is generated as follows. Let a_1 = 2, the first prime. We apply Euclid's argument and consider 2+1; this is the prime 3 so we set a_2 = 3. We apply Euclid's argument and now have 2*3+1 = 7, which is prime, and set a_3 = 7. We apply Euclid's argument again and have 2*3*7+1 = 43, which is prime and set a_4 = 43. Now things get interesting: we apply Euclid's argument and obtain 2*3*7*43 + 1 = 1807 = 13*139, and set a_5 = 13. Thus a_n is the smallest prime not on our list genereated by Euclid's argument at the nth stage. There are a plethora of (I believe) unknown questions about this sequence, the biggest of course being whether or not it contains every prime. This is a great sequence to think about, but it is a computational nightmare to enumerate! I downloaded these terms from the Online Encyclopedia of Integer Sequences (homepage is http://www.research.att.com/~njas/sequences/  and the page for our sequence is http://www.research.att.com/~njas/sequences/A000945 ). You can enter the first few terms of an integer sequence, and it will list whatever sequences it knows that start this way, provide history, generating functions, connections to parts of mathematics, .... This is a GREAT website to know if you want to continue in mathematics. There have been several times I've computed the first few terms of a problem, looked up what the future terms could be (and thus had a formula to start the induction).
    
    
• Tuesday, February 14: Special algebra supplement. The homework problems due Wednesday involve sketching various curves, many of which are famous conic sections. The shapes that arise are often ellipses, hyperbolas, parabolas, lines and circles. The theory of conic sections says that these are all related, and arise as cross sections obtained by having planes intersect a cone at various angles. These shapes arise throughout mathematics and science. Here are just a few examples, which illustrate their importance.
  • Chemisty / Physics: The ideal gas law states that PV = nRT. If we set T equal to a constant, we then get PV is constant (this special case is called Boyle's law). Note that this is an equation of a hyperbola, and thus the isotherms (level sets of constant temperature) are hyperbolas.
  • Physics / Astrophysics: The most common example of conic section is orbits of planets. In three-dimensional space, planets orbiting the sun under a gravitational force proportional to the inverse-square of the distance travel in ellipses, hyperbolas, or parabolas (see here for more details).

• Monday, February 15. 
  •  There are many different coordinate systems we can use; depending on the symmetry of the problem, frequently it is advantageous to use one system over another. Three of the most common are the following:
    • polar coordinates: used most often in the plane where our quantity depends only on the distance from the origin.
    • cylindrical coordinates: generalization of the above where now we live in three dimensional space, but the part depending on x and y only depends on x² + y².
    • spherical coordinates: another generalization to three dimensions, where we only depend on the distance from the origin.
    • Of course, the story does not end in three dimensions. For many problems we need to work with an n-dimensional sphere, and the resulting coordinate system.
  • We discussed level sets today. These occur all the time in real world plots. For example, weather maps constantly show lines of constant temperature; these are called isotherms.
  • Mathematica is a wonderful program to use to plot functions, take derivatives, evaluate integrals, et cetera. I've created a template on how to use Mathematica (and one on how to use LaTeX, to write nicely formatted technical documents). For more information on these, go to the link here.
  • We quickly reviewed finding maxima and minima by searching for critical points and comparing the value of our function there and at the end points. This is fine in one-dimensional calculus, but becomes a bit harder in several variables? Why? What's the difference? The problem is that the boundary of an interval in one dimension is just two points; in other words, the interval [a,b] has just two endpoints, namely a and b. What about an area in the plane, such as B_(0,0)(1), the ball of radius 1 about the origin? The problem is that the boundary here is a one-dimensional object, and there are now infinitely many points to check! We'll see later how to attack this. The key input will be vectors of partial derivatives and figuring out when one vector is in the same direction as another (in other words, using all the material we banked from week 1). This method is known as Lagrange Multipliers.
  • In finding the roots of r^4 - 4r^2 - c = 0 we had to solve a quartic equation; a monumental theorem in mathematics states that there are explicit, closed form solutions for the roots in terms of the coefficients of polynomials of degree one (lines), two (quadratics), three (cubics) and four (quartics), but not for a general polynomial of degree five or higher.

• Friday, February 12.
  • The cross product is related to generalizations of the derivative of a function. It occurs all the time in physics and engineering. Perhaps the most famous set of equations involving it are Maxwell's equations of Electricity and Magnetism. We talked briefly about the operator = (δ/δx, δ/δy, δ/δz). We'll see it greater detail later in the course.The cross product with this symbol arises in Stokes' theorem, a beautiful generalization of the fundamental theorem of calculus to higher dimensions.
  • While we defined the cross product through an abuse of notation, the formula at the end does make sense. It takes two vectors as inputs and gives back another vector. Note the symmetry in the subscripts: v = (v1, v2, v3) and w = (w1, w2, w3) leads to v x w = (v2 w3 - v3 w2, v3 w1 - v1 w3, v1 w2 - v2 w1); each component has a difference of a product of a v and a w, each component is missing the subscript corresponding to it, and the initial subscripts are in cyclic order (1,2,3,1,2,...).
  • We use the right hand screw rule to determine how to orient our axes; while it is not wrong to do so another way, it is counter to all other notation and thus should be avoided.
  • A nice application of the cross product is in determining equations of planes.
  • Our proof that the determinant of a matrix is the area of the parallelogram used crucially the dot product. It is not surprising that this happens. The dot product gives us the cosine of the angle between two vectors; the area requires us to know the sine of the angle (to get the height of the parallelogram), and these are related through the Pythagorean Theorem, which says cos²(x) + sin²(x) = 1. There are many ways to obtain this formula. Perhaps one of the most useful is the Euler - Cotes formula, exp(ix) = cos(x) + isin(x). One can essentially derive all of trigonometry from this relation, with just a little knowledge of the exponential function. Specifically, we have exp(z) = 1 + z + z²/2! + z³/3! + .... It is not at all clear from this definition that exp(z) exp(w) = exp(z+w); this is a statement about the product of two infinite sums equaling a third infinite sum. It is a nice exercise in combinatorics to show that this relation holds for all complex z and w.
    • Taking the above identities, we sketch how to derive all of trigonometry! Let's prove the angle addition formulas. We have exp(ix) = cos(x) + isin(x) and exp(iy) = cos(y) + isin(y). Then exp(ix) exp(iy) = [cos(x) + isin(x)] [cos(y) + isin(y)] = [cos(x) cos(y) - sin(x) sin(y)] + i [sin(x) cos(y) + cos(x) sin(y)]; however, exp(ix) exp(iy) = exp(i(x+y)) = cos(x+y) + i sin(x+y) by Euler's formula. The only way two complex numbers can be equal is if they have the same real and the same imaginary parts. Thus, equating these yields cos(x+y) = cos(x) + isin(x) and sin(x+y) = sin(x) cos(y) + cos(x) sin(y).
    • It is a nice exercise to derive all the other identities. One can even get the Pythagorean theorem! To obtain this, use exp(ix) exp(-ix) = exp(0) = 1.
    • We thus see there is a connection between the angle addition formulas in trigonometry and the exponential addition formula. Both of these are used in critical ways to compute the derivatives of these functions. For example, these formulas allow us to differentiate sine, cosine and the exponential functions anywhere once we know their derivative at just one point. Let f(x) = exp(x). Then f'(x) = lim [f(x+h) - f(x)]/h = lim [exp(x+h) - exp(x)] / h = lim [exp(x) exp(h) - exp(x)] / h = exp(x) lim [exp(h) - 1] / h; as exp(0) = 1, we find f'(x) = exp(x) lim [f(h) - f(0)] / h = exp(x) f'(0); thus we know the derivative of the exponential function everywhere once we know the derivative at 0! One finds a similar result for the derivatives of sine and cosine (again, this shouldn't be surprising as the functions are related to the exponential through Euler's formula).
  • As I've said in class, we could title the first week of the semester Applications of the Pythagorean Theorem. As a number theorist, it's hard for me not to discuss its generalizations. The Pythagorean Theorem says that for a right triangle, a² + b² = c², where a and b are the bases of our triangle and c is the hypotenuse. It is not immediately clear that there are integer solutions to this, but a little inspection turns up a few, such as (3, 4, 5), (5, 12, 13), and of course trivial modifications such as (6, 8, 10). It turns out there are infinitely many solutions in the integers, and there is even a way to generate all of these solutions, which are called Pythagorean triples. Click here for more information on the Pythagorean triples.
  • One can of course ask about generalizations of the Pythagorean Theorem; the most famous is whether or not there are any non-trivial integer solutions to aⁿ + bⁿ = cⁿ, where n > 2 and abc is non-zero. This is the famous Fermat's Last Theorem, solved by Wiles / Taylor-Wiles using elliptic curves and modular forms (on a personal note, I had the pleasure of teaching a class with Wiles on elliptic curves at Princeton in 2001). There are other generalizations, such as the Beal's Conjecture. Another wonderful generalization is Euler's sum of powers conjecture. For a nice occurrence of these in popular culture, see the Homer³ short from Treehouse of Horror VI. Full clip is here, but no sound on my computer. Sound works here, but it's in German, or if you prefer, in Spanish.

   

• Wednesday, February 10.
  • There are many applications of equations of lines, planes and projections. One of my favorites comes from art. The painter Thomas Eakins projected pictures of people onto canvasses; this allowed him to have realistic pictures, and saved him hours of computations. Two pictures frequently mentioned are Arcadia and Mending the Net. He hid what he did; it wasn't until years later that people noticed he had done this. If memory serves, this was discovered when people were looking through photographs in an attic and noticed a picture of four people on a New Jersey highway who were identical to four people in a seascape. Upon closer inspection of the canvass, they noticed marks (which were partly hidden) indicating Eakins projected the image onto the canvass. Click here for more on the subject. See also here for a nice story on the controversy (the use of `technology' such as projectors in art). For a semi-current view on the merits of tracing, watch this video clip.
  • There is an enormous literature on the applications of lines, planes, projections et cetera in art. The wikipedia article is a good starting point.
  • We discussed the equation for the angle between two vectors. Geometrically, it's clear that if we change the lengths of the vectors then we shouldn't change the angle; after a little inspection, we see that our formula satisfies that property. It is a great skill to be able to look at a formula and see behavior like this. There is a rich history of applying intuition like this to problems. One example is dimensional (or unit) analysis,which is frequently seen in physics or chemistry; my favorite / standard example is the simple pendulum.
  • The Cauchy-Schwarz inequality is one of the most important in mathematics; it's used all the time to bound quantities. My favorite application, which is quite advanced, is to the uncertainty principle in quantum mechanics! It turns out one can view the uncertainty principle as a mathematical statement about a function and its Fourier transform. See me if you want more details.
  • There are lots of inequalities in mathematics. Another very useful one is the arithmetic mean - geometric mean; see also my handout with some proofs (written years ago in my Ohio State days)
  • We will not cover determinants in great detail. For us, the most important property is that determinants are related to the volume of the span of the different directions.

• Monday, February 8. A common feature in several variables is to first recall the one variable case, and use that as intuition to describe what's happening. We started by reviewing the three different ways to write the equation of a line in the plane, point-slope, point-point and slope-intercept. We then generalized this to higher dimensions, and then wrote down the definition of a plane (we'll discuss alternate definitions involving normal vectors later in the course; note that planes arose in the Superbowl yesterday as to whether or not the Saints had control when the ball broke the plane during the two point conversion; click here, click here or click here for more on breaking the plane in football).
  • We discussed how there are several different but equivalent ways of writing the same expression. We can do it with vectors, as in (x,y,z) = P + tv, or we can do it as a series of equations, such as x = p₁ + tv₁, y = p₂ + tv₂, z = p₃ + tv₃, or as x_i = p_i + tv_i with i in {1,2,3}.  You should use whichever way is easier for you to visualize. It is possible to get so caught up in reductions and compactifications that the resulting equation hides all meaning. A terrific example is the great physicist Richard Feynman's reduction of all of physics to one equation, U = 0, where U represents the unworldliness of the universe. Suffice it to say, reducing all of physics to this one equation does not make it easier to solve physics problems / understand physics (though, of course, sometimes good notation does assist us in looking at things the right way).
  • A nice problem is to prove the following about perpendicular lines: the product of their slopes is always -1 if neither is parallel to the x- or y-axis. In some sense, this tells us that in the special case when the lines are the x- and y-axes, we should interpret the product of their slopes as -1, or in other words in this case 0 ·∞ = -1.
  • In differential trigonometry, it is essential that we measure angles in radians. If we use radians, then the derivative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x); this is not true if we use degrees. If we use degrees, we have pesky conversion factors of 360/2π to worry about. The proof of these derivatives follow from the angle addition formulas; let me know if you want more details about this (we'll mention this briefly when we do Taylor series of exp(x)).
  • In the proof of the Law of Cosines, the key step was adding an auxiliary line to reduce the problem to the point where we could apply the Pythagorean Theorem. Learning how to add these auxiliary lines is one of the hardest things to do in math. As a good exercise, figure out what auxiliary lines to add to prove the angle addition formula for sine, namely sin(x+y) = sin(x) cos(y) + cos(x) sin(y); click here for the solution. For another example, click here.
  • We ended the day with the definition of the inner or dot product. While our definition only works for vectors, it turns out this is one of the most useful ideas in mathematics, and can be generalized greatly. For example, we can talk about the dot product of functions! We'll see on Wednesday how the dot product is related to angles and lengths, and thus we will find that we can discuss in a sensible manner what the `angle' is between sin(x) and cos(x)!

• Friday, February 5. The main result we proved today was the Pythagorean Theorem, which relates the length of the hypotenuse of a right triangle to the lengths of the sides (President Garfield is credited with a proof). For us, this result is important as gives us a way to compute the length of vectors. While we only proved it in the special case of a vector with two components, the result holds in general. Specifically, if v = (v₁, ..., v_n) then ||v|| = sqrt(v₁² + ... + v_n²). It is a nice exercise to prove this. One way is to use Mathematical Induction (one common image for induction is that of following dominoes); see also my handout on induction. Below are some additional remarks. These relate to material mentioned in class. The comments below are entirely for your personal enjoyment and edification. You do not need to read these for the class. These are meant to show how topics discussed arise in other parts of mathematics / science; these will not be on exams, you are not responsible for learning them, ....
  • We also discussed notation for the natural numbers, the integers, the rationals, the reals and the complex numbers. We will not do too much with the complex numbers in the course, but it is important to be aware of their existence. Generalizations of the complex numbers, the quaternions, played a key role in the development of mathematics, but have thankfully been replaced with vectors (online vector identities here). The quaternions themselves can be generalized a bit further to the octonions (there are also the sedenions, which I hadn't heard of until doing research for today's comments).
  • A natural question to ask is, if all we care about are real numbers, then why study complex numbers? The reason is that certain operations are not closed under the reals. For example, consider quadratic polynomials f(x) = ax² + bx + c with a, b and c real numbers. Say we want to find the roots of f(x) = 0; unfortunately, not all polynomials with real coefficients have real roots, and thus to find the solutions may require us to leave the real. Of course, you could say that if all you care about is real world problems, this won't matter as your solutions will be real. That said, it becomes very useful (algebraically) to allow imaginary numbers such as i = sqrt(-1). The reason is that it allows us a very clean way to manipulate many quantities. Our text has a great discussion of this on pages 54 to 61, especially the top of page 55. There is an explicit, closed form expression for the three roots of a cubic; while it may not be as simple as the quadratic formula, it does the job. Interestingly, if you look at x³ - 15x - 4 = 0, the aforementioned method yields (2 + 11i)^1/3 + (2-11i)^1/3. It isn't at all obvious, but algebra will show that this does in fact equal 4! As you continue further and further in mathematics, the complex numbers play a larger and larger role.
  • Later in the semester we will revisit Monte Carlo Integration, called by many the most important mathematical paper of the 20th century. Sadly, most integrals cannot be evaluated in closed form, and we must resort to approximation methods.
    • Metropolis: The Beginning Of The Monte Carlo Method
    • Metropolis and Ulam: The Monte Carlo Method
    • Note on the origins of the method: http://www.fas.org/sgp/othergov/doe/lanl/pubs/00326866.pdf
  • Sabermetrics is the `science' of applying math/stats reasoning to baseball. The formula I mentioned in class is what's known as the log-5 method; a better formula is the Pythagorean Won - Loss formula (someone linked my paper deriving this from a reasonable model to the wikipedia page). ESPN, MLB.com and all sites like this use the Pythagorean win expectation in their expanded series. My derivation is a nice exercise in multivariable calculus and probability; we will either derive it in class or I'll give a supplemental talk on it.
  • In general, it is sadly the case that most functions do not have a simple closed form expression for their anti-derivative. Thus integration is magnitudes harder than differentiation. One of the most famous that cannot be integrated in closed form is exp(-x²), which is related to calculating areas under the normal (or bell or Gaussian) curve. We do at least have good series expansions to approximate it; see the entry on the erf (or error) function.
    • In class we mentioned that the anti-derivative of ln(x) is x ln(x) - x; it is a nice exercise to compute the anti-derivative for (ln(x))ⁿ for any integer n. For example, if n=4 we get 24 x-24 x Ln[x]+12 x Ln[x]²-4 x Ln[x]³+x Ln[x]⁴.
  • Finally, the quest to understand the cosmos played an enormous role in the development of mathematics and physics. For those interested, we'll go to the rare books library and see first editions of Newton, Copernicus, Galileo, Kepler, .... Some interesting stories below; see also a great article by Isaac Asimov on all of this, titled The Planet That Wasn't.
    • The discovery of Uranus
    • The discovery of Neptune
    • The attempted discovery of Vulcan