Additional Comments on the lectures: click here
Review sheet for the course (definitions, theorems): click here for a review sheet (.ps version    version wi'out images)
Upcoming HW and reading: (previous HW)    (comments/solutions to the HW)  solutions to similar questions
Readings before class: click here for a bullet point summary to help prepare you for class.
Takeaways for undergraduate classes: summarizes key points from this and other undergrad courses

Summary of what we've done in Math 105

We've covered a lot of material in Math 105. Similar to Calc I and Calc II, the course splits into two separate but related areas, differential and integral calculus. The most important skills developed were learning how to compute partial derivatives and setting up and evaluating multiple integrals. These are very important techniques which have a wealth of applications in subjects such as chemistry, economics, engineering, physics and statistics, to name just a few. The main idea of partial derivatives can be summed up in a sentence: to take the partial derivative of f(x,y,z) with respect to x, simply consider y and z constant and use the standard rules from one variable calculus. Similarly, if we are given the regions of integration then multiple integrals is just an extension of previous courses: to evaluate a multiple integral when the regions of integration are given, simply perform the specified integrations one at a time. While these two skills can be quickly stated, implementation can be difficult. In practice one often has to find the bounds of integration, or even worse, there is no nice anti-derivative for the function to be integrated. We discuss below techniques to get around these difficulties. On the differentiation side, it is always easy to apply the standard differentiation rules to find the derivatives; the subtlety is that sometimes a function has all of its partial derivatives existing but is not differentiable. We thus spent a lot of time discussing just what it means to be differentiable, and this paid numerous dividends throughout the course (especially in the Change of Variable Formula).

We began the semester with a short tutorial on vector calculus. This ranged from generalizing the definition of a line to introducing new concepts such as dot products, cross products, matrices and determinants. It was actually awhile before we returned to calculus. The reason is that to generalize the material properly requires the language of vector calculus. Vectors are a concise and illuminating way of encoding information about the world, and are used all the time in physics, chemistry and engineering. We constantly returned to this material to describe the quantities we investigated, especially at the very end when we did Green's Theorem / Stokes' Theorem, the massive generalization of the Fundamental Theorem of Calculus.

The next part of the course was the most technical, where we covered the notion of what it means for a function of several variables to be continuous and differentiable at a point. We saw how limits in several variables are more subtle then those in one variable, as for the limit to exist we must equal the same finite limit for every path. In one dimension there are essentially just two paths: approach from above or from below. In several variables the structure is far richer. We then generalized the standard differentiation rules to several variables; most were straightforward, though it took awhile to deal with the chain rule. There are many applications of differentiation. The first was Lagrange multipliers, which generalizes checking the boundary points in one-dimension when generating candidate points for max/min. Studying this problem involved understanding level sets (which became our constraints) and directional derivatives (which is the gradient, or the vector of partial derivatives in the direction of the coordinate axes, dotted with the direction we care about). The other application was the Method of Least Squares, where we saw how we could use critical points to find best fit parameters. More importantly, we saw how the best fit parameters depend on how we choose to measure our errors! Another interesting application was Taylor's theorem, which allowed us to replace complicated functions with simpler ones. We saw an application in Newton's method to find roots.

Next it was integration theory. We proved the Fundamental Theorem of Calculus in one variable, and discussed its generalization to several variables. This led to several discussion on what we mean by multi-dimensional integrals. We developed techniques for handling the situation of continuous (or mostly continuous) functions on finite, bounded regions. The most important result here was Fubini's theorem, on when we could interchange orders of integration.  A close second is the change of variable formula, which allows us (for many problems) to exploit the natural coordinate system for that problem. Proving this result required a solid foundation in what it means to be differentiable (the tangent plane approximation does a great job). One of the most important applications of computing integrals are with probability. We may frequently interpret the integrals as probabilities. Unfortunately, most integrals cannot be done in closed form, and thus we also discussed Monte Carlo techniques to simulate with high probability the values of these integrals.

We then turned to sequences and their sum, series. One clear reason for the importance of this topic is that some functions, such as exp(x), are often defined by their Taylor series expansion, and thus it is nice to now that this function is well-behaved. We learned several methods to test whether or not a series converges or diverges, and assembled a table with our findings.

Finally, we ended by considering the Generalized Stokes' Theorem (we did the special case in the plane, Green's Theorem). This involved arc length, path and line integrals, operations built up from the del operator, and many concepts that will resurface in physics and engineering. For me one of the most beautiful moments of the semester was seeing how the interior integrals cancel which allows us to reduce to the special case of rectangles.

In summary, we covered a lot of material. We mastered techniques such as computing partial derivatives and integrating over regions in the plane, while at the same time learning about having a healthy skepticism. Additionally, we talked about mathematical techniques, especially the importance of arranging algebra to be enlightening and doable. Common methods to attack problems were adding zero, multiplying by one, and approximating through the Mean Value Theorem or Taylor series.

Math 105: Outline of the Course

Below is a quick outline of key points covered in the course.

• Chapter 1: Geometry of Euclidean Space
  1. Equation of lines, planes, distance (Pythagorean theorem)
  2. Vectors (magnitude and direction)
  3. Dot products and angles
  4. Cross product and areas
  5. Determinants and areas / volumes

   

• Chapter 2: Differentiation
  1. Level sets
  2. Open Sets
  3. Continuity
  4. Limit laws
  5. Differentiability (tangent plane approximation is very good)

                                                              i.      Main Theorem: continuos partials à differentiable à partials exist, converses can fail

                                                            ii.      Rules of differentiation (especially Chain Rule)

                                                          iii.      Directional derivatives:

1.      gradient points in direction of fastest change

2.      gradient is normal (perpendicular) to level sets

6. Parametrized curves c(t), speed is length of c’(t)

• Chapter 3: Higher order derivatives and Max/Min
  1. Taylor’s theorem (replace complicated functions with simpler ones)

                                                              i.      Application: Newton’s Method (when applicable superior to Divide and Conquer)

2. Main Theorem: If f of class C² (derivatives up to 2^nd order continuous) then mixed partials are equal
3. Finding candidates for max/min of f subject to some constraint, say g(x) ≤ c.

                                                              i.      Check interior points: find critical points (those where grad(f) = 0)

                                                            ii.      Check boundary points: if on level set g(x) = c, use Lagrange Multipliers

1.      grad(f) = λ grad(g); often break analysis into λ zero and non-zero, frequently taking ratios of equations helps, sometimes can find point without finding λ, remember to include the constraint when listing all equations.

4. Method of least squares (application to grad(f) = 0).

• Chapter 5: Double and triple integrals
  1. Riemann sums for 1- and 2-dimensional integrals (follows from limits and MVT)
  2. Finding bounds of integration: x-simple, y-simple, simple and elementary regions
  3. Main theorem: Fubini’s theorem: interchanging order of integration
  4. Monte Carlo Integration: most integrals cannot be evaluated in closed form, resort to numerical simulations, this uses probability.
  5. Application of integration: areas / volumes often represent probabilities of events.

   

• Chapter 6: Change of variables formula
  1. Change of variable formula (especially polar, cylindrical, spherical): figure out exchange rate from volume element in one coordinates and volume element in another; need absolute value of the determinant of the derivative of the inverse map (called the Jacobian).

   

• Sequences and Series
  1. Formulas for geometric series: if |r| < 1 then Sum_{n = 1 to ∞} rⁿ = 1 / (1-r).
  2. Divergence of harmonic series Sum_{n = 1 to ∞} 1/n but convergence of Sum_{n = 1 to ∞} 1/n².
  3. Tests of convergence: comparison, ratio, root, integral.

   

• Stokes’ Theorem:
  1. generalizes fundamental theorem of calculus, applications in physics, engineering. Involves other combinations of del operator (divergence, curl).

Additional Comments on the lectures: click here
Review sheet for the course (definitions, theorems): click here for a review sheet (.ps version    version wi'out images)
Upcoming HW and reading: (previous HW)    (comments/solutions to the HW)  solutions to similar questions
Readings before class: click here for a bullet point summary to help prepare you for class.
Takeaways for undergraduate classes: summarizes key points from this and other undergrad courses