INFORMATION ON READING BEFORE CLASS

Below are some comments to help you prepare for each class' lecture. For each section in the book, I'll mention what you should have read for class. In other words, what are the key points. When you come to class, you should have already read the section and have some sense of the definitions of the terms we'll study and the results we'll prove. This does not mean you should know the material well enough to give the lecture; it does mean that you should have a familiarity with the material so that when I lecture on the math, it won't be your first exposure to the terminology or results. Everyone processes and learns material in different ways; for me, I find it very hard to go to a lecture on a subject I'm unfamiliar with and get much out of it. I need to have some sense of what will happen, as otherwise I spend too much time absorbing the definitions, and then I fall behind. I'm hoping the bullet points below will help you in preparing for each lecture. If there is anything else I can do to assist, as always let me know (either email directly, or anonymously through mathephs@gmail.com, passsword 11235813).

Also, you may wish to look at some worked out examples before class that are similar to the HW. These are available online here; I will do many of these problems in class. The reason I want to do these is precisely because I have written up the solution. This way you can sit back a bit more and follow the example without worrying about writing everything down.

CHAPTER 2: DIFFERENTIATION

• Section 2.6: Gradients and Directional Derivatives:
  • The definition of the gradient (page 163). Note the gradient is the derivative of a function from Rⁿ to R; it is a vector with n components, where the i^th component is the partial derivative of f in the direction of the i^th coordinate axis.
  • The definition of the directional derivative (page 164): This generalizes the partial derivatives we've discussed, and allows us to look at how a function is changing along an arbitrary line (but not an arbitrary curve). The definition even suggests a way to compute the directional derivative: use the first special case of the chain rule (Section 2.5, page 153).
  • Theorem 12 (page 165): This is the simplest way to compute the directional derivative; it states that the directional derivative of f in the direction of v at the point x is just the dot product of the gradient of f and v; in other words, the directional derivative is (f)(x) • v.
  • Geometric interpretation of the gradient: Theorem 13 (page 166) states that the gradient points in the direction where f is increasing the fastest, while Theorem 14 (page 167) tells us that the gradient is perpendicular to level surfaces (we'll discuss this in much greater detail in class). These two items will be of great aid in optimization problems in Chapter 3.

CHAPTER 3: HIGHER-ORDER DERIVATIVES: MAXIMA AND MINIMA

• Section 3.1: Iterated Partial Derivatives:
  • The definition of mixed partial derivatives (page 182): Given a function f, we can compute its partial derivatives, such as δf/δx and δf/δy. We can then take the partial derivatives of the partial derivatives: δ(δf/δx)δy and δ(δf/δy)δx. In the first, we first take the derivative with respect to x, and then take the derivative with respect to y; in the second, we take the derivatives in the other order. Does the order matter? We write δ²f/δyδx for δ(δf/δx)δy; thus the derivative symbol on the far right of the denominator is the derivative we take first, and the symbol on the far left is what we take last.
  • The definition of C², the class of twice continuously differentiable functions (page 182): If the function is C², the mixed partial derivatives of second order (i.e., involving at most two derivatives) exist and are continuous. You should compare this to the definition of C¹ on page 138. Just as C¹ functions had nice properties (being C¹ means the partial derivatives exist and are continuous, which implies the function is differentiable), being C² has nice properties.
  • Equality of Mixed Partial Derivatives (Theorem 1, page 183): For a C² function, the order of differentiation does not matter; in other words, δ²f/δyδx = δ²f/δxδy.
  • Notation: f_x means δf/δx, f_xy means (f_x)_y, which is δ²f/δyδx. Note that the order of subscripts is the opposite of the order of differentiation; fortunately if f is C² then the order does not matter!
  • Examples of partial differential equations: The rest of the section is devoted to examples of partial differential equations. Solving these in general are beyond the scope of this course; in fact, most are beyond the scope of humanity! One example is the Millenium Prize for the Navier-Stokes equation (i.e., solve this and earn $1,000,000). You are not responsible for any of this material; it is provided in nice detail in this book for your interest, and to show you what you will see if you continue with mathematics.
• Section 3.2: Taylor's Theorem:
  • In addition to reading our book, see also my lecture notes on Taylor's Theorem in one-variable. I derive Taylor's theorem from (surprise!) the Mean Value Theorem.
  • Taylor's Theorem in one-variable: You should read the statement of Taylor's theorem in one variable (equation 1 on page 194). I will present a proof of a special case using the mean value theorem, and thus you may ignore (if you wish) the proof from page 194 to195).
  • Taylor's Theorem in several variables (Theorems 2 and 3, page 196): You should be aware of the statement. The top of page 197 gives a nice interpretation of the result in terms of matrices. We'll talk a little about this, but as linear algebra is not a pre-requisite, we will not delve into great detail. Knowing the statements of Theorems 2 and 3 is enough; you should return to this section after linear algebra.
  • Ignore all forms of the remainder when reading this section, unless you want to know more about this. Thus you do not need to read pages 198 and the top of 199. Instead of using the integral version of the MVT, I prefer giving proofs using the standard formulation.
  • Examples: You should read the examples from pages 199 to 202. I'll discuss some of these and related problems in class; I'll also show you a nice `trick' that allows you to compute certain Taylor series expansions a lot faster than the standard method; though it only works some of the time, when it does it is quite fast.
• Section 3.3: Extrema of Real-Valued Functions:
  • Fun reading: the historical part at the beginning of the section is for your enjoyment only. This is continued in the historical note starting on page 267.
  • Definition of local maximum / minimum (page 207): You should be comfortable with the definition of a local max/min. Essentially, a point x₀ is a local maximum if there is some ball centered at x₀ such that f(x₀) is at least as large as f(x) for all other x in that ball. For example, f(x,y) = y² sin²(xy) has a local minimum at (x,y) = (0,0). Clearly f(x,y) is never negative, and it is zero at (0,0). Thus (0,0) is a local minimum. Note that f(x,0) is also zero for any choice of x. Thus to be a local minimum we don't need to be strictly less than all other nearby points.
  • First derivative test for local extrema (Theorem 4, page 208): The generalization of the results from one-variable calculus; candidates for max/min are where the first derivative (the gradient) vanishes.
  • Examples (pages 209 - 210): you should look at these examples on how to find max/min.
  • Do not read pages 211 - 220. We will not cover these in class; however, once you know linear algebra these results make more sense (and can be rewritten in a more illuminating manner). We will discuss this in the advanced class.
  • Summary (page 221): Look at the summary on the top of page 221 and the subsequent example. This provides a nice recap of the method to find maxima/minima.
• Section 3.4: Constrained Extrema and Lagrange Multipliers:
  • The method of Lagrange Multipliers (Theorem 8, page 226): This is the key result: it says that if we want to find the extrema for a function f whose input x is the level set of some value for a function g (i.e., find the max/min of f(x) given that g(x) = c for a fixed c), then this happens at points where the direction of the gradient of f is the same as the direction of the gradient of g. We may rewrite this condition and say that x₀ is an extremum for f with our constraints if there is some number λ such that (f)(x₀) = λ (g)(x₀).
  • Examples (pages 228 - 231): Read these examples on how to use Lagrange multipliers.
  • Caveats: Existence of solutions (page 231): You should read the story about Dorothy Sayers' detective. It illustrates in a nice way a common danger: just because we prove that the only candidates for a max/min are the critical points, the function may not have a max/min! Theorem 7 (page 220 of Section 3.3) states when we must have a max/min.
  • I will try and do several examples of applications of Lagrange multipliers.
• Section 3.5: The Implicit Function Theorem:
  • We will not cover this section.
• Special Topic: Method of Least Squares
  • We will give one of the most important applications of partial derivatives and optimization, the Method of Least Squares. This is a technique to allow us to find best fit parameters. Finding such values is central in numerous disciplines. Specifically, we have some data and we want to see if it fits our theory. If you have a data set you'd like analyzed, please let me know.

CHAPTER 4: VECTOR VALUED FUNCTIONS: will not cover this chapter in the main lectures

CHAPTER 5: DOUBLE AND TRIPLE INTEGRALS

• General Comments:
  • As many people have not seen a proof of the Fundamental Theorem of Calculus, I will prove this important result in full detail in class, and merely state what happens in several variables. We will loosely follow the book for this chapter. The reason is that, as we only have 12 weeks, we do not have time to delve fully into the theory of double and triple integrals. Instead, for this chapter we will concentrate on the applications, namely becoming proficient at computing these integrals.
• Section 5.1: Introduction
  • For us, all that matters is the definition of the double integral (the box on page 318). I encourage you to skim the rest of the section, but this is all that matters for us.
• Section 5.2: The Double Integral over a Rectangle
  • The definition of the double integral is very important (pages 327-328); we'll discuss in great depth the corresponding framework in one-dimension.
  • Know Theorem 1, page 329 (any continuous function on a closed rectangle, such as [a,b] x [c,d] with a,b,c,d finite) is integrable. We will discuss the proof of a related, simpler statement. We will not prove this result in full generality, though the proof is in the book if you wish to read it / discuss it with me.
  • Be aware of the four properties of integrals on page 331 (linearity, homogeneity, monotonicity and additivity). The proofs are similar to the proofs in the 1-dimensional case.
  • Know the statement of Fubini's Theorem (page 334) about when you can interchange orders of integration. We will not do the proof in class, though it is in the book.
• Section 5.3: The Double Integral over more general Regions
  • Know the definition of the following terms: boundar, x-simple, y-simple, simple, and elementary region (page 341).
  • Know the definition of the integral over an elementary region (pages 342-343).
  • Know Theorem 4 (Reduction to Iterated Integrals, page 344): this allows us to compute integrals over elementary regions. See also the example on page 345.
• Section 5.4: Changing the Order of Integration
  • The key result of this section is the formula (sadly not boxed) at the top of page 349, which tells us how to switch the order of integration.
  • Read example 1, page 349: great illustration of how changing orders can help.
• Section 5.5: The Triple Integral
  • We will not cover this section in the lectures.
• Special Topic: Monte Carlo Integration
  • Monte Carlo Integration has been called one of the most useful results of 20th century mathematics. We'll discuss how it is done. It is an alternative to standard integration. Normally we look for anti-derivatives; however, in the real world most functions we encounter do not have nice anti-derivatives; Monte Carlo Integration provides a way to approximate these integrals.
  • The lecture notes for this is not the book, but rather my lecture notes (the last three pages of my chapter 3 notes, namely pages 36-38).

CHAPTER 6: CHANGE OF VARIABLE FORMULA AND APPLICATIONS OF INTEGRATION

• Section 6.2: The Change of Variable Theorem
  • Know the definition of Jacobian determinants (page 377); see example 1 of that page.
  • Read the statement of the Change of Variables formula for double integrals on page 382. We will not deal with this theorem in its full generality, but I want you to at least be aware of its statement. We will concentrate on several special cases: polar coordinates (page 383), cylindrical coordinates (page 388), and spherical coordinates (page 389). Feel free to skip the rest of this section (as well as the rest of this chapter).

CHAPTER 10 (Cain and Herod): SEQUENCES, SERIES AND ALL THAT: notes available here.

• 10.1: Introduction
  • Just know that one motivation comes from Taylor series.
• 10.2: Sequences
  • Know the definition of a sequence and some common examples.
  • Know what it means for a sequence to converge.
• 10.3: Series
  • Know the definition of a series.
  • Know what it means for a series to converge.
  • Know the definition of the harmonic series.
  • Know the integral test.
• 10.4: More Series
  • Know the definition of a positive series.
  • Know the comparison test for convergence (it's the method of this section; they don't call it the comparison test till the next section).
• 10.5: Even More Series
  • Know the ratio test for convergence.
• 10.6: A Final Remark
  • Know the alternating test for convergence.

Also, you may wish to look at some worked out examples before class that are similar to the HW. These are available online here; I will do many of these problems in class. The reason I want to do these is precisely because I have written up the solution. This way you can sit back a bit more and follow the example without worrying about writing everything down.