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. Examples
from when I taught the class in 2010 are
available online here, though we used a different book then and covered slightly
different material; 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 11: Vectors, Curves and Surfaces in Space
 Section 11.1: Vectors in
the Plane
 Notation, definition of vectors and
properties.
 Proof of the Pythagorean formula (which is
crucial in determining lengths).
 Section 11.2:
ThreeDimensional Vectors
 Know the definition of the dot product of two
vectors, and the connection of that to the angle between two vectors.
 Section 11.3: The Cross
Product of Vectors
 Know the definition of determinants of 2x2
and 3x3 matrices, and how to compute these.
 The determinant has much geometrical meaning,
denoting the (signed) volume of the parallelpiped spanned by the rows (or
columns).
 Know the definition of the cross product and
how to compute it, as well as some of its properties.
 Section 11.4: Lines and
Planes in Space
 Know the various formulas for writing the
equation of a line.
 There are several ways to write the equation
of a plane; it's similar to writing the equation of a line: depending what
information you are given, some ways are more convenient than others.
 One easy way to find the equation of a plane
is to know the normal direction. This is a great application of the cross
product, as v x w is perpendicular to both v and
w. Unfortunately we don't have the cross product in higher dimensions.
 Section 11.8: Cylindrical
and Spherical Coordinates
 Know the different formulas to convert from
Cartesian to Cylindrical or Spherical coordinates.
CHAPTER
12: Partial
Differentiations
 Section 12.1: Introduction
 Not much here except (what a surprise) that
many functions in the real world depend on several variables.
 Section 12.2: Functions of
Several Variables
 First is when a function is defined on a
domain (usually just making sure the denominator is nonzero).
 The level set (of value c) of a function are
all inputs that are mapped to c. Think of this as all points on a mountain
that are the same height, or on a weathermap all places with the same
temperature.
 Section 12.3: Limits and
Continuity
 Know the definition and basic properties of
limits.
 Caveats: certain operations are not defined:
∞∞, 0 * ∞.
 Section 12.4: Partial
Derivatives
 Know the definition of how to take a partial derivative. Similar to
onevariable calculus, we do not want to have to use this definition in
practice, and thus want to modify our rules of onevariable differentiation to
allow us to take derivatives here.
 Know the formula for computing the tangent plane to z = f(x,y) at a given
point, so long as the partial derivatives exist at that point.
 Iterated Partial Derivatives:
 The definition of mixed partial derivatives: 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 δ^{2}f/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 C^{2}, the class
of twice continuously differentiable functions: If the function
is C^{2}, the mixed partial derivatives of second order (i.e.,
involving at most two derivatives) exist and are continuous. Just as C^{1}
functions had nice properties (being C^{1} means the partial
derivatives exist and are continuous, which implies the function is
differentiable), being C^{2} has nice properties.
 Equality of Mixed Partial Derivatives: For a C^{2} function, the order of
differentiation does not matter; in other words, δ^{2}f/δyδx = δ^{2}f/δxδy.
 Notation: f_{x} means δf/δx, f_{xy}
means (f_{x})_{y}, which is δ^{2}f/δyδx. Note that
the order of subscripts is the opposite of the order of differentiation;
fortunately if f is C^{2} 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
NavierStokes 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 12.5:
Multivariable Optimization Problems
 Advanced result: any continuous function on a nice region that includes
the boundary attains its maximum and minimum.
 Definition of local maximum / minimum: You should be comfortable with the definition of a local max/min.
Essentially, a point x_{0} is a local maximum if there is
some ball centered at x_{0} such that f(x_{0})
is at least as large as f(x) for all other x in that ball. For
example, f(x,y) = y^{2} sin^{2}(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: The generalization of the results from onevariable
calculus; candidates for max/min are where the first derivative (the
gradient) vanishes.
 Important Example: The 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.
 Method of Least Squares

My notes on the Method of Least Squares

There are
many different ways to choose how we measure errors. Different choices lead to
different `best fit' values for parameters. The main advantage of measure
errors by summing the squares of the deviations is that the tools of calculus
and linear algebra are available.
 Section 12.6: Increments
and Linear Approximations

The idea is
to replace complicated functions with simpler ones that are easier to analyze
(in many cases, one can get very good results just by using linear
approximations).

Newton's
method is one of the most important uses of the tangent line. The idea is
based on the fact that locally any function is approximately linear.
 Section 12.7: The
Multivariable Chain Rule
 The most important part of this section is
the statement of the chain rule. A good way to remember what goes where is
through the atom graph.
 Section 12.8: Directional
Derivatives and the Gradient Vector

The
definition of the gradient. Note the gradient is the derivative
of a function from R^{n} 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: 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 chain rule.

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:
the gradient points in the direction where
f is increasing the fastest, and 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.
 Section 12.9: Lagrange
Multipliers and Constrained Optimization
 The method of Lagrange Multipliers: 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 x_{0} is an
extremum for f with our constraints if there is some number λ such that
(f)(x_{0})
= λ
(g)(x_{0}).
 Caveats: Existence of solutions: Just because we've found candidates does
not mean one of them must work! Also, while the idea is straightforward,
frequently the algebra needed to solve the problem can be tedious.
 I will try and do several examples of applications of Lagrange
multipliers.
 Section 12.10: Critical
Points of Functions of Two Variables
 Basically just be aware of Theorem 1, namely that there exist conditions
to classify the nature of critical points. The formulas look quite strange,
and will make a lot more sense after learning about eigenvalues in Linear
Algebra.
CHAPTER 13: Multiple 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 13.1: Double
Integrals
 The definition of the double integral is
very important; we'll discuss in great depth the
corresponding framework in onedimension. One can check the Fundamental
Theorem of Calculus by using Mathematical Induction and limits to find the
area under polynomial functions.
 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 (linearity,
homogeneity, monotonicity and additivity). The proofs are similar to the
proofs in the 1dimensional case.
 Section 13.2: Double
Integrals over more general Regions
 Know the definition of the following terms:
boundary, vertically simple, horizontally simple.
 The main result is that the integral over
the rectangle can be written as an iterated integral (remember the double
integral is defined through boxes and limits).
 Know the statement of Fubini's Theorem about when you can
interchange orders of integration. We will not do the proof in class, though
it is in the book.
 Section 13.3: Area and
Volume by Double Integrals
 Know the formulas to find volumes from
integrating.
 Section 13.4: Double
Integrals in Polar Coordinates
 Know how to convert an integral in (x,y) space to one in (r,theta)
space.
 Unit analysis is a great guide, and suggests dx dy becomes r dr dtheta.
 Section 13.6: Triple
Integrals
 Essentially the same as double integrals.
 Section 13.7: Integration
in Cylindrical and Spherical Coordinates
 Know the change of variables from Cartesian
to Cylindrical and Spherical.
 Know how the volume element changes in
each: r dr dtheta in the first, rho^2 sin(phi) dpho dphi dtheta in the
second.
 Know how to convert Cartesian integrals to
Cylindrical or Spherical.
 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
antiderivatives; however, in the real world most functions we encounter do
not have nice antiderivatives; 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 3638).
 Section 13.9: Change of
Variables in Multiple Integrals
 Know the definition of Jacobian determinants.
 Read the statement of the Change of
Variables formula. 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, cylindrical coordinates, and spherical
coordinates.
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.
From Line Integrals
to Green's Theorem: TBD
Also, you may
wish to look at some worked out examples before class that are similar to the
HW. These are
available online here, those these problems are from the 2010 version of the
class, which used a different book; 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.