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### Math 236 - Multivariable Calculus - Spring 2009

Be sure to check back, because this may change during the semester.

I'll use Maple syntax for mathematical notation on this page. All numbers indicate sections from Multivariable Calculus Early Transcendental Functions, Third Edition by Smith and Minton.

#### For Friday January 23

Section 10.1 Vectors in the Plane
Section 10.2 Vectors in Space
Section 10.3 The Dot Product

Let a, b, and c be vectors, and let a.b denote the dot product.
1. Is (a.b).c the same as a.(b.c)?
2. In what direction does projba point?
3. When is compba equal to compab?

#### For Monday January 26

Section 10.4 The Cross Product

1. How is axb related to a and b geometrically?
2. Why don't we define axb for vectors in the plane?

#### For Wednesday January 28

Section 10.5 Lines and Planes in Space

1. What information about a line L do you need to determine an equation for the line?
2. What information about a plane P do you need to determine an equation for the plane?

#### For Friday January 30

Section 11.1 Vector-Valued Functions

1. Consider the graph of the vector-valued function r(t)=cos(t)i - sin(t)j. Is this the graph of a function y=f(x)? Explain.
2. What are some advantages to using vector-valued functions?

#### For Monday February 2

Section 11.2 The Calculus of Vector-Valued Functions

1. If r(t) is a vector-valued function, what geometric information does r'(t) give you?
2. If the graphs of r(t) and s(t) are the same, will r'(0) be the same as s'(0)? Explain.

#### For Wednesday February 4

Section 11.3 Motion in Space

To read : All. Be sure to understand Examples 3.4 and 3.5.

1. What quantity does the magnitude of the velocity vector give?
2. How would including air resistance in Example 3.4 complicate issues?

#### For Friday February 6

Section 11.4 Curvature

1. Explain the idea of curvature in your own words.
2. If the helix in Example 4.5 were changed to r(t)=< 2sin(t), 2cos(t), 4t2 > , will the curvature still be constant? Don't actually do the calculation, but give an intuitive justification.

#### For Monday February 9

Work on Project 1 today. No Reading Assignment.

#### For Wednesday February 11

Section 11.5 Tangent and Normal Vectors

To read : Through the section Tangential and Normal Components of Acceleration.

1. Suppose you are skiing down a hill that curves left. Describe the directions of the unit tangent, principle unit normal, and binormal vectors.
2. Why do you want the normal component of acceleration to be small when steering a car through a curve?

#### For Friday February 13

Section 11.5 Tangent and Normal Vectors

#### For Monday February 16

Section 10.6 Surfaces in Space

Consider the surface y=x2 + z2
1. What does the trace in the xy-plane look like?
2. What do the traces in the planes y=k look like?

#### For Wednesday February 18

Class rescheduled to February 22. No Reading Assignment.

#### For Friday February 20

Section 12.1 Functions of Several Variables

1. Is a hyperboloid of one sheet the graph of a function of two variables? Explain.
2. How can you identify the local extrema of f(x,y) from its contour plot?

#### For Monday February 23

In-class part of Exam 1. No Reading Assignment.

#### For Wednesday February 25

Section 12.2 Limits and Continuity

1. What is the point of Example 2.5?
2. Why do you think we are studying limits now?

#### For Friday February 27

Section 12.3 Partial Derivatives

1. For f(x,y), what information does fx(1,0) give?
2. How many second-order partial derivatives does g(x,y,z) have?

#### For Monday March 2

Snow Day. Very, very weird.

#### For Wednesday March 4

Section 12.4 Tangent Planes and Linear Approximations

1. If f(x,y) is a well-behaved function and has a local maximum at (a,b), what can you say about the linear approximation to f(x,y) at (a,b)?
2. Let L(x,y) be the linear approximation of f(x,y) at (a,b). What graphical properties of the surface z=f(x,y) would make L(x,y) particularly accurate? particularly inaccurate?

#### For Friday March 6

Class rescheduled (TBA). No Reading Assignment for today.

#### For March 9 - 13

Spring Break. Surprisingly, no Reading Assignments.

#### For Monday March 16

Section 12.5 The Chain Rule

Suppose that w=f(x,y,z) and that x,y,z are all function of r,s,t. How many partial derivatives do you need to calculate in order to determine dw/dt?

#### For Wednesday March 18

This is shifted to be for Friday, March 20.

Section 12.6 The Gradient and Directional Derivatives

1. Explain the idea of a directional derivative your own words.
2. What type of quantity is gradient of a function f(x,y)?

#### For Monday March 23

Section 12.7 Extrema of Functions of Several Variables

1. If the partials fx and fy exist everywhere, at what points (x0, y0) can f have a local max or a local min?
2. Suppose that f is a well-behaved function where fx(3,4)=0, fy(3,4)=0, fxx(3,4)=2, fyy(3,4)=-3, and fxy(3,4)=-2. Will (3,4) be a local max, min, or neither of f? Why?
3. Explain the idea behind the method of steepest ascent in your own words.

#### For Wednesday March 25

Section 12.7 Extrema of Functions of Several Variables

#### For Friday March 27

Section 13.1 Double Integrals

1. If f(x,y) is a function of two variables, what does R f(x,y) dA measure?
2. Explain Fubini's Theorem in your own words. What is its importance?

#### For Monday March 30

Section 13.1 Double Integrals
Reread the section, paying especial attention to Example 1.7, but no Reading Questions for today.

#### For Wednesday April 1

Polar Coordinates: Handout

1. What do the coordinates (r,theta) in polar coordinates measure?
2. Why does finding area by integration in polar coordinates differ from finding area by integration in rectangular coordinates?

#### For Friday April 3

Section 13.3 Double Integrals in Polar Coordinates

1. Why would you ever want to convert a double integral from rectangular to polar coordinates?
2. What is the shape of a polar rectangle?

#### For Monday April 6

Section 13.3 Double Integrals in Polar Coordinates

#### For Wednesday April 8

Section 13.4 Surface Area

1. Give a real-world example where you would want to compute surface area.
2. After partitioning the region R, what object is used to approximate the surface area over each subregion Ri?

#### For Friday April 10

Section 14.1 Vector Fields

1. Explain why Graph B in Example 1.3 is not the actual graph of a vector field but is just a representation of the graph of a vector field.
2. If a particle is dropped onto Figure 14.7a at the point (-1,1), describe the path the particle will follow.

#### For Monday April 13

In-class part of Exam 2. No Reading Assignment.

#### For Wednesday April 15

Section 14.2 Line Integrals

Give two different uses for the line integral.

#### For Friday April 17

Section 14.2 Line Integrals

#### For Monday April 20

Section 14.3 Independence of Path and Conservative Vector Fields

1. Why are conservative vector fields your friend when evaluating line integrals?
2. Why is Theorem 3.2 call the Fundamental Theorem for line integrals?

#### For Wednesday April 22

Section 14.3 Independence of Path and Conservative Vector Fields

#### For Friday April 24

Section 14.4 Green's Theorem

1. What surprises you about Green's Theorem?
2. Give an example of a region R in the plane where Green's Theorem does not hold.

#### For Monday April 27

Section 14.4 Green's Theorem