Math 236 - Multivariable Calculus - Spring 2009

Reading Assignments

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

To read : All

1. Is (a.b).c the same as a.(b.c)?
2. In what direction does proj_ba point?
3. When is comp_ba equal to comp_ab?

For Monday January 26

To read : All

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

To read : All

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

To read : All

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

To read : All

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

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

To read : All

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

For Wednesday February 11

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

Finish reading the section, but no Reading Questions for today.

For Monday February 16

To read : All

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

To read : All

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

For Wednesday February 25

To read : All

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

For Friday February 27

To read : All

1. For f(x,y), what information does f_x(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

To read : All

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

For March 9 - 13

For Monday March 16

To read : All

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.

To read : All

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

For Monday March 23

Section 12.7 Extrema of Functions of Several Variables

To read : All

1. If the partials f_x and f_y 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 f_x(3,4)=0, f_y(3,4)=0, f_xx(3,4)=2, f_yy(3,4)=-3, and f_xy(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
Reread the section, but no Reading Questions.

For Friday March 27

Section 13.1 Double Integrals

To read : All

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

Reading Questions:

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

To read : All

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
Reread the section, but no Reading Questions for today.

For Wednesday April 8

To read : All

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 R_i?

For Friday April 10

To read : All

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

For Wednesday April 15

To read : All

Give two different uses for the line integral.

For Friday April 17

For Monday April 20

To read : All

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
Reread the section, but no Reading Questions for today.

For Friday April 24

Section 14.4 Green's Theorem

To read : All

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

For Wednesday April 29

For Friday May 1