Mon. 10/2 (1.2/1.4) Dot Product, Cross Product, Scalar Triple Product
   pg. 15: #2 (a)-(g), #5, #12 for (a)-(d) decide if both sides of the equation are numbers or vectors.
   (Optional: Compute both sides of 12(a)-(d) to show that the identity is true. Notice that you can use (a) to prove the fact from class that |v x w| = |v| |w| sin theta)

Wed. 10/4 (1.3) Lines and Planes in Space
   pg. 15: #6, #7

Fri. 10/6 (2.1)-(2.4) Functions of Several Variables / Limits
   pg. 82: #2 (a),(b) (Optional:(c),(d)); #3 (a), (b); #4 (a),(c); #6

Mon. 10/9 (2.4)-(2.5) Limits / Partial Derivatives
   A problem on limits: pdf or ps
   pg. 89 #1 (a),(b),(d),(e),(f)

Wed. 10/11 (2.5)-(2.6) Partial Derivatives / Total Differential
   pg. 89 #4

Fri. 10/13 (2.6) Total Differential
   pg. 89 #6

Mon. 10/16 (2.7) Jacobian Matrix
   pg. 95 #1, #2 (a),(b),(c)

Wed. 10/18 (2.8) Chain Rules
   pg. 100 #1, #2, #5, #6, #10
   Two chain rule problems: pdf or ps

Fri. 10/20 (2.10) Implicit Functions
   pg. 89 #1 (c), (g), (h)
   pg. 95 #3 (a), (c)
   pg. 116 #1 (c), (d) (in these problems, z is an implicit function of x and y; find both the partial of z with respect to x and the partial of z with respect to y)
   pg. 116 #3 (here, u and v are implicit functions of x and y given by two equations; find the partial of u with respect to x and the partial of u with respect to y)

Mon. 10/23 (2.12) Curvilinear Coordinates / Inverse Functions
   pg. 121 #2, #3

Wed. 10/25 (2.12), (2.15) Curvilinear Coordinates / Higher Derivatives
   pg. 142 #1(a),(b),(c) (Optional: #1(d)), #2
   Problems on polar/cylindrical/spherical coordinates: pdf or ps

Fri. 10/27 (2.16), (2.17), (2.18) Higher Derivatives
   pg. 143 #3, #10, #11
   pg. 116 In #1(a), z is an implicit function of x and y. Find the second derivatives of z (z_xx, z_xy and z_yy)
   Find the Laplacian of the function f(r,theta) = r^2 e^{3 theta} + r
   Optional: Here's a problem involving a function whose mixed partials aren't always equal!

Mon. 10/30 (2.14) Directional Derivatives
   pg. 134 #1(a),(b),(c)
   For f(x,y)= x^2 + xy, find the unit vector v that makes the directional derivative of f in the direction v as large as possible at the point (2,1).

Wed. 11/1 (2.13) Tangent lines and planes
   pg. 127 #1(a),(b), #2, #8 (a),(b),(c), #10    (Optional: #11 (a),(b),(c) and #12 --hint: first parameterize the curves!)

Fri. 11/3 (2.19) Maximum and Minimum Problems
   pg. 158 #4(a),(c),(d),(e),(f),(h); #5(a); #8(b) (Optional: #8(c))
   Let f(x,y) = sin(x) + cos(2y). Find the global minimum and global maximum values of f on the square [0, pi] x [0, pi] (ie, the square is the set of all (x,y) points where 0 <= x <= pi and 0 <= y <= pi). Hint: Check the values of the function at all of the critical points inside the square and check for the maximum and minimum values on each of the four lines that make up the boundary of the square.

Mon. 11/13 (3.1-3.5) Vector Fields / Gradient / Divergence / Curl
   pg.180 #1(b),(d); #2(a); #3(a); #4
   Related to #7: Let f(x,y,z) = F(u) = sin(u)e^(u) where u= x^2 y + z^3. Compute grad f using the chain rule. Show this equals (dF/du)*(grad u) (Optional: answer #7)
   pg. 185 For #11, determine if the right- and left-hand sides of each equation are scalar or vector fields!
   (Optional: pg. 186 #12)

Wed. 11/15 (3.5-3.6) Curl / Combined Operations
   pg. 185 #4, #5, #6, #15 (Hint: At any point, the unit normal n to a sphere is equal to r/|r|. The partial derivative with respect to n is just the directional derivative in the direction n.)
   Optional: #7, #12

Fri. 11/17: (4.1), (4.3) Review of Integration / Double Integrals
   pg. 219 #1, (a), (b), (c) - hint: show that 1/(x-1)(x-2) = -1/(x-1) + 1/(x-2)
   pg. 219 #2 (a), (c), (d) - hint: integrate by parts!; #5 (a), (b)
   pg. 234 #1(c), #2(a),(b)

Mon. 11/20: (4.3) Double Integrals
   pg. 234 #2(c),(d); #3; #5

Wed. 11/22: (4.3)-(4.6) Double Integrals / Change of Variables
   pg. 234 #1(b), #4(a), #10
   pg. 241 #4(a), #4(b) [Hint: Draw the region in the x-y plane! Theta will go from 0 to pi/4, and the limits of integration for r will have to depend on theta.], #11, #12

Mon. 11/27: (4.6), (4.9) Change of Variables/ Leibnitz' Rule
   pg. 241 #4(c), #4(d), #5, #6, #7
   pg. 256 #1(a),(b),(c); #2
   (Optional: pg. 256, #6, #7)

Wed. 11/29: (5.1),(5.2) Line Integrals
   Consider the curve parameterized by x=2t^2, y=t^3+1 for 0 <= t <= 1. Find the length of the curve.
   pg. 248 #1 (Hint: for (b), what values of the parameter do you need to go all the way around the circle?)
   pg. 278 #1, #2

Fri. 12/1: (5.2) Line Integrals
   pg. 278 #3

Mon. 12/4: (5.3)-(5.5) Green's Theorem
In class, we showed that for v = x^2 i + x y^2 j, the line integral integral of v_T with respect to arclength counterclockwise around the triangle with vertices (0,0), (1,0), and (1,1) is integral(v_T ds) = integral(x^2 dx + x y^2 dy) = 1/12. (We actually integrated along two different curves; put these results together to get the answer for integrating the whole way around the triangle!) Use Green's theroem to verify this result by doing a double integral over the triangle.
   pg 286 #1; #7; #5(a),(b),(c),(d),(h); for #5(e), compute the integral of [2xy dx + (x^2+y^2) dy] along C; for #5(f), compute the integral of [-2y dx + 2(x-2) dy] along C.

Wed. 12/6: (5.6) Independence of Path
pg. 300 #1; #2; #3(a),(b),(c),(e); #6