Homepage for Math 417: Advanced Calculus

SECTION 002 ONLY!!


Here is the syllabus (for section 2 only!).

HOMEWORK

(I include notes after each problem so you can make sure that the numbering in your edition matches that in mine.)

HW 1 (due Friday, 9/6):

  1. 1-7: norm-preserving vs. inner product-preserving.
  2. 1-10: There is a number M....
  3. 1-12: Dual spaces: Show someting is 1-1, linear, and something is unique.
  4. 1-13: perpendicular (orthogonal)
  5. 1-14: Union of open sets is open....
  6. 1-16: Exterior, interior, boundary of 3 sets.
HW 2 (due Friday, 9/13):
  1. 1-21: PARTS a AND c ONLY!! Distance between a point and a set.
  2. 1-22: U open, subset C is compact, there exists D such that....
    READ (don't do) 1-23, 1-24.
  3. 1-25: Use problem 1-10. (!)
  4. 1-29: Every continuous function on a compact set A takes on a max and a min value.
    READ (don't do) 2-1.
HW 3 (due Friday, 9/27):
  1. 2-7
    READ (don't do, unless you want to) 2-10
  2. 2-16
  3. 2-19
  4. 2-23a
  5. 2-38 (Mean Value Theorem could help for (a).)
HW 4 (due Friday, 10/11):
  1. 3-1 (use Theorem 3-3)
  2. 3-2 (feel free to use Theorem 3-8)
  3. 3-5
  4. 3-12 (use the hint in the book; no need to prove 1-30)
  5. 3-14
  6. 3-36
HW 5 (due Monday, 10/21):
  1. 3-41 EXCEPT part b. Just that one (5-1=4-part) problem! I suggest reading through the full (page-long) problem statement so that you can see the final target of all the parts before you begin.

(Exam 1 given in late October.)

HW 6 (due Monday, 11/11):

  1. Prove that (w1+w2)^w3 = w1^w3 + w2^w3 (where ^ is the wedge product; w1 and w2 are alternating k-tensors; and w3 is an alternating m-tensor).
  2. Prove that v x (aw) x z = a(v x w x z) (where x is the cross product; a is a real number; and v, w, and z are vectors).
  3. 4-1a. Read b, but don't bother doing it.
  4. 4-9ab (not c-e).
HW 7 (due Friday, 11/22):
  1. 4-13b.
    READ 4-14 through 4-16.
  2. 4-17b.
  3. 4-23.
HW 8 (due Friday, 12/13):
  1. 5-5.
  2. 5-6: backwards direction only.
  3. 5-8a: think about thickening the manifold by an arbitrarily small amount.