CAAM 502 Analysis II

Spring 2016

Time and room

Instructor

Syllabus and other materials

Syllabus describes course policies, goals, and intended outcomes.

Grading

• 40% on unpledged assignments (roughly 8);
• 45% on three pledged assignments/midterms, 15% each;
• 15% on class participation, as assessed by the instructor.

Itinerary

Week 1. Derivatives in Rⁿ, part 1

Reading: Lang, XV.1 and XV.2; Spivak, Ch. 2, first four sections

Key Ideas:

• derivative is a linear map
• derivative of a linear map is itself
• Jacobian matrix represents the derivative
• derivative implies partial derivatives, but not (ncessarily) other way round
• calculus works (sum rule, chain rule, some versions of Leibnitz' rule)
• first order necessary condition for an extremum
• partial derivatives commute

Class Topics:

• Lang XV.1, 2, Spivak Ch. 2 first section. Are the definitions of "differentiable" given on Lang p. 380 and Spivak p. 16 the "same"? Logically equivalent? Explain the remark on the bottom of Lang p. 381: Lang makes several assertions, prove them all.
• Think up an example of a function whose derivative you can compute both directly and by using the chain rule, and show that you get the same result - its domain and range must both be Rⁿ's with n > 1. Can you think of a quicker way to prove Thm 2-3 (5)? For both, you can use the fact that the Jacobian matrix represents the derivative (Spivak 2-7) - when it exists. Note the cool proof of the sum rule via the chain rule (Cor. 2-4). Prove the product and quotient rules the same way. What is the simplest example you can think of showing that partial derivatives may exist even when the derivative does not? Be prepared to discuss the proofs of the uniqueness of the derivative and the C¹ condition (Lang XV.2.1, Spivak 2-7, 2-8).

Homework (due Thurs. 21 Jan):

1. Lang, XV.1.1
2. Lang, XV.1.7
3. Lang, XV.2.1
4. Lang, XV.2.4 or Spivak 2-34
5. Spivak 2-1
6. Spivak 2-6
7. Spivak 2-9(a)
8. Spivak 2-13 (part d is really simple)
9. Spivak 2-17 (only parts labeled by vowels)
10. Spivak 2-16
11. Spivak 2-19
12. Spivak 2-22 (a good tool is the Mean Value Theorem of one-variable calculus, Lang Thm III.2.3 p. 71)
13. Spivak 2-24 (why does this not contradict Thm 2-5 or Lang Thm XV.1.1?)
14. Lang XV.2.11

Week 2. Derivatives in Rⁿ, part 2

Reading: Lang, XVII 1-4; Spivak, Ch. 2, first two pages of fifth section

Class Topics:

• Come to class prepared to prove (1) an Rⁿ-valued function is differentiable if and only if its components are differentiable, (2) the Chain Rule (Lang p. 471 or Spivak Thm 2-2) and (3) the Mean Value Theorem (Lang Theorem 4.2). Examine Spivak Lemma 2-10. We can do better! f(y)-f(x) = F(1) - F(0) where F(t)=f(ty+(1-t)x). Compute F' by the chain rule. Apply the FTC to each component of F' to get F(1)-F(0) = int₀¹ F'. Apply the integral version of the triangle inequality, which is good for any matrix norm. What do you get...the Mean Value Theorem, integral form (like 2-10 but better!).

Homework (due Thurs. 28 Jan):

1. Lang XV.2.11 - what does this say about existence of all directional derivatives and differentiability?
2. Lang XVII.1.4
3. Lang XVII.3.1
4. Lang XVII.3.2
5. According to Spivak Thm 2-8, if f:Rⁿ → R^m has continuous partial derivatives in a neighborhood of a in Rⁿ, then f is differentiable at a. Produce an example showing that the converse of this statement is false. [Hint: n=m=1 suffices.] [Remark: if you replace ``differentiable at a'' with ``continuously differentiable in a neighborhood of a'' then the converse becomes true.]
6. Lang XVII.4.1
7. Spivak 2-26 - will be VERY useful
8. Define F:C⁰([0,1]) → R by

   F(u) = ∫₀¹ u³(x) dx

   Show that F is differentiable at any u ∊ C⁰([0,1]). Compute its derivative, and demonstrate explicitly that any value of the derivative is a continuous linear map: C⁰([0,1]) → R.

Week3, Inverse Function Theorem

Reading: Lang, XVIII.1-3; Spivak, Ch. 2, pp. 34-39 (as background)

Class Topics:

• Be prepared to present/discuss the proofs of the Shrinking Lemma (Lang, XVIII.1) and the linear case of the IFT (Lang, XVIII.2) on Tuesday, 26 Jan. The role of the "shrinking" hypothesis in the Shrinking Lemma (K<1) is clear, but what about the condition that the closed subset M be mapped to itself? Is that essential? Can you think of a counterexample to the statement of the Lemma with this condition removed? The series in Theorem XVIII.2.1 is widely used under the name "Neumann series". NB: A common synonym for "Shrinking Lemma" is "Contraction Mapping Theorem/Principle".
• Be prepared to present the proof of the IFT itself (Lang, XVIII.3) on Thursday, 28 Jan. Can you spot the essential difference between the proof in Spivak and the Shrinking Lemma proof given in Lang? What is the difference in scope? Read and work through Lemma XVIII.3.3 carefully - it is actually useful, and we will need it later.

Homework (due Thurs. 4 Feb):

1. First Qualifying Exam Dry Run

Week 4, Implicit Function Theorem

Reading: Spivak, pp. 40-43 (main reading), Lang, XVIII.4-5

Note: No class on Tuesday 2 Feb

Class Topics:

• Logical path from Inverse Function Theorem to Implicit Function Theorem - how the latter constructs parametrized families of solutions to systems of equations. Concept of coordinate chart on a level set.

Homework (due Thurs. 11 Feb):

1. Re-do Lang XVII.3.1
2. Spivak 2-36
3. Spivak 2-37
4. Spivak 2-38
5. Spivak 2-39

Week 5, Implicit Function Theorem (really) and Multivariate Integration

Reading: Spivak, pp. 40-43 and 46-49 (main reading), Lang, XVIII.4-5

Class Topics:

• Logical path from Inverse Function Theorem to Implicit Function Theorem - how the latter constructs parametrized families of solutions to systems of equations. Concept of coordinate chart on a level set.
• Lightning review of the Riemann integral, Rⁿ version: partitions, upper and lower sums, integrability, wierd examples.

Homework (due Thurs. 25 Feb):

1. Spivak, 2-40 (a "do" rather than a "re-do", for you)
2. Spivak, 2-41
3. Spivak, 3-2
4. Spivak, 3-4
5. Spivak, 3-7

Week 6, Measure/Content Zero, Sets over which you can integrate, Fubini's Theorem

Reading: Spivak, pp. 50-61 (main reading)

Class Topics:

• Review of measure and content, Jordan measureability
• Lebesgue's Theorem: what it means, examples (proof: your responsibility!)
• Fubini's Theorem: proof for the continuous case

Homework (due Tues. 8 March):

1. Spivak, 3-10
2. Spivak, 1-30
3. Spivak, 3-12
4. Spivak, 3-15
5. Spivak, 3-16
6. Spivak, 3-17
7. Show that for any finite collection {R_i} of subrectangles of a rectangle A, there is a partition P of A so that for any S in P, either (1) the interior of S is a subset of the interior of R_i for some i, or (2) the interior of S does not intersect the interior of R_i for any i. Then show how the conclusion of Spivak 3-21 follows from Spivak 3-10.
8. Spivak, 3-23
9. Spivak, 3-29

Week 7, Change of Variables Theorem: linear case

Reading: Lang, XX.4

Class Topics:

• Linear algebra preliminaries: Gaussian elimination in terms of elementary row/column ops
• Volume of a block
• Change of variables, version 1

Homework: last week's was enough!

Week 8, Change of Variables Theorem: linear case

Reading: Lang, XX.4

Class Topics:

• Linear algebra preliminaries: Gaussian elimination in terms of elementary row/column ops
• Volume of a block
• Change of variables, version 1

Week 9, Change of Variables Theorem: nonlinear case

Reading: Lang, XX.4; Spivak, pp. 63-66 - BE PREPARED TO PRESENT ALL RESULTS IN THIS SECTION!!!

Class Topics:

• end game of linear case
• nonlinear case by linear approximation
• Partitions of Unity

Homework (due Thurs. 17 March):

1. Second Qualifying Exam Dry Run
2. Second Qualifying Exam Solutions

Week 10 Exterior Algebra and Differential Forms

Reading:

Class Topics:

• Introduction to exterior algebra
• the determinant as an n-alternating tensor, orientation, volume elements
• vector fields, differential forms
• the exterior derivative, relation to div, curl, grad

1. Spivak 4-1
2. Spivak 4-2

Week 11, Vector Fields and Differential Forms

Reading: Spivak, Ch. 4, first two sections

Class Topics (March 29 only - Thursday is Spring Recess):

• Poincare' Lemma
• Pullback of forms by a map
• Integration of n-forms
• Open surfaces and integration of (n-1)-forms

1. Spivak 4-3
2. Spivak 4-4
3. Spivak 4-19 (b), (c) ((a) was done in class on 24 March). For (c) you will use the Poincare' Lemma (Theorem 4-11).
4. Spivak 4-20
5. Spivak 4-21

Week 12, Integration on surfaces

Reading: Spivak, Ch. 4, second section, Ch. 5, first section; Notes on surface integration

Class Topics (7 April only - instructor absent on 5 April):

• Integration of (n-1)-forms on surfaces: consistency of the definition

1. Third Qualifying Exam Dry Run

Week 13, Stokes' Theorem I

Reading: Notes on surface integration

Class Topics (12, 14 April):

• Integration on orientable surfaces in Rⁿ via partition of unity
• Local Stokes Theorem
• Global Stokes via Partition of Unity

1. Several assertions in the Notes on surface integration are highlighted in blue - most of them have been topics of class discussion already. Please write them up in clear, well-organized proofs, and turn them in as the last homework of the semester.

Week 14, Stokes' Theorem II

Reading: Spivak, Ch. 5

Class Topics (19, 21 April):

• The classical theorems: Divergence, Green's, Stokes'.

1. You gotta be kidding, school's out!