Time and room
Tuesday and Thursday, 10:50 am  12:05 in Mech Lab 254.
Class attendance is very important  any absences should be discussed beforehand with the instructor.
Instructor
William W. Symes (symes_AT_rice.edu), Duncan Hall 2009,
Syllabus and other materials
Syllabus describes course policies, goals, and intended outcomes.
Grading
Grades will be based on homework and class participation:
 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 R^{n}, 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 R^{n}'s with n >
1. Can you think of a quicker way to prove Thm 23 (5)? For both, you
can use the fact that the Jacobian matrix represents the derivative
(Spivak 27)  when it exists. Note the cool proof of the sum rule
via the chain rule (Cor. 24). 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 C^{1} condition (Lang
XV.2.1, Spivak 27, 28).
Homework (due Thurs. 21 Jan):
 Lang, XV.1.1
 Lang, XV.1.7
 Lang, XV.2.1
 Lang, XV.2.4 or Spivak 234
 Spivak 21
 Spivak 26
 Spivak 29(a)
 Spivak 213 (part d is really simple)
 Spivak 217 (only parts labeled by vowels)
 Spivak 216
 Spivak 219
 Spivak 222 (a good tool is the Mean Value Theorem of onevariable
calculus, Lang Thm III.2.3 p. 71)
 Spivak 224 (why does this not contradict Thm 25 or Lang Thm XV.1.1?)
 Lang XV.2.11
Week 2. Derivatives in R^{n}, part 2
Reading: Lang, XVII 14; Spivak, Ch. 2, first two pages of fifth section
Class Topics:

Come to class prepared to prove (1) an R^{n}valued function is differentiable if and only if its components are differentiable, (2) the Chain Rule (Lang p. 471 or Spivak Thm 22) and (3) the Mean Value Theorem (Lang Theorem 4.2). Examine Spivak Lemma 210. We can do better! f(y)f(x) =
F(1)  F(0) where F(t)=f(ty+(1t)x). Compute F' by the chain
rule. Apply the FTC to each component of F' to get F(1)F(0) =
int_{0}^{1} 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 210 but better!).
Homework (due Thurs. 28 Jan):
 Lang XV.2.11  what does this say about existence of all directional derivatives and differentiability?
 Lang XVII.1.4
 Lang XVII.3.1
 Lang XVII.3.2
 According to Spivak Thm 28, if f:R^{n} → R^{m} has continuous partial derivatives in a neighborhood of a in R^{n}, 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.]
 Lang XVII.4.1
 Spivak 226  will be VERY useful

Define F:C^{0}([0,1]) → R by
F(u) = ∫_{0}^{1} u^{3}(x) dx
Show that F is differentiable at any u ∊ C^{0}([0,1]). Compute its derivative, and demonstrate explicitly that any value of the derivative is a continuous linear map: C^{0}([0,1]) → R.
Week3, Inverse Function Theorem
Reading: Lang, XVIII.13; Spivak, Ch. 2, pp. 3439 (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):
 First Qualifying Exam Dry Run
Week 4, Implicit Function Theorem
Reading: Spivak, pp. 4043 (main reading), Lang, XVIII.45
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):
 Redo Lang XVII.3.1
 Spivak 236
 Spivak 237
 Spivak 238
 Spivak 239
Week 5, Implicit Function Theorem (really) and Multivariate Integration
Reading: Spivak, pp. 4043 and 4649 (main reading), Lang, XVIII.45
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, R^{n} version: partitions, upper and lower sums, integrability, wierd examples.
Homework (due Thurs. 25 Feb):
 Spivak, 240 (a "do" rather than a "redo", for you)
 Spivak, 241
 Spivak, 32
 Spivak, 34
 Spivak, 37
Week 6, Measure/Content Zero, Sets over which you can integrate, Fubini's Theorem
Reading: Spivak, pp. 5061 (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):
 Spivak, 310
 Spivak, 130
 Spivak, 312
 Spivak, 315
 Spivak, 316
 Spivak, 317
 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 321 follows from Spivak 310.
 Spivak, 323
 Spivak, 329
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. 6366  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):
 Second Qualifying Exam Dry Run
 Second Qualifying Exam Solutions
Week 10 Exterior Algebra and Differential Forms
Reading:
Class Topics:
 Introduction to exterior algebra
 the determinant as an nalternating tensor, orientation, volume elements
 vector fields, differential forms
 the exterior derivative, relation to div, curl, grad
Homework (due Thurs. 24 March):
 Spivak 41
 Spivak 42
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 nforms
 Open surfaces and integration of (n1)forms
Homework (due Thurs 7 April):
 Spivak 43
 Spivak 44
 Spivak 419 (b), (c) ((a) was done in class on 24 March). For (c) you will use the Poincare' Lemma (Theorem 411).
 Spivak 420
 Spivak 421
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 (n1)forms on surfaces: consistency of the definition
Homework (due Thurs 14 April):
 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 R^{n} via partition of unity
 Local Stokes Theorem
 Global Stokes via Partition of Unity
Homework (due Thurs 21 April):
 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, wellorganized 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'.
Homework:
 You gotta be kidding, school's out!
Any student with a disability requiring accommodation in this course is encouraged
to contact the instructor during the first week of class, and also to contact a
Disability Support Services in the Ley Student Center.
