Math 31BH  Honors Multivariable Calculus
Welcome to Math 31BH!
Course description:
 This course is the second in the Mathematics Department's Honors
sequence. We will cover multivariable calculus: pointset topology, limits
and continuity, derivatives in several variables, tangent spaces, Taylor
polynomials, Min/max in several variables, Lagrange multipliers,
manifolds.
 The three courses in the Honors sequence, Math 31ABC, cover
essentially the same material as do Math 20F, 20C, and 20E, respectively,
but at a more sophisticated conceptual level. The Honors sequence
emphasizes proofs, so students completing it will be exempt from taking
Math 109 (Mathematical Reasoning).
 The Honors sequence is intended for mathematics majors and
prospective mathematics majors (although others are very welcome) and
gives a much better view of what upperdivision mathematics is like than
does the standard Math 20 sequence.
Instructor: Dragos Oprea, doprea "at" math.youknowwhere.edu,
AP&M 6101.
Lectures: MWF, 2pm2:50pm, AP&M B412.
Office hours:
 Monday 34, Friday 34 in
AP&M
6101.
 I
am
available for questions after lecture or by appointment. Also, feel free
to drop in if you see me in my office.
Teaching Assistant:
 Daniel Vallieres
dvallieres "at" you know where.
 Daniel's Office Hours: Thursday
12  2pm, Office: APM 5132.
Prerequisities:
 A grade of B or better in Math 31AH.
Textbook: J. H. Hubbard and B. B.
Hubbard, Vector
Calculus, Linear Algebra, and Differential Forms: A
Unified Approach, Fourth Edition.
Grading:
 Homework (20%), two midterms (20% each), final
(40%).
Homework:
 There will be weekly homework assignments,
due in section on Tuesdays, or
before then in the drop box on the sixth floor of AP&M. Students are
allowed to discuss the homework assignments among themselves, but are
expected to turn in their own work  copying someone else's is not
acceptable
Important Dates and Holidays:

 Monday, January 3: First lecture
 Monday, January 17: Martin Luther King Holiday
 Friday, January 28: Midterm I
 Saturday, January 29: Drop deadline
 Monday, February 21: Presidents' Day
 Friday, February 25: Midterm II
 Sunday, March 5: Withdrawal date
 Friday, March 11: Last lecture
 Monday, March 14: Final exam (36pm).
Lecture Summaries
 Lecture
1: Course overview. Notions of pointset topology. Open sets. Closed
sets. Interior. Closure. Boundary. Examples.
 Lecture
2: Definition of limits using epsilon and deltas, and some
examples.
 Lecture 3: Continuity of vector valued
functions. Epsilondelta definition and examples.
 Lecture 4: Linear transformations are continuous. In fact,
linear transformations are uniformly continuous. Example of continuous
functions which are not uniformly continuous.
 Lecture 5: Preimages of open/closed subsets under continuous
maps are open/closed; examples. Sequences and convergence. Sequences and
continuity. Inf and sup of a set.
 Lecture 6: Sequentially compact sets. Compact sets are closed
and bounded. Images of compact sets under continuous functions are
compact.
Continuous functions on compact sets assume minimum and maximum.
 Lecture 7: Functions of several variables. Graphs. Cross
sections. Level curves.
 Lecture 8: Partial derivatives. Jacobian matrix. Total
derivative. Differentiable functions are continuous.
 Lecture 9: Pathological functions. Function which admits
partial derivatives, but is not differentiable, not even continuous. For
differentiable functions, the total derivative is the Jacobian. If the
partial derivatives exist and are continuous, the function is
differentiable.
 Lecture 10: Gradient. Directional derivatives.
Gradinet is the direction of
steepest increase. Tangent spaces to level surfaces and graphs.
 Lecture 11: More examples of tangent spaces. Second order
derivatives.
 Lecture 12: Taylor polynomials in several variables. C^k,
C^{\infty} and real analytic functions.
 Lecture 13: Chain rule in several variables and examples.
Matrix chain rule.
 Lecture 14: Derivatives of functions
between abstract vector spaces. Definition and some preliminary
inequalities.
 Lecture 15: More on norm/length of matrices. Derivative of the
function that sends a matrix A to its square A^2.
 Lecture 16: Derivative of the function that sends a matrix A
to its inverse A^{1}.
 Lecture 17: Critical points. Examples. Least squares.
 Lecture 18: Second derivative test. Hessian matrix. Examples.
 Lecture 19: Functions on compact sets. Lagrange multipliers
and examples.
 Lecture 20: More examples of Lagrange multipliers.
Several constraints.
 Lecture 21: Introduction to smooth manifolds and motivation.
Implicit function theorem.
 Lecture 22: Inverse function theorem. Locally invertible
functions. Complex exponential and logarithm.
 Lecture 23: Smooth manifolds and examples via the implicit
function theorem.
 Lecture 24: Tangent spaces to manifolds. Critical points of functions on manifolds.
 Lecture 25: Proof of Lagrange multipliers. Parametrizations of
manifolds. Tangent spaces to parametrized manifolds.
 Lecture 26: Integration. Definition of the integral and first
propreties.
 Lecture 27: Fubini's theorem. Examples of changing order of
integration. What lies ahead; conclusion.
Homework
 Homework 1: PDF due
January 11. Problem 1.5.24 is now extra credit.
 Homework 2: PDF due
January 25.
 Practice problems for Midterm 1 with solutions: PDF
Also solve problems 1(i)(ii), 2, and 3 from Homework 3.
 Midterm 1: PDF
 Homework 3: PDF due
February 2.
 Homework 4: PDF due
February 8. Problems 5 and 6 will now be part of the next homework.
Also, problem 1.3.10(b) is optional.
 Homework 5: PDF due
February 15.
 Homework 6: PDF due
February 24.
 Practice problems for Midterm 2: PDF
 Midterm 2: PDF
 Homework 7: PDF due
March 3. Problem 3.1.11 will be part of the next homework.
 Homework 8: PDF due
March 8. Do not solve problem 3.8.1 since we did not cover the
material needed for it.
 Practice problems for Final: PDF
 Final: PDF