Instructor |
Dragos Oprea
|
Lectures: |
MWF, 12pm-12:50pm, APM B-412. |
Course Assistant |
Christian Woods
|
Course Content |
This course is the second in the Mathematics Department's Honors
sequence.
It covers multivariable differentiation: point-set 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 upper-division mathematics is like than
does the standard Math 20 sequence.
|
Prerequisities: |
A grade of B- or better in Math 31AH. |
Grade Breakdown | The grade is computed as the
following
weighed average:
Homework 20%, Midterm I 20%, Midterm II
20%,
Final Exam 40%. |
Textbook: | J. H. Hubbard and B. B.
Hubbard, Vector
Calculus, Linear Algebra, and Differential Forms: A
Unified Approach, Fourth Edition. |
Readings | Reading the sections of the textbook
corresponding to
the assigned homework exercises is considered part of the homework
assignment. You are responsible for material in the assigned reading
whether or not it is discussed in the lecture. It will be expected that
you read the assigned material in advance of each lecture. |
Homework |
Homework problems will be assigned on the
course
homework
page. There will be 8 problem sets, due certain Fridays at
4:30PM in
the TA's mailbox. The tentative due dates are January 10, 17, 24,
February 7, 14, 21, March 5, 14.
You may work together with your
classmates
on your
homework
and/or ask the TA (or myself) for
help on assigned homework problems. However, the work you turn in must be
your own. No late homework assignments will be accepted.
|
---|
Midterm Exams | There will be two midterm exams given
on January 31 and February 28. There will be no makeup
exams. |
Final Exam | The final examination will be held on
Wednesday, March 19, 11:30-2:30. There is no
make up final
examination. It is your responsability
to ensure that you do not have a schedule conflict during the final
examination; you should not enroll in this class if you cannot
sit for the final examination at its scheduled time. |
Announcements and
Dates |
- Monday, January 6: First lecture
- Monday, January 20: MLK Day. No
class
- Friday, January 31: Midterm I
- Monday, February 17: Presidents' Day. No
class
- Friday, February 28: Midterm II
- Friday, March 14: Last Lecture
- Wednesday, March 19 : FINAL EXAM,
11:30-2:30
|
Exams |
- Preparation for Midterm 1:
- Preparation for Midterm 2:
- Preparation for the Final
|
Lecture Summaries | -
Lecture 1: Course overview. Notions of point-set topology. Open
sets. Closed sets. Interior. Closure. Boundary. Examples.
- Lecture 2: Limits of sequences. epsilon-delta definitions.
Limits of functions. Properties. Examples.
- Lecture 3: Continuity via epsilon-delta's. Properties of
continuous functions. Examples.
- Lecture 4: Linear transformations are continuous. Continuity
and open/closed sets. Open sets of matrices.
- Lecture 5: Uniform continuity and examples. Compact sets.
Sequentially compact sets. Properties and examples.
- Lecture 6: Images of compact sets under continuous maps are
compact. Inf and sup, min and max. Continuous functions over compact sets
achieve min and max.
- Lecture 7: Min and max via derivatives. Mean value theorem
and proof. Applications of the
mean value theorem: monotonic functions via sign of derivatives. Functions
of
several variables.
- Lecture 8: Graphs, level contours, cross sections.
Paraboloids, saddles, cones. Partial derivatives.
- Lecture 9: Partial derivatives and tangent plane to graphs.
Jacobian matrix. Total derivative. Comparison between the total derivative
and the Jacobian. Differentiability versur continuity.
- Lecture 10: Directional derivative. Total derivative and the
directional derivative. Direction of steepest increase.
- Lecture 11: Pathological functions. Function which admits
partial derivatives but not a total derivative. Functions of class C^1.
Mean value theorem in several variables.
- Lecture 12: Rules for calculating derivatives: sums, products,
quotients, composition. Chain rule.
- Lecture 13: Examples of the chain rules in matrix form and
partial
derivative form. Polar coordinates. Gradient is perpendicular to level
sets.
- Lecture 14: Tangent planes to level sets. Second order
derivatives. Functions of class C^k, C^{\infty}. Equality of mixed partial
derivatives.
- Lecture 15: Example of a function for which the order of
differentiation matters. Taylor polynomials and functions of class
C^{omega}.
- Lecture 16: Taylor polynomials in several variables. Examples
of Taylor polynomials. Shortcuts to finding Taylor polynomials.
- Lecture 17: Critical points. Second
derivative test. Saddle points, local min and local max.
- Lecture 18: Min and max of functions on compact sets: example
when the boundary is a rectangle. More general boundaries and Lagrange
multipliers.
- Lecture 19: Examples of Lagrange multipliers. One or
several constraints.
- Lecture 20: Introduction to the implicit function theorem.
Motivation and statement.
- Lecture 21: More on the implicit function theorem. Examples.
The solution set looks like the graph of a function. The calculation of
the derivative of the implicit function.
- Lecture 22: The inverse function theorem. Locally invertible
functions. Examples. Complex exponential and logarithm.
- Lecture 23: Smooth manifolds. Examples from the implicit
function theorem. Parametrization of manifolds. Graphs are naturally
parametrized.
- Lecture 24: Reparametrization. Standard tangent planes to
manifolds. Examples: tangent space of manifolds given by equations,
tangent space of manifolds given by parametrization.
- Lecture 25: More examples of tangent spaces. Critical points
of functions over manifolds. Critical points over parametrized manifolds.
- Lecture 26: Critical points over manifolds given by equations.
Proof of Lagrange multipliers with several constraints.
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