Instructor |
Dragos Oprea
|
Lectures: |
MWF, 12pm-12:50pm, APM B-412. |
Course Assistant |
Gautam Wilkins
|
Course Content |
This course is the third in the Mathematics Department's Honors
sequence.
It covers multivariable integration: Fubini's theorem, change of variables
formula, differential forms, exterior derivative, generalized
Stokes' theorem, conservative vector fields, potentials.
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 31BH. |
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.
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 April 23 and May 23. There will be no makeup
exams. |
Final Exam | The final examination will be held on
Wednesday, June 11, 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, March 31: First lecture
- Wednesday, April 23: Midterm I
- Friday, May 23: Midterm II
- Monday, May 26: Memorial Day. No class
- Friday, June 6: Last Lecture
- Wednesday, June 11 : FINAL EXAM,
11:30-2:30
|
Exams |
- Preparation for Midterm 1:
- Preparation for Midterm 2:
- Preparation for the Final
|
Lecture Summaries | -
Lecture 1: Introduction. Integrating functions over R^n. Dyadic cubes.
Lower and upper Riemann sums. Lower and upper Riemann integrals.
Integrable functions.
- Lecture 2: Examples of integrals
calculated via Riemann sums. Properties of the integral. Geometric
interpretation of the integral of a continuous function. Volume of sets.
Sets of volume zero.
- Lecture 3: Two integrable
functions which differ only on a set of volume zero have the same
integral. Sets of measure zero. Definition and examples. Riemann
integrability theorem.
- Lecture 4: Cavalieri's principle
and volume by slicing. Volume of the ball in R^n.
- Lecture
5: Fubini's theorem and iterated integrals. Separation of variables
for integrals. Switching order of integration. Examples of double
integrals.
- Lecture 6: Setting up the limits of
integration in R^3 and examples.
- Lecture 7: Review of
determinants. Linear change of coordinates and proof of the change of
variables formula. Volume of the ellipsoid. Arbitrary change of
coordinates.
- Lecture 8: Polar coordinates. Cylindrical
coordinates. Spherical coordinates. Examples.
- Lecture 9:
More examples of coordinate changes. What lies ahead: integration over
manifolds.
- Lecture 10: Strategy for integrating over
manifolds. Volume of k-dimensional parallelograms in R^n. Parametrizing
manifolds and issues with the old conventions about parametrizations.
- Lecture 11: Relaxed parametrizations. Graphs. Surfaces of
revolution such as the torus. Setting up the volume of k-dimensional
manifolds.
- Lecture 12: Examples of volumes of manifolds.
Arclength of parametrized curves and the astroid. Surface areas of graphs
and the paraboloid. Surface area of surfaces of revolution and the
torus.
- Lecture 13: Area in spherical and cylindrical
coordinates. k-forms in R^n as multilinear and alternating functions of
k-vectors. The standard k-forms.
- Lecture 14: All forms can be expressed in terms of the
elementary forms. Wedge product and examples. Properties of wedge
product. Forms with non-constant coefficients and their
values on parallelograms.
- Lecture 15: Integration of forms over parametrized
manifolds as the integral of the pullback of the form under the
parametrization. Geometric definition of pullbacks in terms of
parallelograms, and method of calculation.
- Lecture 16: The need of orientations for integration of forms.
Orientations of R^n and
of subspaces of R^n. Examples. Orientation of manifolds.
- Lecture 17: Orienting manifolds via nonvanishing top forms.
Constructing non-vanishing top forms from normal vectors. Orientation of
manifolds defined by global equations. Generalization of the right hand
rule.
- Lecture 18: Summary of orientations. Non orientable manifolds.
Integrating forms over oriented manifolds. Orientation preserving
parametrizations. Examples.
- Lecture 19: Forms in R^3 and physical meaning: vector fields,
work, flux, mass forms. The gravitational field and its work form, path
independence.
- Lecture 20: Flux in cylindrical and
spherical coordinates, examples. Physical meaning of forms in R^n. Faraday
2-form in R^4.
- Lecture 21: Exterior derivative of forms.
Exterior derivative and wedge products. Square of the exterior derivative.
- Lecture 22: Exterior derivative of functions, work forms and
flux forms. Gradient, curl, divergence and identities connecting them. Example: the
gravitational field has zero
curl and divergence. Gravitational potential.
- Lecture 23: Pieces of manifolds and their boundary. Boundary
orientations. Discussion for dimension 1, 2, 3, and the general case.
Orientation of parallelograms and of their boundary.
- Lecture 24: Generalized Stokes. Exact forms integrate
to zero over closed manifolds. Proof of generalized Stokes. Checking the
assumptions in Stokes is important: the flux of the
gravitational field through the sphere.
- Lecture 25: Applications of Stokes' theorem.
(i) Fundamental
theorem of calculus for line integrals. Path independence. Zero-curl
fields and potentials. (ii) Green's theorem
for
work and
flux forms.
- Lecture 26: Applications of Stokes' theorem. (iii) Classical
Stokes' theorem in R^3 and examples. (iv) Divergence theorem and examples.
Flux of the
gravitational field over an arbitrary surface;
isolating the
singularity.
- Lecture 27: Physical interpretation of divergence and curl.
Review.
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