Math 241, Calculus III, Honors section
Spring 2008
Course Description:
The focus of this course is vector calculus, which concerns functions
of several variables and functions whose values are vectors rather
than just numbers. In this broader context, we will revisit notions
like continuity, derivatives, and integrals, as well as their
applications (like finding minimal and maxima). We'll explore new
geometric objects such as vector fields, curves, and surfaces in
3-space and study how these relate to differentiation and
integration. The highlight of the course will be theorems of Green,
Stokes, and Gauss, which relate seemingly disparate types of integrals
in surprising ways.
Grading:
Your course grade will be based on:
- Weekly Homework (15%). Homework will be
assigned during each lecture and due at the beginning of class
each Tuesday. No late homework will be accepted; however, your lowest
homework grade will be dropped so you are effectively allowed one
infinitely late assignment. Collaboration on homework is permitted,
nay encouraged. However, you must write up your solutions individually
and understand them completely. You may use a computer or calculator
on the HW for experimentation and to check your answers, but may not
refer to it directly in the solution, e.g. "by Mathematica" is not an
acceptable justification for deriving one equation from
another. (Also, computers and calculators will not be allowed on the
exams, so it's best not to get too dependent on them.)
- Three in class exams (20% each). These will be closed-book,
calculator-free exams, though you will be allowed to bring one piece
of paper with handwritten formulas. They will be on Thursdays, in
particular on February 14, March 6, and April 10.
- A final exam (25%) Our exam is scheduled for Monday, May 5
from 1:30-4:30.
Textbook
The required text for this course is:
- M. Lovric, Vector Calculus, Wiley, 2007
A change of perspective is sometimes helpful to clear up confusion.
Here are two other vector calculus sources you might find helpful.
- H. M. Schey, Div, Grad, Curl, and All That, W. W. Norton. A classic informal account of vector calculus from a physics point of view.
- Adams, Thompson, and Hass, How to ace the rest of calculus, the
streetwise guide, Freeman.
Homework Assignments
- HW #1 (PDF): Due Wednesday,
January 23.
- HW #2 (PDF): Due Tuesday, January 29.
- HW #3 (PDF): Due Tuesday, February 5.
- HW #4: Due Tuesday, February 12.
- Section 2.4: #13, 23, 29, 38, 44, 62, 64.
- Section 2.5: #2, 3, 10, 39.
- Section 2.6: #21, 25, 26.
- Section 2.7: #1, 2, 6, 28, 40.
- Exam 1 review problems.
- HW #5 (PDF): Due Tuesday, February 26.
- HW #6 (PDF): Due Tuesday, March 4.
- Exam 2 review problems.
- HW #7 (PDF): Due Thursday, March 13.
- HW #8: Due Tuesday, April 1.
- Section 6.4: #7, 13, 18, 29, 30.
- Section 6.5: #3, 5, 11, 24, 29, 31.
- Section 7.1: #3, 9, 12, 13, 22.
- Section 7.3: #5.
- HW #9 (PDF): Due Thursday, March 13.
- Exam 3 review problems.
- HW #10 (PDF): Due Tuesday, April 22.
- HW #11 (PDF): Due Tuesday, April 29.
- FINAL REVIEW PROBLEMS (PDF): Not to be turned in.
Lecture notes
Here scans of my lecture notes, in PDF format.
- Jan 14: Introduction and outline of
course.
- Jan 15: Vectors and the dot
product.
- Jan 16: More on the dot product.
- Jan 17: Matrices and linear
transformations.
- Jan 22: Linear transformations and
matrix multiplication.
- Jan 23: Properties of matrix
multiplication; the determinant.
- Jan 24: Cross product.
- Jan 28: Functions of several variables:
graphs and level sets.
- Jan 29: Level sets in 3-dimensions.
- Jan 30: Limits in several
variables.
- Jan 31: Limits and continuity.
- Feb 4: Derivatives.
- Feb 5: More on derivatives.
- Feb 6: The chain rule.
- Feb 7: Derivative miscellanae.
- Feb 11: More on the gradient.
- Feb 12: Introduction to min/max.
- Feb 15:
Review.
- Feb 18: Taylor series.
- Feb 19: Unconstrained min/max.
- Feb 20: Extreme Value Theorem; Intro to
constrained min/max.
- Feb 21: Lagrange Multipliers; Partial
Differential Equations. More on the
sinking of the Sleipner A.
- Feb 25: Linear Programming. Interesting
applications: Trucking,
Biology.
- Feb 26: Curves and their lengths.
- Feb 27: Integration over paths.
- Feb 28: More on integration over paths.
- Mar 3: Conservative vector fields I.
- Mar 4: Conservative vector fields II.
- Mar 5: Midterm Review.
- Mar 10: Curl and conservativity; multivariable integration.
- Mar 11: More on multivariable integration.
- Mar
12: Change of variables I.
- Mar 13:
Change of variables II.
- Mar 24: Triple integrals.
- Mar 25: Change of coordinates in 3-dimensions.
- Mar 26: Surfaces in R^{3}.
- Mar 27: Surface area and integration on surfaces.
- Mar 31: More on integrating over
surfaces.
- Apr 1: Integrating vector
fields over surfaces.
- Apr 2: Green's Theorem. Aside: Planimeters and Green's Theorem.
- Apr 3: The Divergence Theorem.
- Apr 7: Gauss's Law.
- Apr 8: More applications of the divergence theorem.
- Apr 9: Midterm review.
- Apr 14: Stokes Theorem I.
- Apr 15: Stokes Theorem II.
- Apr 16: Conservative vector fields and
Stokes Theorem.
- Apr 17: Surfacesbounded by knots; Maxwell's equations.
- Apr 21: Differential Forms I.
- Apr 22: Differential Forms II.
- Apr 23: Differential Forms III.
- Apr 24: Stokes' Theorem for manifolds.
- Apr 28: Why Stoke's Theorem works.
- Apr 29: Cohomology to cosmology.
Additional resources on cosmology: Jeff Week's really excellent book
and website, especially
the "torus games" and "curved spaces".
- Apr 30: Final Review.