# 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.

• 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.

### Lecture notes

Here scans of my lecture notes, in PDF format.

1. Jan 14: Introduction and outline of course.
2. Jan 15: Vectors and the dot product.
3. Jan 16: More on the dot product.
4. Jan 17: Matrices and linear transformations.
5. Jan 22: Linear transformations and matrix multiplication.
6. Jan 23: Properties of matrix multiplication; the determinant.
7. Jan 24: Cross product.
8. Jan 28: Functions of several variables: graphs and level sets.
9. Jan 29: Level sets in 3-dimensions.
10. Jan 30: Limits in several variables.
11. Jan 31: Limits and continuity.
12. Feb 4: Derivatives.
13. Feb 5: More on derivatives.
14. Feb 6: The chain rule.
15. Feb 7: Derivative miscellanae.
16. Feb 11: More on the gradient.
17. Feb 12: Introduction to min/max.
18. Feb 15: Review.
19. Feb 18: Taylor series.
20. Feb 19: Unconstrained min/max.
21. Feb 20: Extreme Value Theorem; Intro to constrained min/max.
22. Feb 21: Lagrange Multipliers; Partial Differential Equations. More on the sinking of the Sleipner A.
23. Feb 25: Linear Programming. Interesting applications: Trucking, Biology.
24. Feb 26: Curves and their lengths.
25. Feb 27: Integration over paths.
26. Feb 28: More on integration over paths.
27. Mar 3: Conservative vector fields I.
28. Mar 4: Conservative vector fields II.
29. Mar 5: Midterm Review.
30. Mar 10: Curl and conservativity; multivariable integration.
31. Mar 11: More on multivariable integration.
32. Mar 12: Change of variables I.
33. Mar 13: Change of variables II.
34. Mar 24: Triple integrals.
35. Mar 25: Change of coordinates in 3-dimensions.
36. Mar 26: Surfaces in R3.
37. Mar 27: Surface area and integration on surfaces.
38. Mar 31: More on integrating over surfaces.
39. Apr 1: Integrating vector fields over surfaces.
40. Apr 2: Green's Theorem. Aside: Planimeters and Green's Theorem.
41. Apr 3: The Divergence Theorem.
42. Apr 7: Gauss's Law.
43. Apr 8: More applications of the divergence theorem.
44. Apr 9: Midterm review.
45. Apr 14: Stokes Theorem I.
46. Apr 15: Stokes Theorem II.
47. Apr 16: Conservative vector fields and Stokes Theorem.
48. Apr 17: Surfacesbounded by knots; Maxwell's equations.
49. Apr 21: Differential Forms I.
50. Apr 22: Differential Forms II.
51. Apr 23: Differential Forms III.
52. Apr 24: Stokes' Theorem for manifolds.
53. Apr 28: Why Stoke's Theorem works.
54. Apr 29: Cohomology to cosmology.
Additional resources on cosmology: Jeff Week's really excellent book and website, especially the "torus games" and "curved spaces".
55. Apr 30: Final Review.