Syllabus Math 317

• Textbook: "Multivariable Calculus" by James Stewart, edition 7E.
• Synopsis: The usual fundamental theorem of calculus (which we all should be very familiar with by now) is a relationship between the integral of df /dx over a region (in this case the region is an interval) and the integral of f over the boundary of the region (which in this case is two endpoints and this amounts to evaluating f at these two points).This theorem has analogues in every dimension. In dimension 3 these go by the name of the divergence and Stokes’ theorem. They are quite fundamental and have important applications. For example, basic results in electrodynamics are essentially applications of these theorems. The aim of this course is to get to the point of understanding the statements and contents of these theorems and how to apply them.

•  Topics:

•  Vector valued functions of one variable (Chapter 13): Parameterized curves, velocity, acceleration, arc length (includes curvature, normal and binormal vectors, tangential and normal components of acceleration).
•  Vector valued functions of several variables (Chapter 16): Vector fields, line integrals, conservative fields, fundamental theorem of line integrals, Green's theorem, gradient, curl, divergence, parameterized surfaces, suface area, surface integrals, Stoke's theorem, divergence theorem.

• Rough course schedule, subject to later adjustments:

• Week 1 Jan. 3-7: Outline 13.1 Vector functions and space curves. 13.2 Derivatives and tangent vectors. Differentiation rules, integrals.

• Week 2 Jan. 10-14:  13.3 Arclength and curvature. Normal and Binormal vectors.

• Week 3 Jan. 17-21: 13.3 Normal and osculating plane, osculating circle. 13.4 Velocity and Acceleration.

• Week 4 Jan. 24-28: 16.1 Vector fields. 16.2 Center of mass, line integrals with respect of x and y. Line integrals in space and of vector fields.

• Week 5 Feb. 1-5: 16.3 When a line integral is independant of path. Equivalent statements of conservative vector fields. Necessary and sufficient conditions in terms of partial derivatives.

• Week 6 Feb. 8-12: Feb. 8, Family Day  Midterm 1 on Wednesday Feb 10 in class  End of 16.3

• Feb. 15-19: Winter Break

• Week 7 Feb. 22-26: 16.4 Green's theorem: statement and examples. 16.5 algebraic definition of curl F.

• Week 8 Feb. 29-Mar. 4: 16.5 Algebraic definition of div F, geometric meanings, vector forms of Green's theorem. 16.6 parametric surfaces, tangent planes, surface area.

• Week 9 Mar. 7-11: 16.6 Surface area, area of graphs and surfaces of revolution. 16.7  Surface integrals, flux integral of a vector field through a surface.

• Week 10 Mar. 14-18: 16.7 Examples, 16.8 Stokes theorem

• Week 11 Mar. 21-25: 16.7 16.8 More examples Midterm 2 on Wednesday March 23rd in class Good Friday

• Week 12 Mar. 28-Apr. 1: Easter Monday 16.8 More examples

• Week 13 Apr. 4-8:  16.9 Divergence theorem and examples.

• Final Exam : some time between April 12 and April 27.