Syllabus Math 317
- Textbook: "Multivariable Calculus" by James Stewart, edition 7E.
- Synopsis: The usual
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.
- 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:
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
- 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.