Department
of Mathematics
Math 67. Vector Calculus
Fall 2019
Course Outline
Time
and Place: MWF
11:00 am - 11:50 am Millikan 2393
Instructor:
Dr. Adolfo J. Rumbos
Office: Andrew 2287
Phone / e-mail: ext. 18713
/
Office Hours:
TuTh 9:00 am - 10:00 am, or by
appointment
Course Notes: http://pages.pomona.edu/~ajr04747/
Prerequisites: Math
60 (Linear Algebra) or equivalent course.
Course
Description. The main goal of this course is the development
of differential and integral calculus ideas, which students learned in a
single-variable calculus courses, in dimensions higher than 1. The main objects of study are functions from n-dimensional Euclidean space to m-dimensional Euclidean space (also
known as Vector Fields) and their
differentiability and integrability properties.
We will also be concerned with the study of subsets of Euclidean space
on which those functions act. The
culmination of the course will be the multivariable version of the Fundamental Theorem of Calculus (also
known as the generalized Stokes’ Theorem). In the process leading to Stokes’ Theorem,
the machinery of differentiable
manifolds and differential forms
will be introduced.
The
specific topics to be covered are listed in the attached Tentative Schedule of Lectures and Examinations.
Assigned
Grading
Policy. Grades will be based on the homework, two 50-minute
examinations, plus a comprehensive final examination. The grades will be computed as follows:
homework
20%
Two 50-minute
exams 50%
final examination 30%
Final
Examination.
Time: Monday,
December 16, 2019 9:00 am
Place: Millikan
2393
Math 67-Rumbos Fall
2019
Tentative Schedule of Lectures and
Examinations
Date Topic
W Sep.
4 n-Dimensional Euclidean Space
F Sep.
6 Spans, lines and
planes
M Sep.
9 Dot
product and Euclidean norm
W Sep. 11 Orthogonality
and projections
F Sep. 13 The cross product
M Sep.
16 Functions on Euclidean space
W Sep. 18 Open
subsets of Euclidean space
F Sep. 20 Continuous functions
M Sep.
23 Continuous functions
(continued)
W Sep. 25 Limits
and continuity
F Sep. 27 Differentiability
M Sep.
30 The derivative map
W Oct.
2 The derivative map (continued)
F Oct.
4 Sufficient conditions for
differentiability
M Oct. 7 Sufficient
conditions for differentiability (continued)
W Oct. 9 Derivatives of compositions
F Oct. 11 Derivatives of
compositions (continued)
M Oct.
14 Review
W Oct. 16 Exam 1
F Oct. 18 Path integrals
M Oct.
21 Fall Recess
W Oct. 23 Path
integrals (continued)
F Oct. 25 Line integrals
M Oct.
28 Gradient
fields
W Oct. 30 Flux across plane curves
F Nov.
1 Differential
forms
M Nov. 4 Calculus of differential forms
W Nov.
6 Calculus
of differential forms (continued)
F Nov.
8 Evaluating
2-forms: Double integrals
M Nov.
11 Green’s Theorem
W Nov. 13 Fundamental
Theorem of Calculus in two dimensions
F Nov. 15 Change of variables Theorem
M Nov.
18 Change of variables Theorem
(continued)
W Nov. 20 Triple
integrals
F Nov. 22 Surface integrals
Math
67-Rumbos Fall
2019
Date Topic
M Nov.
25 Surface integrals (continued)
W Nov. 27 Thanksgiving Recess
F Nov. 29 Thanksgiving Recess
M Dec. 2 Stokes’
Theorem
W Dec.
4 Review
F Dec. 6 Exam
2
M Dec.
9 Review
W Dec. 11 Review
M Dec.
16 Final Examination