Pomona College

Department of Mathematics

 

Mathematics 107. Vector Calculus

 

Fall 2008

 

Course Outline

 

Time and Place:         MWF 10:00 am - 10:50 am    Millikan  218

 

Instructor:                  Dr. Adolfo J. Rumbos

 

Office:                         Andrew 259

 

Phone / e-mail:           ext.  18713  / arumbos@pomona.edu

 

Office Hours:             MWF 9:15 am-9:50 am or by appointment

 

Text:                           Second Year Calculus  by David M. Bressoud

Undergraduate Texts in Mathematics, Springer 2000

 

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 differentiation and integration 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 and developed.

The specific topics to be covered are listed in the attached Tentative Schedule of Lectures and Examinations.

Assigned Readings and Problems.   Readings and problem sets will be assigned at every lecture.  Homework assignments will be collected on an alternate basis.  Students are strongly encouraged to work on every assigned problem.  Late homework assignments will not be graded.

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:    Wednesday, December 17       9:00 am

Place:   Millikan 218

 

 

Math 107                                                                                           Fall  2008

 

Tentative Schedule of Lectures and Examinations

Date                            Topic

 

W        Sep.     3          n-Dimensional Euclidean Space

F          Sep.     5          Spans, lines and planes

 

M         Sep.     8          Dot product and Euclidean norm

W        Sep.   10          Orthogonality and projections

F          Sep.   12          The cross product

 

M         Sep.   15          Functions on Euclidean space

W        Sep.   17          Open subsets of Euclidean space

F          Sep.   19          Continuous functions

 

M         Sep.   22          Continuous functions (continued)

W        Sep.   24          Limits and continuity

F          Sep.   26          Differentiability  

 

M         Sep.   29          The derivative map

W        Oct.     1          The derivative map (continued)

F          Oct.     3          Sufficient conditions for differentiability

 

M         Oct.     6          Sufficient conditions for differentiability (continued)

W        Oct.     8          Derivatives of compositions

F          Oct.   10          Derivatives of compositions (continued)

 

M         Oct.   13          Review

W        Oct.   15          Exam 1

F          Oct.   17          Problems and Examples

 

M         Oct.   20          Fall recess: No Classes

W        Oct.   22          Path integrals

F          Oct.   24          Path integrals (continued)

 

M         Oct.   27          Line integrals      

W        Oct.   29          Gradient fields

F          Oct.   31          Flux across plane curves

 

M         Nov.    3          Differential forms

W        Nov.    5          Calculus of differential forms

F          Nov.    7          Calculus of differential forms (continued)

 

 

Date                            Topic

 

M         Nov.  10          Evaluating 2-forms: Double integrals

W        Nov.  12          Green’s Theorem

F          Nov.  14          Fundamental Theorem of Calculus in two dimensions

 

M         Nov.  17          Change of variables Theorem

W        Nov.  19          Change of variables Theorem (continued)

F          Nov.  21          Triple integrals

 

M         Nov.  24          Surface integrals

W        Nov.  26          Stokes’ Theorem

F          Nov.  28          Thanksgiving recess

 

M         Dec.    1           Problems and examples

W        Dec.    3           Review

F          Dec.    5           Exam 2

 

M         Dec.    8           Review

W        Dec.  10           Review

 

 

W        Dec.  17           Final Examination