This is a course in the calculus for functions of several variables; but the material will be taught by combining linear algebra with multivariable calculus to emphasize the interconnectedness of these two subjects.

Each topic in the calculus will be re-introduced for functions of a single variable and then extended to functions of several variables: limits, continuity, derivatives, integrals and the Fundamental Theorem of Calculus. Topics in linear algebra will be introduced as needed to understand the calculus for functions of several variables: vectors, matrices, linear independence, basis and dimension. Major goals of the course are to understand the Inverse Function Theorem, the Implicit Function Theorem, and Green’s Theorem, which is a multidimensional analogue of the Fundamental Theorem of Calculus.

There are no specific prerequisites for the course; all the essential topics will be introduced in the course itself, and the course should be accessible to any student with a modest background in mathematics. Our text will be Multivariable Mathematics / Linear Algebra, Multivariable Calculus, and Manifolds by Theodore Shifrin. Grades will be based on regularly assigned problem sets that will be roughly an equal mixture of computational problems and theoretical problems. (Problems will be assigned at one class, we’ll go over them at the next class and the written solutions will be due at the class after that.)

Multivariable Mathematics - Linear Algebra, Multivariable Calculus, and Manifolds

By Theodore Shifrin

Publisher: John Wiley & Sons, Inc.

ISBN:   0-471-52638-X

            9780471526384

CLASS #1:

Vectors in Rⁿ, dot product, subspaces of Rⁿ, linear transformations, matrix algebra.

Text: Chapter 1, sections 1,2,3,4.

 

CLASS #2:

Determinants, cross product, scalar-valued functions, vector-valued functions, limits, continuity.

Text: Chapter 1, section 5.  Chapter 2

 

CLASS #3:

Derivatives, partials derivatives, directional derivative.

Text: Chapter 3, sections 1,2.

 

CLASS #4:

Differentiability, differentiation rules, gradient; Extreme Values in higher dimensions

Text: Chapter 3, sections 3,4 (more if time)

 

CLASS #5:

Some Linear Algebra material from Chapter 4. Max/min problems.

Text: Some of Chapter 4 very quickly; Chapter 5, sections 1,2.

 

CLASS #6:

The Inverse Function Theorem and The Implicit Function Theorem

Text: Some of Chapter 6.

 

CLASS #7:

Integration, multiple integrals, iterated integrals, Fubini’s Theorem (more if time).

Text: Chapter 7: sections 1 and 2 (more if time).

 

CLASS #8:

More on integration, introduction of Green’s Theorem

Text: Parts of Chapter 7 and parts of Chapter 8

 

CLASS #9:

Proof of Green’s Theorem. Discussion of Stokes’s Theorem.

Text: Chapter 8, section 3; some of sections 4 and 5.

 

CLASS #10:

Catching up since we will surely fall behind! Final thoughts to summarize the course.

GRADING AND HOMEWORK:

To help students practice, at each class “homework exercises” will be assigned. These will be routine problems designed to give the students practice with computational problems. The homework exercises do not need to be written up or handed in. Solutions to the homework exercises will be given at the class after which they are assigned.

Grades in this class will be based entirely on Problems Sets. A Problem Set will be assigned at each of the first 9 classes. In the next class, students can ask questions and get help on the Problem Set, and the Problem Set will be due at the class after that (second class after it was assigned).  We’ll make an exception for Problem Set #9 which will be due at Class #10.

Irene Mulvey (B.A., Stonehill College; Ph.D., Wesleyan University) is professor of mathematics at Fairfield University. Her publications include "Symbolic Representation for a Class of Unimodal Cycles," Topology and Its Applications (2002), and "Multi-modal Cycles with Linear Map Having Exactly One Fixed Point," International Journal of Mathematics and Mathematical Sciences (2001). Click here for more information about Irene Mulvey.