John Zweck

MATH 390, Spring 2011
Special Topics Course

Vector Calculus with Linear Algebra and Applications

The goal of the course is to improve students' working knowledge of Vector Calculus through

• Formulation of the fundamental concepts and theorems using the language of Linear Algebra;
• Careful exposition of the material from the last third of Math 251 (specifically aiming at Stokes' Theorem and the Divergence Theorem);
• Application of Vector Calculus to an introductory study of the PDEs that govern electromagnetism and/or continuum mechanics.

Vector Calculus is a huge subject that cannot be adequately covered in a single-semester sophomore-level course. Mastery of Vector Calculus is invaluable for students studying related fields such as complex analysis, differential geometry, optimization, numerical linear algebra, electromagnetism, PDEs, computer graphics, continuum mechanics, and image processing.

The course will be geared to undergraduates concentrating in Theoretical and in Applied Mathematics, and Math minors majoring in Physics or Mechanical Engineering.

Cute Fact we will learn: The derivative of a function is a linear transformation.

Prerequisites: Math 221, 251, 301.

Textbook: "Vector Calculus" by Peter Baxandall and Hans Liebeck, Dover Publications, 2008 ($21 on Amazon).

Partial List of Topics:

• Real-valued functions of several variables: level sets and graphs, linear approximation, differentiability, tangent planes, Taylor's Theorem, all via Linear Algebra;
• Vector-valued functions of several variables: vector fields, differentiability, general chain rule, change of variables theorem for double integrals;
• Inverse and Implicit Function Theorems, focusing on geometric and linear algebraic interpretations and examples (The Inverse Function Theorem will not be proved as this is usually covered in Math 302);
• Parametrized surfaces, Surface Integrals, Stokes and Divergence Theorems, Physical Interpretations of Curl and Divergence;
• Maxwell's equations (Electromagnetism), focusing on (1) What a mathematician needs to know about basic physics of electromagnetism; (2) derivation of differential form of Maxwell's equations from integral versions using Stokes and Divergence Theorems, and (3) Mathematical explanation of why light can be regarded as coupled electric and magnetic fields.