# Multivariable Calculus

Math 233H is the honors section of Math 233, the third semester of calculus at UNC. It focuses on Multivariable Calculus. The text for this section will be:

Multivariable Calculus

Here is the syllabus

In outline, here are the contents of the text:
Chapter 1. Basic one variable calculus
Chapter 2. Multidimensional spaces
Chapter 3. Curves in Euclidean space
Chapter 4. Multivariable differential calculus
Chapter 5. Multivariable integral calculus
Chapter 6. Calculus on surfaces
Appendix A. Foundational material on the real numbers
Appendix B. Sequences and series of continuous functions
Appendix C. Supplementary material on linear algebra
Appendix D. Green’s theorem and complex differentiable functions
Appendix E. Polynomials and the fundamental theorem of algebra

Chapter 1 presents a brisk review of the basics in one variable calculus: definitions and elementary properties of the derivative and integral, the fundamental theorem of calculus, and power series. One might skim over this introductory chapter, to refresh one’s memory of this material.

Multidimensional calculus is done on multidimensional spaces, and Chapter 2 introduces tools useful for this study. It discusses n-dimensional Euclidean space and general vector spaces, linear transformations, determinants, and a study of the cross product on 3D space. We proceed to a study of curves in Euclidean space in Chapter 3.

Chapter 4 treats the derivative of a function of several variables, including higher derivatives and multivariable power series, and the inverse and implicit function theorems. Chapter 5 develops the integral of functions of several variables. A key result is the change of variable formula for multiple integrals.

Chapter 6 studies smooth surfaces in Euclidean space, and differential and integral calculus on such surfaces, and establishes a trio of fundamental integral identities known as the formulas of Gauss, Green, and Stokes.

There are also five appendices, dealing with supplementary material.

This course prepares one for our advanced calculus sequence, Math 521–522. The eager reader can peek into the texts for these courses:

Introduction to Analysis in One Variable

Introduction to Analysis in Several Variables

The text for Math 233H has introductory material on linear algebra. A more complete linear algebra text, part of which is sometimes used in Math 577, can be found here:

Linear Algebra

As a response to the health crisis of 2020 and the need for distance instruction, I have compiled a set of worksheets on the material covered in Math 233H in Fall 2020, which can be found here:

Worksheets for Math 233H