Preface

I took this class in the summer of 2024. These notes are just for me to refresh the material and for anyone else looking to review multivariable calculus.

Topics

1. Parametric Equations and Polar Coordinates - Alternative ways to describe curves
2. Vectors - The language of multivariable calculus
3. Vector Functions and Space Curves - Curves in 3D, motion in space
4. Differential Calculus of Several Variables - Partial derivatives, gradients, optimization
5. Integral Calculus of Several Variables - Double and triple integrals
6. Line Integrals - Integrating along curves
7. Surface Integrals - Integrating over surfaces
8. Vector Calculus Theorems - Green’s, Stokes’, and Divergence theorems

Why Multivariable Calculus?

Single-variable calculus deals with functions $f: \mathbb{R} \to \mathbb{R}$. But most real-world quantities depend on multiple variables:

• Temperature depends on position $(x, y, z)$ and time $t$
• Electric fields are vector functions of position
• Fluid flow involves velocity fields in 3D space

Multivariable calculus provides the tools to analyze these situations.