# Multivariable calculus

Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.

## Thinking about multivariable functions

The only thing separating multivariable calculus from ordinary calculus is this newfangled word "multivariable". It means we will deal with functions that can have either multiple inputs or multiple outputs, like f(x, y) = (xy, x-3y). Before diving into the many new topics in calculus this seemingly small shift brings about, we take a moment in this tutorial to go through the different ways one can think about and visualize multivariable functions.

## Double and triple integrals

Volume under a surface with double integrals. Triple integrals as well.

## Partial derivatives, gradient, divergence, curl

Thinking about forms of derivatives in multi-dimensions and for vector-valued functions: partial derivatives, gradient, divergence and curl.

## Line integrals and Green's theorem

Line integral of scalar and vector-valued functions. Green's theorem and 2-D divergence theorem.

- Line integrals for scalar functions
- Position vector functions and derivatives
- Line integrals in vector fields
- Green's theorem
- 2D divergence theorem

## Surface integrals and Stokes' theorem

Parameterizing a surface. Surface integrals. Stokes' theorem.

- Parameterizing a surface
- Surface integrals
- Flux in 3D and constructing unit normal vectors to surface
- Stokes' theorem intuition and application
- Proof of Stokes' theorem

## Divergence theorem

Divergence theorem intuition. Divergence theorem examples and proofs. Types of regions in 3D.