# Multivariable calculus

In multivariable calculus, we progress from working with numbers on a line to points in space. It gives us the tools to break free from the constraints of one-dimension, using functions to describe space, and space to describe functions.

## 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 whose inputs or outputs live in two or more dimensions. Here we lay the foundations for thinking about and visualizing multivariable functions.

## Derivatives of multivariable functions

What does it mean to take the derivative of a function whose input lives in multiple dimensions? What about when its output is a vector? Here we go over many different ways to extend the idea of a derivative to higher dimensions, including partial derivatives, directional derivatives, the gradient, vector derivatives, divergence, curl, etc.

- Partial derivatives
- Gradient and directional derivatives
- Differentiating vector-valued functions
- Approximating multivariable functions
- Optimizing multivariable functions
- Constrained optimization

## Line integrals and Green's theorem

Learn how to extend the idea of integration to functions with multiple inputs. This includes line integrals, both in scalar and vector fields, as well as double integrals. These lead to a two-dimensional version of the fundamental theorem of calculus: Green's theorem.

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

## Surface integrals

Learn how to integrate over a two-dimensional surface in three-dimensional space. These higher dimensional integrals lead to some powerful extensions of the fundamental theorem of calculus, known as Stokes' theorem and the divergence theorem.