Multivariable Calculus
A brief introduction to 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 onedimension, using functions to describe space, and space to describe functions.
Content
 Thinking about multivariable functions
 Derivatives of multivariable functions
 Applications of multivariable derivatives
 Integrating multivariable functions
 Green's, Stokes', and the divergence theorems
1. 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.

2.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.

3.Applications of multivariable derivatives
The tools of partial derivatives, the gradient, etc. can be used to optimize and approximate multivariable functions. These are very useful in practice, and to a large extent this is why people study multivariable calculus.

4.Integrating multivariable functions
There are many ways to extend the idea of integration to multiple dimensions: Line integrals, double integrals, triple integrals, surface integrals, etc. Each one lets you add infinitely many infinitely small values, where those values might come from points on a curve, points in an area, points on a surface, etc. These are all very powerful tools, relevant to almost all realworld applications of calculus. In particular, they are an invaluable tool in physics.

5.Green's, Stokes', and the divergence theorems
Here we cover four different ways to extend the fundamental theorem of calculus to multiple dimensions. Green's theorem and the 2D divergence theorem do this for two dimensions, then we crank it up to three dimensions with Stokes' theorem and the (3D) divergence theorem.
