Learn multivariable calculus—derivatives and integrals of multivariable functions, application problems, and more.

An overview of multivariable functions, with a sneak preview of what applying calculus to such functions looks like

An introduction to multivariable functions, and a welcome to the multivariable calculus content as a whole.

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 …

The only thing separating multivariable calculus from ordinary calculus is this newfangled word "multivariable". It means we'll deal with functions whose inputs or outputs live in two or more …

Each one lets you add infinitely many infinitely small values, where those values might come from points on a curve, points in an area, or points on a surface. These are all very powerful tools, relevant to …

The partial derivative symbol (∂) is used in multivariable calculus to indicate that you are taking a derivative with respect to one variable while keeping other variables constant. So, "dx/∂x" doesn't …

As with single variable functions, it is not enough for the gradient to be zero to ensure that a point is a local maximum or minimum. For one thing, you can still have something similar to an inflection point: …

The tools of partial derivatives, like the gradient and other concepts, can be used to optimize and approximate multivariable functions. These are very useful in the real world—this is one of the main …

These physical interpretations can become particularly helpful as we venture into the topics of multivariable calculus, in which one runs the risk of learning concepts and operations symbolically …