Vector fields let you visualize a function with a two-dimensional input and a two-dimensional output. You end up with, well, a field of vectors sitting at various points in two-dimensional space.

Vector fields represent fluid flow (among many other things). They also offer a way to visualize functions whose input space and output space have the same dimension.

Explore the concepts of gradient and contour maps in multivariable calculus with this educational video tutorial.

Interpret a vector field as representing a fluid flow. The divergence is an operator, which takes in the vector-valued function defining this vector field, and outputs a scalar-valued function …

See an example of how you can start to understand how the formula for a three-dimensional vector field relates to the way it looks.

One more feature of conservative vector fields Background: Conservative vector fields If F (x, y) is a conservative vector field, all line integrals over closed loops are 0 . Looking at the integral …

Because this, this is a scale of valued quantity and yet the vector field that I'm showing, with all these blue vectors indicating rotation, these are vectors. And because the rotation is …

The gradient stores all the partial derivative information of a multivariable function. But it's more than a mere storage device, it has several wonderful interpretations and many, many uses.

Not vector fields in general, but gradient fields, and then again, only scalar-valued functions have gradient fields and the gradient usually doesn't directly give the slope (see the videos on …

A neat way to interpret a vector field is to imagine that it represents some kind of fluid flow.