Intuitive

The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector (a direction to move) that

• Points in the direction of greatest increase of a function (intuition on why)
• Is zero at a local maximum or local minimum (because there is no single direction of increase)

The term "gradient" is typically used for functions with several inputs and a single output (a scalar field). Yes, you can say a line has a gradient (its slope), but using "gradient" for single-variable functions is unnecessarily confusing. Keep it simple. Vector Calculus: Understanding the Gradient by Kalid Azad

Concrete

• Vector Calculus: Understanding the Gradient by Kalid Azad
• Understanding Pythagorean Distance and the Gradient by Kalid Azad

Abstract

Why is it interesting?

The gradient, denoted $\nabla$ (spoken "Nabla"), is a tool that enables us to calculate how much a given function changes in different directions. A "normal" function $f(x)$, lives in a boring one-dimensional space, and the derivative is just another function: $\partial_x f(x)$. This function is the rate of change of $f(x)$. Our real world is three-dimensional and hence in physics we often encounter functions that depend on all spatial directions: $f(x,y,z)$. The gradient of such a function:

$$\nabla f(x,y,z) = \begin{pmatrix} \partial_x f(x,y,z) \\ \partial_y f(x,y,z)\\ \partial_z f(x,y,z) \end{pmatrix}$$

is a vector and tells us how much $f(x,y,z)$ changes in each direction.

FAQ

Why should we view the derivative as a vector?

See http://mathinsight.org/gradient_vector