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:

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

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

The gradient is especially important in electrodynamics, because the fundamental equations of electrodynamics (called Maxwell equations) contain the curl of the electric and magnetic fields.

Layman

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

Student

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

Researcher

Examples

Example1

Example2:

FAQ

–> Why should we view the derivative as a vector?#

See http://mathinsight.org/gradient_vector

History