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The gradient is a fancy word for derivative, or the rate of change of a function. Itâs a vector (a direction to move) that. 1  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)
Why view the derivative as a vector? Viewing the derivative as the gradient vector is useful in a number of contexts. The geometric view of the derivative as a vector with a length and direction helps in understading the properties of the directional derivative. In another context, we can think of the gradient as a function $\nabla f: \R^n \to \R^n$, which can be viewed as a special type of vector field. When one takes line integrals of this vector field, one discovers the line integral is path-independent, the central idea of the gradient theorem for line integrals.