gradient vector

The vector $\nabla f=\langle f_x,f_y\rangle$ of a function's partial derivatives, whose dot product with a unit vector gives the directional derivative in that direction.

The gradient $\nabla f$ is a VECTOR FIELD, assigning the arrow $\langle f_x(x,y),f_y(x,y)\rangle$ to every point $(x,y)$ in the domain, becoming one concrete vector once evaluated at a specific point. The dot-product identity $D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u}$ packages every direction's rate of change into one object: compute $\nabla f$ once at a point, and every subsequent direction is a two-second dot product.

Examples

For $f(x,y)=x^3-2xy^2$ , $\nabla f(2,1)=\langle10,-8\rangle$ , and $D_{\mathbf u}f(2,1)=\langle10,-8\rangle\cdot\mathbf u$ .