Review of the Gradient

1.

Suppose $f\,(x,y)$ is a scalar-valued function of two variables, so f$:\mathbf{R}^{2}\longrightarrow\mathbf{R}$. The gradient of f is the vector of functions: $grad(f)=\nabla f=(f_{x},f_{y})=\left\langle \frac{\partial f}{\partial x},\frac... ...ial x}\right) \mathbf{i}+\left( \frac{\partial f}{\partial y}\right) \mathbf{j}$.

2.

The gradient of f at the point P=(a,b) is the vector:

$$\nabla f\,(a,b)=\left\langle f_{x}(a,b),\,f_{y}(a,b)\right\r... ...c{\partial f}{\partial y}\right\vert _{(a,b)}\right) \mathbf{j}$$

3.

$\nabla f\,(a,b)$ points in the direction, from (a,b), in which the function f increases most rapidly.

4.

$\nabla f\,(a,b)$ is perpendicular to the contour or level curve for f through (a,b).