The gradient vector with a surface
Suppose is a real-valued function of two variables. The gradient of is the vector-valued function denoted and defined by
.
If is a point in the domain of , then the gradient vector of at the point is the vector
.
The gradient vector has two important properties:
1) It points in the direction of greatest increase of , and
2) Its magnitude is the rate of change of in that direction.
In the accompanying interactive figure, move the red point to set a point in the domain of . The gradient vector is drawn and you can confirm the two properties above. Note that while the surface is 3-dimensional, the gradient vector lies in the -plane.