The gradient vector and level curves
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.
By point (1), the gradient vector at must be perpendicular to any level curve passing through . In the figure, create and a sequence of level curves. Then observe the behavior of the gradient vector at various points; see how this behavior relates to the underlying level curves. By clicking the checkbox to show the gradient field, you can see (scaled) gradient vectors at various points throughout the plane.
Developed for use with Thomas' Calculus, published by Pearson.