Offers an intuition about the gradient vector. Some of the graphics were made more attractive by code borrowed from the Calculus & Mathematica program in the Department of Mathematics at the University of Illinois at Urbana-Champaign.

See also Directional Derivatives.

We start with a surface, in this example and a point .

If we stood at that point, there are infinitely many directions in which we could walk. Each direction has a certain steepness associated with it. Here we want to know which direction is the steepest.

To this end we define the gradient vector of f

This is the vector (shown in blue) whose direction gives the direction in which f is increasing fastest and whose magnitude gives the rate of change in that direction ("slope" of the line tangent to the trace of f in the vertical plane that contains the gradient vector).

To prove that the gradient vector gives the direction and magnitude of the greatest rate of change of f, we first write the directional derivative using this gradient notation:

This can be rewritten as

where is the angle between and .

But = 1 since is a unit vector, and the maximum value of occurs when so . This means that and are pointing in the same direction and the maximum value of is

Q.E.D.

The gradient vector of f at the point (a, b) is perpendicular to the level curve of f through (a,b), and points in the direction of increasing f. The magnitude of the gradient vector is large when the contours are close together and small when they are far apart.

The diagrams below show both the gradient vector for our example above and the vector which we used in the Directional Derivatives file.

This work is part of the Multivariable Calculus with Mathematica Project at the University of Oklahoma.