Functions of Three Variables

Since we do not have four dimensions to graph such a function, we try to see what it likes by looking at its contours: level surfaces.

The level surfaces for the function corresponding to the values . Darker colors are for higher function values.

The ellipsoids are graphed with slices removed so that you can see different layers.

Level surfaces for

Level surfaces for the function corresponding to the values .

The coloring is like a map. Green is for . Darker blues are more negative. Darker browns are more positive values.

Again, the surfaces are graphed with slices removed so that you can see different layers.

Level surfaces for

Partial derivatives and directional derivatives

It is not possible to see these on graphs-again 4th dimension problem- but we can say what they are.

is the rate of change when we move from in the positive -direction keeping and constant. To calculate we treat and like numbers and differentiate with respect to with the usual rules of differentiation. Similary, we define and .

For a unit vector we can calculate the directional derivative by

The gradient of is the vector

It has the following important properties:

• The gradient is a vector in -space.

• The gradient is always perpendicularto the contours (level surfaces).

• The gradient points in the direction of maximum increase for

• It's length is the maximum possible directional derivative at that point.

• As the contours get closer, the size of the gradient increases.

Below are graphs, contour diagrams and gradients of several functions.

Note that:

• The gradient is always perpendicular to the contours (level surfaces).

• As the contours get closer, the size of the gradient increases.

For linear functions, all the contours are parallel and equally spaced. Also, the gradient vector is the same at every point.

Level surfaces for

Level surfaces for