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## Functions of 3 variables.

The situation here is very similar to the case of two variables, but we cannot draw the graph of a function.

Definition 6.18   Let f be a function of three real variables, defined on a neighborhhod of thepoint P0 (x0,y0,z)). Suppopse that f has partial derivatives of first order at P0. The vector is called the gradient of f at P0..

Example 6.19   Let f(x,y,z)=x2y+y2z-3xyz. Then

Thus, the gradient of f at P0(1,-1,2) is: .

Definition 6.20   We use the settings of Def  6.18; let be a unit vector. The directional derivative of f at P0 in the direction of is the number:

Example 6.21   Let f and P0 be as in Example  6.19.

Take (please check that ). Then:

Question: Which information do we get from a directional derivative?

Answer: The directional derivative plays the role of the first derivative for a function of a single variable, i.e. gives information about the increase/decrease of a function, according to the point and to the direction.

Proposition 6.22   Let f, P0, and be as above (see Def  6.20). We denote by the angle of the vectors and . Then:
1.
The directional derivative has its greatest value when , i.e. when the direction of and the direction of the gradient are identical. This means that f increases most fastly at a point P0 in the direction of the gradient.
2.
The directional derivative has its least value when , i.e. when the direction of and the direction of the gradient are opposite. This means that f decreases most fastly at a point P0 in the direction opposite to the gradient's direction.
3.
If is orthogonal to the gradient , then the directional derivative is equal to 0.

Example 6.23   Let f(x,y,z)=x2-y2-z2-1.

Proposition 6.24   Let f, P0, and be as in Prop.  6.22. Then we have:
1.
.
2.
.
3.
.
4.
.

Next: Lagrange's multipliers. Up: Directional derivative and Gradient Previous: Functions of 2 variables.
Noah Dana-Picard
2001-05-30