文档介绍：Chapter Eight
f : R n ® R
Introduction
We shall now turn our attention to the very important special case of functions that
are real, or scalar, valued. These are sometimes called scalar fields. In the very, but
important, special subcase in which the dimension of the domain space is 2, we can
actually look at the graph of a function. Specifically, suppose f :Rn ® R . The
3
collection S = {(x1, x2 , x3 ) Î R : f (x1 , x2 ) = x3} is called the graph of f. If f is a
reasonably nice function, then S is what we call a surface. We shall see more of this later.
Let us now return to the general case of a function f : R n ® R . The derivative of f is just
é¶ f ¶ f ¶ f ù
a row vector f '(x) = ê L ú. It is frequently called the gradient of f
ë¶ x1 ¶ x2 ¶ xn û
and denoted grad f or Ñf .
The Directional Derivative
In the applications of scalar fields it is of interest to talk of the rate of change of the
function in a specified direction. Suppose, for instance, the function T(x, y, z) gives the
temperature at points (x, y,z) in space, and we might want to know the rate at which the
temperature changes as we move in a specified direction. Let f :Rn ® R , let a Î Rn ,
and let u Î R n be a vector such that |u|= 1. Then the directional derivative of f at a in
the direction of the vector u is defined to be
d
D f (a) = f (a + tu) .
u dt t =0
Now that we are experts on the Chain Rule, we know at once how pute such a
thing. It is simply
d
D f (a) = f (a + tu) = Ñf ×u .
u dt t =0
Example
The surface of a mountain is the graph of f ( x, y) = 700 - x 2 - 5y2 . In other words, at
the point (x, y), the height is f (x, y). The positive y-axis points North, and, of course,
then the positive x-axis points East. You are on the mountain side above the point (2, 4)
and begin to walk Southeast. What is the slope of the path at the starting point? Are you
going uphill or downhill? (Which!?).
The answers to these questions call for the