Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 6] Derivative in a Given Direction 195
tial derivatives of the given function:
dz , , dz
The derivative of the given function in the direction / is connected with
the gradient of the function by the following formula:
~
That is, the derivative in a given direction is equal to the projection of the
gradient of the function on the direction of differentiation.
The gradient of a function at each point is directed along the normal to
the corresponding level line of the function. The direction of the gradient of
the function at a given point is the direction of the maximum rate of increase
of the function at this point, thlft is, when /=grad z the derivative -^
on its greatest value, equal to
(3)
takes
In similar fashion we define the gradient of a function of three variables,
u -=/(*, y, z):
du .
,
The gradient of a function of three variables at each point is directed along
the normal to the level surface passing through this point.
Example 2. Find and construct the gradient of the function z~x*y at
the point P(I, 1).
X-
2
du .
Fig. 68
,
du .
J X
Solution. Compute the partial derivatives and their values at the point P.
dz s
y'*
^T" \ .
Hence, grad z = 2t+J (Fig. 68).
7*
_ i 9.
dxjp- 2 -