Page 195 - Schaum's Outline of Theory and Problems of Advanced Calculus

186 APPLICATIONS OF PARTIAL DERIVATIVES [CHAP. 8

The result can be extended to determine the envelope of a one-parameter family of surfaces
ðx; y; z; Þ. This envelope can be found from
ðx; y; z; Þ¼ 0; ðx; y; z; Þ¼ 0 ð12Þ
Extensions to two- (or more) parameter families can be made.

DIRECTIONAL DERIVATIVES
Suppose Fðx; y; zÞ is defined at a point ðx; y; zÞ on a given space curve C. Let
Fðx þ x; y þ y; z þ zÞ be the value of the function at a neighboring point on C and let s denote
the length of arc of the curve between those points. Then
F Fðx þ x; y þ y; z þ zÞ Fðx; y; zÞ
lim ¼ lim
s!0 s s!0 s ð13Þ
if it exists, is called the directional derivative of F at the point ðx; y; zÞ along the curve C and is given by
dF @F dx @F dy @F dz
ds ¼ @x ds þ @y ds þ @z ds ð14Þ
In vector form this can be written

dF @F @F @F dx dy dz dr
¼rF T ð15Þ
i þ
k
¼
ds @x @y @z ds ds ds ds
j þ
k ¼rF
j þ
i þ
from which it follows that the directional derivative is given by the component of rF in the direction of
the tangent to C.
In the previous chapter we observed the following fact:
The maximum value of the directional derivative is given by jrFj.
These maxima occur in directions normal to the surfaces Fðx; y; zÞ¼ c (where c is any constant)
which are sometimes called equipotential surfaces or level surfaces.
DIFFERENTIATION UNDER THE INTEGRAL SIGN
ð
u 2
Let ð Þ¼ f ðx; Þ dx a @ @ b ð16Þ
u 1
where u 1 and u 2 may depend on the parameter . Then
d ð u 2 @ f du 2 du 1
d ¼ @ dx þ f ðu 2 ; Þ d f ðu 1 ; Þ d ð17Þ
u 1
for a @ @ b,if f ðx; Þ and @ f =@ are continuous in both x and in some region of the x plane
including u 1 @ x @ u 2 , a @ @ b and if u 1 and u 2 are continuous and have continuous derivatives for
a @ @ b.
In case u 1 and u 2 are constants, the last two terms of (17) are zero.
The result (17), called Leibnitz’s rule,is often useful in evaluating definite integrals (see Problems
8.15, 8.29).

INTEGRATION UNDER THE INTEGRAL SIGN
If ð Þ is defined by (16) and f ðx; Þ is continuous in x and in a region including
u 1 @ x @ u 2 ; a @ x @ b, then if u 1 and u 2 are constants,