Page 203 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 203
194 APPLICATIONS OF PARTIAL DERIVATIVES [CHAP. 8
dF dr dr
is the projection of rF in the direction . This projection is a maximum when rF and
ds ¼rF ds ds
dr=ds have the same direction. Then the maximum value of dF=ds takes place in the direction of rF, and
the magnitude is jrFj.
2
3
8.14. (a) Find the directional derivative of U ¼ 2x y 3y z at Pð1; 2; 1Þ in a direction toward
Qð3; 1; 5Þ. (b)In what direction from P is the directional derivative a maximum?
(c)What is the magnitude of the maximum directional derivative?
3
2
2
rU ¼ 6x yi þð2x 6yzÞj 3y k ¼ 12i þ 14j 12k at P:
ðaÞ
The vector from P to Q ¼ð3 1Þi þð 1 2Þj þ½5 ð 1Þk ¼ 2i 3j þ 6k.
2i 3j þ 6k 2i 3j þ 6k
:
The unit vector from P to Q ¼ T ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼
2 2 2 7
ð2Þ þð 3Þ þð6Þ
Then
2i 3j þ 6k 90
Directional derivative at P ¼ð12i þ 14j 12kÞ ¼
7 7
i.e., U is decreasing in this direction.
(b)From Problem 8.13, the directional derivative is a maximum in the direction 12i þ 14j 12k.
(c) From Problem 8.13, the value of the maximum directional derivative is j12i þ 14j 12kj¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
144 þ 196 þ 144 ¼ 22:
p
DIFFERENTIATION UNDER THE INTEGRAL SIGN
8.15. Prove Leibnitz’s rule for differentiating under the integral sign.
ð
u 2 ð Þ
Let ð Þ¼ f ðx; Þ dx: Then
u 1 ð Þ
ð ð
u 2 ð þ Þ u 2 ð Þ
f ðx; Þ dx
¼ ð þ Þ ð Þ¼ f ðx; þ Þ dx
u 1 ð þ Þ u 1 ð Þ
ð ð ð
u 1 ð Þ u 2 ð Þ u 2 ð þ Þ
f ðx; þ Þ dx
¼ f ðx; þ Þ dx þ f ðx; þ Þ dx þ
u 1 ð þ Þ u 1 ð Þ u 2 ð Þ
ð
u 2 ð Þ
f ðx; Þ dx
u 1 ð Þ
ð ð ð
u 2 ð Þ u 2 ð þ Þ u 1 ð þ Þ
f ðx; þ Þ dx
¼ ½ f ðx; þ Þ f ðx; Þ dx þ f ðx; þ Þ dx
u 1 ð Þ u 2 ð Þ u 1 ð Þ
By the mean value theorems for integrals, we have
ð ð
u 2 ð Þ u 2 ð Þ
½ f ðx; þ Þ f ðx; Þ dx ¼ f ðx; Þ dx ð1Þ
u 1 ð Þ u 1 ð Þ
ð
u 1 ð þ Þ
f ðx; þ Þ dx ¼ f ð 1 ; þ Þ½u 1 ð þ Þ u 1 ð Þ ð2Þ
u 1 ð Þ
ð
u 2 ð þ Þ
f ðx; þ Þ dx ¼ f ð 2 ; þ Þ½u 2 ð þ Þ u 2 ð Þ ð3Þ
u 2 ð Þ
where is between and þ , 1 is between u 1 ð Þ and u 1 ð þ Þ and 2 is between u 2 ð Þ and u 2 ð þ Þ.
