Page 182 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 173

3 4 2 3 3 4 2 3
ðdÞ div ð AÞ¼r ð AÞ¼r ðx yz i x y z j þ 2x y z kÞ
@ 3 4 @ 2 3 3 @ 4 2 3
@x @y @z
¼ ðx yz Þþ ð x y z Þþ ð2x y z Þ
4
2
4 2 2
2 2 3
¼ 3x yz 3x y z þ 6x y z
3 4 2 3 3 4 2 3
ðeÞ curl ð AÞ¼r ð AÞ¼ r ðx yz i x y z j þ 2x y z kÞ

i j k

¼ @=@x @=@y

@=@z
3 4 2 3 3
x yz x y z 2x y z
4 2 3
4 3 2 3 2 3 3 3 2 3 3 3 3 4
¼ð4x yz 3x y z Þi þð4x yz 8x y z Þj ð2xy z þ x z Þk
7.35. Prove r ð AÞ¼ ðr Þ A þ ðr AÞ.
r ð AÞ¼r ð A 1 i þ A 2 j þ A 3 kÞ
@ @ @
@x @y @z
¼ ð A 1 Þþ ð A 2 Þþ ð A 3 Þ
@ @ @ @A 1 @A 2 @A 3
A 3 þ
¼ A 1 þ A 2 þ þ þ
@x @y @z @x @y @z
@ @ @

¼ i þ j þ k ðA 1 i þ A 2 j þ A 3 kÞ
@x @y @z
@ @ @

þ i þ j þ k ðA 1 i þ A 2 j þ A 3 kÞ
@x @y @z
¼ðr Þ A þ ðr AÞ
7.36. Express a formula for the tangent plane to the surface ðx; y; zÞ¼ 0at one of its points
P 0 ðx 0 ; y 0 ; z 0 Þ.
Ans: ðr Þ ðr r 0 Þ¼ 0
0
2
2
7.37. Find a unit normal to the surface 2x þ 4yz 5z ¼ 10 at the point Pð3; 1; 2Þ.
By Problem 7.36, a vector normal to the surface is
2
2
rð2x þ 4yz 5z Þ¼ 4xi þ 4zj þð4y 10zÞk ¼ 12i þ 8j 24k at ð3; 1; 2Þ
12i þ 8j 24k 3i þ 2j 6k
Then a unit normal to the surface at P is q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ :
2 2 2 7
ð12Þ þð8Þ þð 24Þ
3i þ 2j 6k
Another unit normal to the surface at P is :
7

2
2
3
7.38. If ¼ 2x y xz , find (a) r and (b) r .
@ @ @ 3 2 2
k ¼ð4xy z Þi þ 2x j 3xz k
@x @y @z
ðaÞ r ¼ i þ j þ
@ @ @
2 3 2 2
ðbÞ r ¼ Laplacian of ¼r r ¼ ð4xy z Þþ ð2x Þþ ð 3xz Þ¼ 4y 6xz
@x @y @z