Page 199 - Schaum's Outline of Theory and Problems of Advanced Calculus

190 APPLICATIONS OF PARTIAL DERIVATIVES [CHAP. 8

8.2. In what point does the normal line of Problem 8.1(b) meet the plane x þ 3y 2z ¼ 10?
Substituting the parametric equations of Problem 8.1(b), we have
1 6t þ 3ð2 þ 11tÞ 2ð14t 1Þ¼ 10 or t ¼ 1
Then x ¼ 1 6t ¼ 7; y ¼ 2 þ 11t ¼ 9; z ¼ 14t 1 ¼ 15 and the required point is ð7; 9; 15Þ.

2
3
8.3. Show that the surface x 2yz þ y ¼ 4is perpendicular to any member of the family of surfaces
2
2
2
x þ 1 ¼ð2 4aÞy þ az at the point of intersection ð1; 1; 2Þ:
Let the equations of the two surfaces be written in the form
2
2
3
2
2
F ¼ x 2yz þ y 4 ¼ 0 and G ¼ x þ 1 ð2 4aÞy az ¼ 0
Then
2
rF ¼ 2xi þð3y 2zÞj 2yk; rG ¼ 2xi 2ð2 4aÞyj 2azk
Thus, the normals to the two surfaces at ð1; 1; 2Þ are given by
N 1 ¼ 2i j þ 2k; N 2 ¼ 2i þ 2ð2 4aÞj 4ak
Since N 1 N 2 ¼ð2Þð2Þ 2ð2 4aÞ ð2Þð4aÞ 0, it follows that N 1 and N 2 are perpendicular for all a,
and so the required result follows.
8.4. The equation of a surface is given in spherical coordinates by Fðr; ; Þ¼ 0, where we suppose
that F is continuously differentiable. (a) Find an equation for the tangent plane to the surface at
the point ðr 0 ; 0 ; 0 Þ.(b) Find an equation for the tangent plane to the surface r ¼ 4 cos at the
p ffiffiffi
point ð2 2; =4; 3 =4Þ.(c) Find a set of equations for the normal line to the surface in (b)at the
indicated point.
(a) The gradient of in orthogonal curvilinear coordinates is
1 @ 1 @ 1 @
e 3
r ¼ e 1 þ e 2 þ
h 1 @u 1 h 2 @u 2 h 3 @u 3
1 @r 1 @r 1 @r
where e 1 ¼ ; e 2 ¼ ; e 3 ¼
h 1 @u 1 h 2 @u 2 h 3 @u 3
(see Pages 161, 175).

In spherical coordinates u 1 ¼ r; u 2 ¼ ; u 3 ¼ ; h 1 ¼ 1; h 2 ¼ r; h 3 ¼ r sin and r ¼ xi þ yjþ
zk ¼ r sin cos i þ r sin sin j þ r cos k.
Then
8
e 1 ¼ sin cos i þ sin sin j þ cos k
<
e 2 ¼ cos cos i þ cos sin j sin k ð1Þ
e 3 ¼ sin i þ cos j
:
and
@F 1 @F 1 @F
e 3
@r r @ r sin @
rF ¼ e 1 þ e 2 þ ð2Þ
As on Page 183 the required equation is ðr r 0 Þ rFj P ¼ 0.
Now substituting (1) and (2), we have

@F 1 @F sin 0 @F
i

@r P r 0 @ P r 0 sin 0 @ P
rFj P ¼ sin 0 cos 0 þ cos 0 cos 0

@F 1 @F cos 0 @F
j
@r P r 0 @ P r 0 sin 0 @ P
þ sin 0 sin 0 þ cos 0 sin 0 þ

@F 1 @F
sin 0 k
þ cos 0
@r P r 0 @ P