Page 193 - Schaum's Outline of Theory and Problems of Advanced Calculus

184 APPLICATIONS OF PARTIAL DERIVATIVES [CHAP. 8

In rectangular form this is

@F @F @F
ðz z 0 Þ¼ 0
ðx x 0 Þþ ðy y 0 Þþ ð2Þ
@x P @y P @z P
In case the equation of the surface is given in orthogonal curvilinear coordinates in the form
Fðu 1 ; u 2 ; u 3 Þ¼ 0, the equation of the tangent plane can be obtained using the result on Page 162 for
the gradient in these coordinates. See Problem 8.4.
2. Normal Line to a Surface. Suppose we require equations for the normal line to the surface S at
Pðx 0 ; y 0 ; z 0 Þ i.e., the line perpendicular to the tangent plane of the surface at P.If we now let r be the
vector drawn from O in Fig. 8-1 to any point ðx; y; zÞ on the normal N 0 ,we see that r r 0 is collinear
with N 0 and so the required condition is
ðr r 0 Þ N 0 ¼ðr r 0 Þ rFj P ¼ 0 ð3Þ

By expressing the cross product in the determinant form

i j k

x x 0 y y 0

z z 0
F x j P F y j P F z j P

we find that
x x 0 y y 0 z z 0

¼ ¼ ð4Þ
@F @F @F

@x P @y P @z P
Setting each of these ratios equal to a parameter (such as t or u)and solving for x, y; and z yields the
parametric equations of the normal line.
The equations for the normal line can also be written when the equation of the surface is expressed
in orthogonal curvilinear coordinates. (See Problem 8.1(b).)
3. Tangent Line to a Curve. Let the parametric equations of curve C of Fig. 8-2 be
x ¼ f ðuÞ; y ¼ gðuÞ; z ¼ hðuÞ; where we shall suppose, unless otherwise indicated, that f , g; and h are
continuously differentiable. We wish to find equations for the tangent line to C at the point Pðx 0 ; y 0 ; z 0 Þ
where u ¼ u 0 .

Fig. 8-2