Page 194 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 8] APPLICATIONS OF PARTIAL DERIVATIVES 185

dR
.If r 0 and r

du P
If R ¼ f ðuÞi þ gðuÞj þ hðuÞk,a vector tangent to C at the point P is given by T 0 ¼
denote the vectors drawn respectively from O to Pðx 0 ; y 0 ; z 0 Þ and Qðx; y; zÞ on the tangent line, then since
r r 0 is collinear with T 0 we have

dR
¼ 0
ðr r 0 Þ T 0 ¼ðr r 0 Þ ð5Þ
du
P
In rectangular form this becomes
x x 0 y y 0 z z 0
¼ ¼ ð6Þ
0 0 0
f ðu 0 Þ g ðu 0 Þ h ðu 0 Þ
The parametric form is obtained by setting each ratio equal to u.
If the curve C is given as the intersection of two surfaces with equations Fðx; y; zÞ¼ 0 and
Gðx; y; zÞ¼ 0 observe that rF rG has the direction of the line of intersection of the tangent planes;
therefore, the corresponding equations of the tangent line are
x x 0 y y 0 z z 0
¼ ¼
ð7Þ

F y F z F x
F z F x F y

G y G z P G z G x P G x G y P
Note that the determinants in (7) are Jacobians. A similar result can be found when the surfaces are
given in terms of orthogonal curvilinear coordinates.
4. Normal Plane to a Curve. Suppose we wish to find an equation for the normal plane to curve C
at Pðx 0 ; y 0 ; z 0 Þ of Fig. 8-2 (i.e., the plane perpendicular to the tangent line to C at this point). Letting r be
the vector from O to any point ðx; y; zÞ on this plane, it follows that r r 0 is perpendicular to T 0 . Then
the required equation is

dR
¼ 0

ðr r 0 Þ T 0 ¼ðr r 0 Þ ð8Þ
du P
When the curve has parametric equations x ¼ f ðuÞ; y ¼ gðuÞ; z ¼ hðuÞ this becomes
f ðu 0 Þðx x 0 Þþ g ðu 0 Þð y y 0 Þþ h ðu 0 Þðz z 0 Þ¼ 0 ð9Þ
0
0
0
Furthermore, when the curve is the intersection of the implicitly defined surfaces
Fðx; y; zÞ¼ 0and Gðx; y; zÞ¼ 0
then

F y F z F x
F z F x F y ðz z 0 Þ¼ 0
ðx x 0 Þþ ð y y 0 Þþ ð10Þ

G y G z P G z G x P G x G y P
5. Envelopes. Solutions of differential equations in two variables are geometrically represented by
one-parameter families of curves. Sometimes such a family characterizes a curve called an envelope.
For example, the family of all lines (see Problem 8.9) one unit from the origin may be represented by
2
2
x sin y cos 1 ¼ 0, where is a parameter. The envelope of this family is the circle x þ y ¼ 1.
If ðx; y; zÞ¼ 0isa one-parameter family of curves in the xy plane, there may be a curve E which is
tangent at each point to some member of the family and such that each member of the family is tangent
to E.If E exists, its equation can be found by solving simultaneously the equations
ðx; y; Þ¼ 0; ðx; y; Þ¼ 0 ð11Þ

and E is called the envelope of the family.