Page 190 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 181

dT dB dN
¼ N; ¼ N; ¼ B T
ds ds ds
These are called the Frenet-Serret formulas and are of fundamental importance in differential geometry.In
these formulas is called the curvature, is called the torsion; and the reciprocals of these, ¼ 1= and
¼ 1= ,are called the radius of curvature and radius of torsion, respectively.

7.96. (a)Prove that the radius of curvature at any point of the plane curve y ¼ f ðxÞ; z ¼ 0 where f ðxÞ is differ-
entiable, is given by

02 3=2
ð1 þ y Þ
¼
y 00

(b)Find the radius of curvature at the point ð =2; 1; 0Þ of the curve y ¼ sin x; z ¼ 0.
ffiffiffi
p
Ans. (b)2 2
7.97. Prove that the acceleration of a particle along a space curve is given respectively in (a)cylindrical,
(b)spherical coordinates by
ð € _ Þe þð € þ 2 _ _ z
2

Þe þ € ze z
2
2
2
2

r

r

r

ð€ r r _ r _ sin Þe r þðr € þ 2_ r _ r _ sin cos Þe þð2_ r _ sin þ 2r _ _ cos þ r € sin Þe
where dots denote time derivatives and e ; e ; e z ; e r ; e ; e are unit vectors in the directions of increasing
; ; z; r; ; , respectively.
7.98. Let E and H be two vectors assumed to have continuous partial derivatives (of second order at least) with
respect to position and time. Suppose further that E and H satisfy the equations
1 @H 1 @E
r E ¼ 0; r H ¼ 0; ;
c @t r H ¼ c @t ð1Þ
r E ¼
prove that E and H satisfy the equation
2
1 @
2
r ¼ 2 2 ð2Þ
c @t
where is a generic meaning, and in particular can represent any component of E or H.
[The vectors E and H are called electric and magnetic field vectors in electromagnetic theory. Equations (1)
are a special case of Maxwell’s equations. The result (2) led Maxwell to the conclusion that light was an
electromagnetic phenomena. The constant c is the velocity of light.]
7.99. Use the relations in Problem 7.98 to show that
@ 2 2
1
f ðE þ H Þg þ cr ðE HÞ¼ 0
@t 2
7.100. Let A 1 ; A 2 ; A 3 be the components of vector A in an xyz rectangular coordinate system with unit vectors
i 1 ; i 2 ; i 3 (the usual i; j; k vectors), and A 1 ; A 2 ; A 3 the components of A in an x y z rectangular coordinate
0
0 0 0
0
0
system which has the same origin as the xyz system but is rotated with respect to it and has the unit vectors
i 1 ; i 2 ; i 3 . Prove that the following relations (often called invariance relations) must hold:
0
0
0
0 0 0 n ¼ 1; 2; 3
A n ¼ l 1n A 1 þ l 2n A 2 þ l 3n A 3
where i m i n ¼ l mn .
0
7.101. If A is the vector of Problem 7.100, prove that the divergence of A, i.e., r A,isaninvariant (often called a
scalar invariant), i.e., prove that
0 0 0
@A 1 @A 2 @A 3 @A 1 @A 2 @A 3
@x 0 þ @y 0 þ @z 0 ¼ @x þ @y þ @z