Page 158 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 6] PARTIAL DERIVATIVES 149

@ðx; y; zÞ @ðx; y; zÞ @ðu; v; wÞ
¼
@ðr; s; tÞ @ðu; v; wÞ @ðr; s; tÞ
6.110. Given the equations F j ðx 1 ; ... ; x m ; y 1 ; .. . ; y n Þ¼ 0 where j ¼ 1; 2; ... ; n. Prove that under suitable condi-
tions on F j ,
,
@y r @ðF 1 ; F 2 ; ... ; F r ; ... ; F n Þ @ðF 1 ; F 2 ; .. . ; F n Þ
¼
@x s @ð y 1 ; y 2 ; ... ; x s ; ... ; y n Þ @ð y 1 ; y 2 ; .. . ; y n Þ
2
2
2
2 @ F @ F 2 @ F
6.111. (a)If Fðx; yÞ is homogeneous of degree 2, prove that x þ 2xy þ y ¼ 2F:
@x 2 @x @y @y 2
2
(b) Illustrate by using the special case Fðx; yÞ¼ x lnðy=xÞ:
Note that the result can be written in operator form, using D x @=@x and D y @=@y,as
2
ðxD x þ yD y Þ F ¼ 2F.[Hint: Differentiate both sides of equation (1), Problem 6.25, twice with respect
to .]
6.112. Generalize the result of Problem 6.11 as follows. If Fðx 1 ; x 2 ; ... ; x n Þ is homogeneous of degree p,thenfor
any positive integer r,if D xj @=@x j ,
r
Þ F ¼ pðp 1Þ ... ðp r þ 1ÞF
ðx 1 D x 1 þ x 2 D x 2 þ þ x n D x n
3
6.113. (a)Let x and y be determined from u and v according to x þ iy ¼ðu þ ivÞ . Prove that under this
transformation the equation
2
2
2
2
@ @ ¼ 0 is transformed into @ @ ¼ 0
@x 2 þ @y 2 @u 2 þ @v 2
(b)Isthe result in ða) true if x þ iy ¼ Fðu þ ivÞ?Prove your statements.