Page 155 - Schaum's Outline of Theory and Problems of Advanced Calculus

146 PARTIAL DERIVATIVES [CHAP. 6

2
2
2
2
2
2
@w @w @w 2 @ w 2 @ w 2 @ w @ w @ w @ w
x þ y þ z ¼ 0; ðbÞ x þ y þ z þ 2xy þ 2xz þ 2yz ¼ 0:
ðaÞ 2 2 2
@x @y @z @x @y @z @x @y @x @z @y @z
Indicate possible exceptional points.
DIFFERENTIALS
3
2
6.71. If z ¼ x xy þ 3y ,compute (a) z and (b) dz where x ¼ 5, y ¼ 4, x ¼ 0:2, y ¼ 0:1. Explain why
z and dz are approximately equal. (c)Find z and dz if x ¼ 5, y ¼ 4, x ¼ 2, y ¼ 1.
Ans.(a) 11:658; ðbÞ 12:3; ðcÞ z ¼ 66; dz ¼ 123
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5 2 3
6.72. Computer ð3:8Þ þ 2ð2:1Þ approximately, using differentials.
Ans.2.01
3
3
2
2
2 3
6.73. Find dF and dG if (a) Fðx; yÞ¼ x y 4xy þ 8y ; ðbÞ Gðx; y; zÞ¼ 8xy z 3x yz, (c) Fðx; yÞ¼ xy 2
lnðy=xÞ.
2
2
3
2
ð3x y 4y Þ dx þðx 8xy þ 24y Þ dy
Ans. ðaÞ
2
3
2 2
2 3
2
ð8y z 6xyzÞ dx þð16xyz 3x zÞ dy þð24xy z 3x yÞ dz
ðbÞ
2
2
f y lnðy=xÞ y g dx þf2xy lnðy=xÞþ xyg dy
ðcÞ
2
6.74. Prove that (a) dðUVÞ¼ UdV þ VdU; ðbÞ dðU=VÞ¼ðVdU UdVÞ=V ; ðcÞ dðln UÞ¼ ðdUÞ=U,
2
(d) dðtan 1 VÞ¼ ðdVÞ=ð1 þ v Þ where U and V are differentiable functions of two or more variables.
6.75. Determine whether each of the following are exact differentials of a function and if so, find the function.
2
2
ð2xy þ 3y cos 3xÞ dx þð2x y þ sin 3xÞ dy
ðaÞ
2
y
2
ð6xy y Þ dx þð2xe x Þ dy
ðbÞ
2
2
3
ðz 3yÞ dx þð12y 3xÞ dy þ 3xz dz
ðcÞ
2 2
2
3
Ans. ðaÞ x y þ y sin 3x þ c; ðbÞ not exact, ðcÞ xz þ 4y 3xy þ c
DIFFERENTIATION OF COMPOSITE FUNCTIONS
t
2
2
2
6.76. (a)If Uðx; y; zÞ¼ 2x yz þ xz , x ¼ 2 sin t, y ¼ t t þ 1, z ¼ 3e , find dU=dt at t ¼ 0. (b)if
2
3
3
2
Hðx; yÞ¼ sinð3x yÞ, x þ 2y ¼ 2t , x y ¼ t þ 3t, find dH=dt.
!
2
2
2
2
36t y þ 12t þ 9x 6t þ 6x t þ 18
Ans: ðaÞ 24; ðbÞ cosð3x yÞ
2
6x y þ 2
2
2
6.77. If Fðx; yÞ¼ ð2x þ yÞ=ðy 2xÞ, x ¼ 2u 3v, y ¼ u þ 2v, find (a) @F=@u; ðbÞ @F=@v; ðcÞ @ F=@u ,
2
2
2
(d) @ F=@v ; ðeÞ @ F=@u @v, where u ¼ 2, v ¼ 1.
Ans.(a)7, (b) 14; ðcÞ 21; ðdÞ 112; ðeÞ 49
2
6.78. If U ¼ x Fðy=xÞ,show that under suitable restrictions on F, xð@U=@xÞþ yð@U=@yÞ¼ 2U.
6.79. If x ¼ u cos v sin and y ¼ u sin þ v cos , where is a constant, show that
2 2 2 2
ð@V=@xÞ þð@V=@yÞ ¼ð@V=@uÞ þð@V=@vÞ
6.80. Show that if x ¼ cos , y ¼ sin ,the equations
@u @v @u @v @u 1 @v @v 1 @u
; becomes ;
@x ¼ @y @y ¼ @x @ ¼ @ @ ¼ @
6.81. Use Problem 6.80 to show that under the transformation x ¼ cos , y ¼ sin ,the equation
2
2
2
2
@ u @ u @ u 1 @u 1 @ u
¼ 0 becomes ¼ 0
2
@x 2 þ @y 2 @ 2 þ @ þ @ 2
IMPLICIT FUNCTIONS AND JACOBIANS
6.82. If Fðx; yÞ¼ 0, prove that dy=dx ¼ F x =F y .