Page 156 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 6] PARTIAL DERIVATIVES 147

2
2
3
3
6.83. Find (a) dy=dx and (b) d y=dx if x þ y 3xy ¼ 0.
2
2
2
Ans. (a) ðy x Þ=ðy xÞ; ðbÞ 2xy=ðy xÞ 3
2 2
3
2
@u @v v 3xu v þ 4 2xu þ 3y 3
3
2
3
6.84. If xu þ v ¼ y ,2yu xv ¼ 4x, find (a) ; ðbÞ . Ans: ðaÞ ; ðbÞ
2
2
2
2
@x @y 6x uv þ 2y 3x uv þ y
@u @x @v @x
6.85. If u ¼ f ðx; yÞ, v ¼ gðx; yÞ are differentiable, prove that þ ¼ 1. Explain clearly which variables
@x @u @x @v
are considered independent in each partial derivative.
@y @r @y @s
6.86. If f ðx; y; r; sÞ¼ 0, gðx; y; r; sÞ¼ 0, prove that þ ¼ 0, explaining which variables are independent.
@r @x @s @x
What notation could you use to indicate the independent variables considered?
2
2
d y F xx F y 2F xy F x F y þ F yy F x 2
6.87. If Fðx; yÞ¼ 0, show that ¼
dx 2 F y 3
2
2
2
3
2
6.88. Evaluate @ðF; GÞ if Fðu; vÞ¼ 3u uv, Gðu; vÞ¼ 2uv þ v . Ans.24u v þ 16uv 3v 3
@ðu; vÞ
2
2
3
2
6.89. If F ¼ x þ 3y z , G ¼ 2x yz, and H ¼ 2z xy,evaluate @ðF; G; HÞ at ð1; 1; 0Þ. Ans. 10
@ðx; y; zÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffi
p
p
2
2
6.90. If u ¼ sin 1 x þ sin 1 y and v ¼ x 1 y þ y 1 x ,determine whether there is a functional relationship
between u and v, and if so find it.
2
2
2
6.91. If F ¼ xy þ yz þ zx, G ¼ x þ y þ z , and H ¼ x þ y þ z,determine whether there is a functional relation-
2
ship connecting F, G, and H, and if so find it. Ans. H G 2F ¼ 0.
6.92. (a)If x ¼ f ðu; v; wÞ, y ¼ gðu; v; wÞ, and z ¼ hðu; v; wÞ,prove that @ðx; y; zÞ @ðu; v; wÞ ¼ 1provided
@ðu; v; wÞ @ðx; y; wÞ
6¼ 0. (b)Give an interpretation of the result of (a)in terms of transformations.
@ðx; y; zÞ
@ðu; v; wÞ
6.93. If f ðx; y; zÞ¼ 0 and gðx; y; zÞ¼ 0, show that
dx dy dz
¼ ¼
@ð f ; gÞ @ð f ; gÞ @ð f ; gÞ
@ðy; zÞ @ðz; xÞ @ðx; yÞ
giving conditions under which the result is valid.
2
2
@x @ x @ x
2
2
2
6.94. If x þ y ¼ u, y þ z ¼ v, z þ x ¼ w, find (a) ; ðbÞ ; ðcÞ assuming that the equations
@u @u 2 @u @v
define x; y; and z as twice differentiable functions of u, v; and w.
2
2 2
2 2
2
1 16x y 8yz 32x z 16y z 8xz 32x y
Ans: ðaÞ ; ðbÞ ; ðcÞ
1 þ 8xyz 3 3
ð1 þ 8xyzÞ ð1 þ 8xyzÞ
6.95. State and prove a theorem similar to that in Problem 6.35, for the case where u ¼ f ðx; y; zÞ, v ¼ gðx; y; zÞ,
w ¼ hðx; y; zÞ.
TRANSFORMATIONS, CURVILINEAR COORDINATES
6.96. Given the transformation x ¼ 2u þ v, y ¼ u 3v.(a) Sketch the region r of the uv plane into which the
0
region r of the xy plane bounded by x ¼ 0, x ¼ 1, y ¼ 0, y ¼ 1is mapped under the transformation.
(b) Compute @ðx; yÞ . (c) Compare the result of (b)with the ratios of the areas of r and r . 0
@ðu; vÞ
Ans. (b) 7