Page 157 - Schaum's Outline of Theory and Problems of Advanced Calculus

148 PARTIAL DERIVATIVES [CHAP. 6

6.97. (a)Prove that under a linear transformation x ¼ a 1 u þ a 2 v, y ¼ b 1 u þ b 2 v (a 1 b 2 a 2 b 1 6¼ 0Þ lines and circles
in the xy plane are mapped respectively into lines and circles in the uv plane. (b) Compute the Jacobian J of
the transformation and discuss the significance of J ¼ 0.

6.98. Given x ¼ cos u cosh v, y ¼ sin u sinh v.(a) Show that in general the coordinate curves u ¼ a and v ¼ b in

@ðx; yÞ
the uv plane are mapped into hyperbolas and ellipses, respectively, in the xy plane. (b) Compute .

@ðu; vÞ
@ðu; vÞ
(c) Compute .

@ðx; yÞ
2
2
2
2
2
2
2
2
Ans.(b) sin u cosh v þ cos u sinh v; ðcÞðsin u cosh v þ cos u sinh vÞ 1
6.99. Given the transformation x ¼ 2u þ 3v w, y ¼ 2v þ w, z ¼ 2u 2v þ w. (a) Sketch the region r of the
0
uvw space into which the region r of the xyz space bounded by x ¼ 0; x ¼ 8; y ¼ 0; y ¼ 4; z ¼ 0; z ¼ 6is
mapped. (b) Compute @ðx; y; zÞ .(c) Compare the result of (b)with the ratios of the volumes of r and r . 0
@ðu; v; wÞ
Ans.(b)1
6.100. Given the spherical coordinate transformation x ¼ r sin cos , y ¼ r sin sin , z ¼ r cos , where r A 0,
0 @ @ ,0 @ < 2 . Describe the coordinate surfaces (a) r ¼ a; ðbÞ ¼ b, and (c) ¼ c,
where a; b; c are any constants. Ans. (a)spheres, (b)cones, (c)planes
¼ r sin .
@ðx; y; zÞ 2
6.101. (a)Verify that for the spherical coordinate transformation of Problem 6.100, J ¼
(b)Discuss the case where J ¼ 0. @ðr; ; Þ
MISCELLANEOUS PROBLEMS

@P @T @P @P @T @V
6.102. If FðP; V; TÞ¼ 0, prove that (a) ; ¼ 1.
@T @V ¼ @V ðbÞ @T @V @P
V P T V P T
These results are useful in thermodynamics, where P; V; T correspond to pressure, volume, and temperature
of a physical system.
6.103. Show that Fðx=y; z=yÞ¼ 0satisfies xð@z=@xÞþ yð@z=@yÞ¼ z.
2
2
6.104. Show that Fðx þ y z; x þ y Þ¼ 0satisfies xð@z=@yÞ yð@z=@xÞ¼ x y.
@v 1 @y
6.105. If x ¼ f ðu; vÞ and y ¼ gðu; vÞ,prove that ¼ where J ¼ @ðx; yÞ .
@x J @u @ðu; vÞ
6.106. If x ¼ f ðu; vÞ, y ¼ gðu; vÞ, z ¼ hðu; vÞ and Fðx; y; zÞ¼ 0, prove that
dz ¼ 0
@ðy; zÞ @ðz; xÞ @ðx; yÞ
dx þ dy þ
@ðu; vÞ @ðu; vÞ @ðu; vÞ
6.107. If x ¼ ðu; v; wÞ, y ¼ ðu; v; wÞ and u ¼ f ðr; sÞ, v ¼ gðr; sÞ, w ¼ hðr; sÞ,prove that
@ðx; yÞ @ðx; yÞ @ðu; vÞ @ðx; yÞ @ðv; wÞ @ðx; yÞ @ðw; uÞ
¼ þ þ
@ðr; sÞ @ðu; vÞ @ðr; sÞ @ðv; wÞ @ðr; sÞ @ðw; uÞ @ðr; sÞ

a b e f ae þ bg af þ bh
6.108. (a)Prove that ,thus establishing the rule for the product of two

c d gh ¼ ce þ dg cf þ dh
second order determinants referred to in Problem 6.43. (b)Generalize the result of ðaÞ to determinants
of 3; 4 .. . .
6.109. If x; y; and z are functions of u; v; and w, while u; v; and w are functions of r; s; and t,prove that