Page 148 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 6] PARTIAL DERIVATIVES 139

2
1 þ y 1 þ x 2

2 2

u x
@ðu; vÞ u y ð1 xyÞ ð1 xyÞ

¼ 0 if xy 6¼ 1:
ðaÞ ¼ ¼

@ðx; yÞ v x v y 1 1

1 þ x 2 1 þ y 2
(b)ByProblem 6.35, since the Jacobian is identically zero in a region, there must be a functional relation-
ship between u and v. This is seen to be tan v ¼ u, i.e., ðu; vÞ¼ u tan v ¼ 0. We can show this
directly by solving for x (say) in one of the equations and then substituting in the other. Thus, for
example, from v ¼ tan 1 x þ tan 1 y we find tan 1 x ¼ v tan 1 y and so
x ¼ tanðv tan 1 yÞ¼ tan v tanðtan 1 yÞ ¼ tan v y
1 þ tan v tanðtan 1 yÞ 1 þ y tan v
Then substituting this in u ¼ðx þ yÞ=ð1 xyÞ and simplifying, we find u ¼ tan v.
2
2
3
2
6.37. (a)If x ¼ u v þ w, y ¼ u v w and z ¼ u þ v, evaluate the Jacobian @ðx; y; zÞ and
(b) explain the significance of the non-vanishing of this Jacobian. @ðu; v; wÞ

x u x v 1
1 1
x w
2
2

@ðx; y; zÞ y w ¼ 2u 2v 2w ¼ 6wu þ 2u þ 6u v þ 2w
¼ y u y v
ðaÞ

@ðu; v; wÞ 2
z u z v z w
3u 1 0
(b) The given equations can be solved simultaneously for u; v; w in terms of x; y; z in a region r if the
Jacobian is not zero in r.
TRANSFORMATIONS, CURVILINEAR COORDINATES
6.38. A region r in the xy plane is bounded by x þ y ¼ 6, x y ¼ 2; and y ¼ 0. (a) Determine the
region r in the uv plane into which r is mapped under the transformation x ¼ u þ v, y ¼ u v.
0
(b) Compute @ðx; yÞ .(c) Compare the result of (b) with the ratio of the areas of r and r .
0
@ðu; vÞ
(a) The region r shown shaded in Fig. 6-9(a)below is a triangle bounded by the lines x þ y ¼ 6, x y ¼ 2,
and y ¼ 0 which for distinguishing purposes are shown dotted, dashed, and heavy respectively.

Fig. 6-9
Under the given transformation the line x þ y ¼ 6istransformed into ðu þ vÞþðu vÞ¼ 6, i.e.,
2u ¼ 6or u ¼ 3, which is a line (shown dotted) in the uv plane of Fig. 6-9(b) above.