Page 133 - Schaum's Outline of Theory and Problems of Advanced Calculus

124 PARTIAL DERIVATIVES [CHAP. 6

we may, in general, consider u and v as functions of x and y.In this case, we have (see Problem 6.31)
@ðF; GÞ @ðF; GÞ @ðF; GÞ @ðF; GÞ
@u @u @v @v
; ¼ ; ¼ ; ¼
@ðx; vÞ @ðy; vÞ @ðu; xÞ @ðu; yÞ
¼
@x @ðF; GÞ @y @ðF; GÞ @x @ðF; GÞ @y @ðF; GÞ
@ðu; vÞ @ðu; vÞ @ðu; vÞ @ðu; vÞ
The ideas are easily extended. Thus if we consider the simultaneous equations
Fðu; v; w; x; yÞ¼ 0; Gðu; v; w; x; yÞ¼ 0; Hðu; v; w; x; yÞ¼ 0
we may, for example, consider u, v, and w as functions of x and y. In this case,

@ðF; G; HÞ @ðF; G; HÞ
@u @ðx; v; wÞ @w @ðu; v; yÞ
;
@x ¼ @ðF; G; HÞ @y ¼ @ðF; G; HÞ
@ðu; v; wÞ @ðu; v; wÞ
with similar results for the remaining partial derivatives (see Problem 6.33).

THEOREMS ON JACOBIANS
In the following we assume that all functions are continuously differentiable.
1. A necessary and sufficient condition that the equations Fðu; v; x; y; zÞ¼ 0, Gðu; v; x; y; zÞ¼ 0

can be solved for u and v (for example) is that @ðF; GÞ is not identically zero in a region r.
@ðu; vÞ
Similar results are valid for m equations in n variables, where m < n.
2. If x and y are functions of u and v while u and v are functions of r and s, then (see Problem 6.43)
@ðx; yÞ @ðx; yÞ @ðu; vÞ
¼ ð9Þ
@ðr; sÞ @ðu; vÞ @ðr; sÞ
This is an example of a chain rule for Jacobians. These ideas are capable of generalization (see
Problems 6.107 and 6.109, for example).
3. If u ¼ f ðx; yÞ and v ¼ gðx; yÞ, then a necessary and sufficient condition that a functional relation
of the form ðu; vÞ¼ 0 exists between u and v is that @ðu; vÞ be identically zero. Similar results
@ðx; yÞ
hold for n functions of n variables.
Further discussion of Jacobians appears in Chapter 7 where vector interpretations are employed.

TRANSFORMATIONS
The set of equations

x ¼ Fðu; vÞ
ð10Þ
y ¼ Gðu; vÞ
defines, in general, a transformation or mapping which establishes a correspondence between points in the
uv and xy planes. If to each point in the uv plane there corresponds one and only one point in the xy
plane, and conversely, we speak of a one-to-one transformation or mapping. This will be so if F and G
are continuously differentiable with Jacobian not identically zero in a region. In such case (which we
shall assume unless otherwise stated) equations (10) are said to define a continuously differentiable
transformation or mapping.