Page 131 - Schaum's Outline of Theory and Problems of Advanced Calculus

122 PARTIAL DERIVATIVES [CHAP. 6

3. The expression Pðx; yÞdx þ Qðx; yÞdy or briefly Pdx þ Qdy is the differential of f ðx; yÞ if and
@P @Q
only if ¼ .In such case Pdx þ Qdy is called an exact differential.
@y @x
2
2
@P @Q @ f @ f
Note: Observe that ¼ implies that ¼ .
@y @x @y @x @x @y
4. The expression Pðx; y; zÞ dx þ Qðx; y; zÞ dy þ Rðx; y; zÞ dz or briefly Pdx þ Qdy þ Rdz is the
@P @Q @Q @R @R @P
differential of f ðx; y; zÞ if and only if ¼ ; ¼ ; ¼ . In such case
@y @x @z @y @x @z
Pdx þ Qdy þ Rdz is called an exact differential.
Proofs of Theorems 3 and 4 are best supplied by methods of later chapters (see Chapter 10,
Problems 10.13 and 10.30).

DIFFERENTIATION OF COMPOSITE FUNCTIONS

Let z ¼ f ðx; yÞ where x ¼ gðr; sÞ, y ¼ hðr; sÞ so that z is a function of r and s. Then
@z @z @x @z @y @z @z @x @z @y
;
@r ¼ @x @r þ @y @r @s ¼ @x @s þ @y @s ð7Þ
In general, if u ¼ Fðx 1 ; ... ; x n Þ where x 1 ¼ f 1 ðr 1 ; ... ; r p Þ; ... ; x n ¼ f n ðr 1 ; ... ; r p Þ, then
@u @u @x 1 @u @x 2 @u @x n
k ¼ 1; 2; ... ; p
¼ þ þ þ ð8Þ
@r k @x 1 @r k @x 2 @r k @x n @r k
If in particular x 1 ; x 2 ; ... ; x n depend on only one variable s, then
du @u dx 1 @u dx 2 @u dx n
ds ¼ @x 1 ds þ @x 2 ds þ þ @x n ds ð9Þ
These results, often called chain rules, are useful in transforming derivatives from one set of variables
to another.
Higher derivatives are obtained by repeated application of the chain rules.

EULER’S THEOREM ON HOMOGENEOUS FUNCTIONS
A function represented by Fðx 1 ; x 2 ; ... ; x n Þ is called homogeneous of degree p if, for all values of the
parameter and some constant p,wehave the identity
p
Fð x 1 ; x 2 ; ... ; x n Þ¼ Fðx 1 ; x 2 ; ... ; x n Þ ð10Þ

4
3
4
EXAMPLE. Fðx; yÞ¼ x þ 2xy 5y is homogeneous of degree 4, since
4 3 4 4 4 3 4 4
Fð x; yÞ¼ ð xÞ þ 2ð xÞð yÞ 5ð yÞ ¼ ðx þ 2xy 5y Þ¼ Fðx; yÞ
Euler’s theorem on homogeneous functions states that if Fðx 1 ; x 2 ; ... ; x n Þ is homogeneous of degree
p then (see Problem 6.25)
@F @F @F
x 1 þ x 2 þ þ x n ¼ pF ð11Þ
@x 1 @x 2 @x n