Page 129 - Schaum's Outline of Theory and Problems of Advanced Calculus

120 PARTIAL DERIVATIVES [CHAP. 6

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@f @f @f @f
or f x ; f x ðx; yÞ; and or f y ; f y ðx; yÞ; , respectively, the latter notations being used when
@x x @y @y
y x
it is needed to emphasize which variables are held constant.
By definition,
@f f ðx þ x; yÞ f ðx; yÞ @f f ðx; y þ yÞ f ðx; yÞ
¼ lim ; ¼ lim
@x x!0 x @y y!0 y ð1Þ
when these limits exist. The derivatives evaluated at the particular point ðx 0 ; y 0 Þ are often indicated by

@f @f
¼ f x ðx 0 ; y 0 Þ and ¼ f y ðx 0 ; y 0 Þ, respectively.

@x @y
ðx 0 ;y 0 Þ ðx 0 ;y 0 Þ
2
3
2
EXAMPLE. If f ðx; yÞ¼ 2x þ 3xy ,then f x ¼ @f =@x ¼ 6x þ 3y 2 and f y ¼ @f =@y ¼ 6xy. Also, f x ð1; 2Þ¼
2
2
6ð1Þ þ 3ð2Þ ¼ 18, f y ð1; 2Þ¼ 6ð1Þð2Þ¼ 12.
If a function f has continuous partial derivatives @f =@x, @f =@y in a region, then f must be continuous
in the region. However, the existence of these partial derivatives alone is not enough to guarantee the
continuity of f (see Problem 6.9).
HIGHER ORDER PARTIAL DERIVATIVES
If f ðx; yÞ has partial derivatives at each point ðx; yÞ in a region, then @f =@x and @f =@y are themselves
functions of x and y, which may also have partial derivatives. These second derivatives are denoted by
2
2
2
2
@ @f @ f @ @ f @ @ f @ @f @ f
@f
@f
¼ f xx ; ¼ ¼ f yy ; ¼ ¼ f yx ; ¼ ¼ f xy ð2Þ
¼
@x @x @x 2 @y @y @y 2 @x @y @x @y @y @x @y @x
If f xy and f yx are continuous, then f xy ¼ f yx and the order of differentiation is immaterial; otherwise they
may not be equal (see Problems 6.13 and 6.41).
3
2
EXAMPLE. If f ðx; yÞ¼ 2x þ 3xy (see preceding example), then f xx ¼ 12x, f yy ¼ 6x, f xy ¼ 6y ¼ f yx .In suchcase
f xx ð1; 2Þ¼ 12, f yy ð1; 2Þ¼ 6, f xy ð1; 2Þ¼ f yx ð1; 2Þ¼ 12.
3
@ f
In a similar manner, higher order derivatives are defined. For example ¼ f yxx is the derivative
2
@x @y
of f taken once with respect to y and twice with respect to x.
DIFFERENTIALS
(The section of differentials in Chapter 4 should be read before beginning this one.)
Let x ¼ dx and y ¼ dy be increments given to x and y, respectively. Then
z ¼ f ðx þ x; y þ yÞ f ðx; yÞ¼ f ð3Þ
is called the increment in z ¼ f ðx; yÞ.If f ðx; yÞ has continuous first partial derivatives in a region, then
@f @f @z @z
dy þ 1 dx þ 2 dy ¼ f
z ¼ x þ y þ 1 x þ 2 y ¼ dx þ ð4Þ
@x @y @x @y
where 1 and 2 approach zero as x and y approach zero (see Problem 6.14). The expression
@z @z @f @f
dy or dy
dz ¼ dx þ df ¼ dx þ ð5Þ
@x @y @x @y
is called the total differential or simply differential of z or f ,or the principal part of z or f . Note that
z 6¼ dz in general. However, if x ¼ dx and y ¼ dy are ‘‘small,’’ then dz is a close approximation of
z (see Problem 6.15). The quantities dx and dy, called differentials of x and y respectively, need not be
small.