Page 132 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 6] PARTIAL DERIVATIVES 123

IMPLICIT FUNCTIONS
In general, an equation such as Fðx; y; zÞ¼ 0 defines one variable, say z,asa function of the other
two variables x and y. Then z is sometimes called an implicit function of x and y,as distinguished from a
so-called explicit function f, where z ¼ f ðx; yÞ, which is such that F½x; y; f ðx; yފ 0.
Differentiation of implicit functions requires considerable discipline in interpreting the independent
and dependent character of the variables and in distinguishing the intent of one’s notation. For
example, suppose that in the implicit equation F½x; y; f ðx; zފ ¼ 0, the independent variables are x and
@f @f
y and that z ¼ f ðx; yÞ.In order to find and ,we initially write (observe that Fðx; t; zÞ is zero for all
@x @y
domain pairs ðx; yÞ,in other words it is a constant):
0 ¼ dF ¼ F x dx þ F y dy þ F z dz
and then compute the partial derivatives F x ; F y ; F z as though y; y; z constituted an independent set of
variables. At this stage we invoke the dependence of z on x and y to obtain the differential form
@f @f
dy.Upon substitution and some algebra (see Problem 6.30) the following results are
@x @y
dz ¼ dx þ
obtained:
@f F x @f F y
;
@x ¼ F z @y ¼ F z

2
2
2
2
2
EXAMPLE. If 0 ¼ Fðx; y; zÞ¼ x z þ yz þ 2xy z 3 and z ¼ f ðx; yÞ then F x ¼ 2xz þ 2y , F y ¼ z þ 4xy.
2
2
F z ¼ x þ 2yz 3z . Then
2
2
@f ð2xz þ 2y Þ @f ðz þ 4xyÞ
;
2
2
@x ¼ x þ 2yz 3z 2 @y ¼ x þ 2yz 3x 2
Observe that f need not be known to obtain these results. If that information is available then (at
least theoretically) the partial derivatives may be expressed through the independent variables x and y.
JACOBIANS
If Fðu; vÞ and Gðu; vÞ are differentiable in a region, the Jacobian determinant,or briefly the Jacobian,
of F and G with respect to u and v is the second order functional determinant defined by
@F

@F

@u @v F u
@ðF; GÞ F v

¼ ¼ ð7Þ
@G @G
@ðu; vÞ G u G v

@u @v

Similarly, the third order determinant

F u F v

F w
@ðF; G; HÞ

¼ G u G v G w

@ðu; v; wÞ

H u H v H w
is called the Jacobian of F, G, and H with respect to u, v, and w. Extensions are easily made.
PARTIAL DERIVATIVES USING JACOBIANS
Jacobians often prove useful in obtaining partial derivatives of implicit functions. Thus, for
example, given the simultaneous equations
Fðx; y; u; vÞ¼ 0; Gðx; y; u; vÞ¼ 0