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EXAMPLE.
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F x; y x4 2xy3 5y4 is homogeneous of degree 4, since F x; y x 4 2 x y 3 5 y 4 4 x4 2xy3 5y4 4 F x; y
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Euler s theorem on homogeneous functions states that if F x1 ; x2 ; . . . ; xn is homogeneous of degree p then (see Problem 6.25) x1 @F @F @F x2 xn pF @x1 @x2 @xn 11
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IMPLICIT FUNCTIONS In general, an equation such as F x; y; z 0 de nes one variable, say z, as a 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 f x; y , which is such that F x; y; f x; y  0. Di erentiation 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 F x; y; f x; z 0, the independent variables are x and @f @f y and that z f x; y . In order to nd and , we initially write (observe that F x; t; z is zero for all @x @y domain pairs x; y , in other words it is a constant): 0 dF Fx dx Fy dy Fz dz and then compute the partial derivatives Fx ; Fy ; Fz 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 di erential form @f @f dx dy. Upon substitution and some algebra (see Problem 6.30) the following results are dz @x @y obtained: @f F x; @x Fz Fy @f @y Fz
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EXAMPLE. If 0 F x; y; z x2 z yz2 2xy2 z3 and z f x; y then Fx 2xz 2y2 , Fy z2 4xy. Fz x2 2yz 3z2 . Then @f 2xz 2y2 2 ; @x x 2yz 3z2 @f z2 4xy 2 @y x 2yz 3x2
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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.
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JACOBIANS If F u; v and G u; v are di erentiable in a region, the Jacobian determinant, or brie y the Jacobian, of F and G with respect to u and v is the second order functional determinant de ned by    @F @F      @ F; G  @u @v   Fu Fv      7    @G @G   Gu Gv  @ u; v     @u @v Similarly, the third order determinant  F @ F; G; H  u  Gu  @ u; v; w  Hu Fv Gv Hv  Fw   Gw   Hw  Extensions are easily made.
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is called the Jacobian of F, G, and H with respect to u, v, and w.
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PARTIAL DERIVATIVES USING JACOBIANS Jacobians often prove useful in obtaining partial derivatives of implicit functions. example, given the simultaneous equations F x; y; u; v 0; G x; y; u; v 0 Thus, for
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) @ F; G @u @ x; v ; @ F; G @x @ u; v @ F; G @u @ y; v ; @ F; G @y @ u; v @ F; G @v @ u; x ; @ F; G @x @ u; v @ F; G @v @ u; y @ F; G @y @ u; v
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