The implicitfunction theorem also comes in a more general and powerful version that deals with the conditions under which a set of simultaneous equations
The generalized version of the theorem states that;
Given the equation system (8.24), if (a) the functions Fl, Fn ail have continuous partial derivatives with respect to all they and* variables, and if (b) at a point {v\q> .. m v„o; xuu____satisfying (8.24). the following Jacobian determinant is nonzero:
will assuredly define a set of implicit functions^
f To view ir another way, what these conditions serve to do is to assure us that the n equations in (8.24) can in principie be solved for the n variables—yi,..., yn—even if we may not be able to obtain the solution (8.25) in an explicit form.
then there exists an wdimensional neighborhood uf (xio*..., x^o), .V, in which the variables
_V, yn arc functions of the variables x]} xp, in the form of (8.25). These implicit functions satisfy yw — /'Uio, rWJfl)
They also satisfy (8.24) for e very wtuple (,vj. ..., xtll ) in the neighborhood N thereby giving (8.24) the status of a set of identities as far as this neighborhood 15 concerned, Moreover, the implicit functions / ],.,., f" are continuous and have continuous partial derivatives with respect to all (he x variables.
As in the singleequation case, it is possible 10 find the partial derivatives of the implicit functions directly from the n equations in (8.24), without having to solve them for the y variables. Taking advantage of the fact that, in the neighborhood M the equations in (8.24) have the status of identities, we can take the total differential of each of these, and write dFJ =0(j = 1,2The result is a set of equations involving the differentials dy\,... ? dyn and dx\ ?... ? dxm ♦ Specifically, after transposing the dx{ terms to the right of Ihe equals signs, we have
—— i/>! + — dy2 + ■ ■ + ——dyn =  —dxi + ■ • • + —dxm 9 vi dy2 ¿yn \ t)jr] dxm /
—dy! + ~dy2 h ■ h —dy, =  —dx\ +  ■ + —dxa,
—dy{ + ——</)2 + ■■■ + ~—dyn ^  —dxl 4  ■ ■ + —dxm 8>'i dyi dyn \ 0X\ dxm /
Moreover, from (8.25), we can write the differentials of the yj variables as
OX{ 0X2 QXm dy 2 = —dx i 4 dx 2 + ■ ■ + ~~dxm
dyn dyn Syn dv„ = —dxi + —dx2 H+ —dxm dx2 3xm and these can be used to eliminate the dy} expressions in (8.26). But since the result of substitution would be unmanageably messy, let us simplify matters by considering only what would happen when x\ alone changes while all the other variables ■ ■ remain constant. Letting dx\ i= 0, but setting d.v2 = ■ ■ ■ = dxm = 0 in (8.26) and (8.27), then substituting (8.27) into (8.26) and dividing through by dx\ ^ 0, we obtain the equation system df[ /l)y{ \ i)F] / dvj \ dF] /3v„\ _
(*F* />A  dF" /'dy2\   OF" f dyn\ _ BFfl 'dyi \dX] / dyi \i)xj J \3x\ J 9x}
Even this result—for the ease where alone changes—looks Formidably complcx. because it is full of derivatives. But its structure is actually quite easy to comprehend, oncc we learn to distinguish between the two types of derivatives that appear in (8.28). One type, which we have parenthesized for visual distinction, consists of the partial derivatives of the implicit functions with rcspect to that we are seeking. These, therefore, should be viewed as the "variables" to be solved for in (8.28). The other type, on the other hand, consists of the partial derivatives of the F' functions given in (8,24), Since they would all take specific values when evaluated at the point (yiq, ..., y^: jrl(),..., xmi))—the point around which the implicit functions are defined they appear here not as derivative functions but as derivative values. As such, they can be treated as given constants. This fact makes (8.28) a linear equation system, with a structure similar to (4.1). What is interesting is that such a linear system has arisen during the process of analysis of a problem that is not necessarily linear in itself, since no linearity restrictions have been placcd on the equation system (8.24), Thus we have here an illustration of how linear algebra can comc into play even in nonlinear problems.
Being a linear equation system, (8.28} can be written in matrix notation as
3F2 3F2 3F1
dyi dy2 Syn
Since the determinant of the coefficient matrix in (8.28') is nothing but the particular Jacobian determinant 7 which is known to be nonzero under conditions of the implicitfunction theorem, and since the system must be nonhomogeneous (why?), there should be a unique nontrivial solution to (8.28 ). By Cramer's rule, this solution may be expressed analytically as follows:
By a suitable adaptation of this procedure, the partial derivatives of the implicit functions with rcspcct to the other variables, can also be obtained. It is a nice feature of this procedure that, cach time wc allow a particular Xi variable to change, we can obtain in
'dFl ~  
WJ 
dxi  
/>2\ 
SF2  
UJ 
— 
ajr,  
SF"  
WJ 
one fell swoop the partial derivatives of all the implicit functions f \ ..., fn with respect to thai particular xf variable. Similarly, to the implicitfunction rule (8.23) for the singleequation case, the procedure just described calls only for the use of the partial derivatives of the F functions—evaluated at the point { vio? •   • • , xmo)—in the calculation of the partial derivatives of the implicit functions f[ , /". Thus the matrix equation (8.28') and its analytical solution (8.29) are in effect a statement of the simultaneousequation version of the implicitfunction rule, Note that the requirement \ rules out a zero denominator in (8.29), just as the requirement Fv ^ 0 did in the implicitfunction rule (8.23) and (8.23'). Also, the role played by the condition \J\ / 0 in guaranteeing a unique (albeit implicit) solution (8.23) to the general (possibly nonlinear) system (8.24) is very similar to the role of the nonsingularity condition \ A\ ^ 0 in a linear system Ax = d. Example 5 The following three equations xy  w — 0 F1 = (x, y, w; z) = 0 are satisfied at point P: {x, y, w; z) = 4,1,1), The F functions obviously possess continuous derivatives. Thus, if the Jacobian  / is nonzero at point P, we can use the implicitfunction theorem to find the comparativestatic derivative (dxfdz). To do this, we can first take the total differential of the system: Moving the exogenous differential (and ib coefficients) to the righthand side and writing in matrix form, we get dz
where the coefficient matrix on the lefthand side is the Jacobian
At the point Pf the Jacobian determinant /  = 4 (^ 0). Therefore, the implicitfunction rule applies and 2w 1z2 Using Cramer's rule to find an expression for (ax/ttz), we obtain

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