Our study of partial derivatives was motivated solely by comparative-static considerations. But partial derivatives also provide a means of testing whether there exists functional (linear or nonlinear) dependence among a set of n functions in n variables. This is related to the notion of Jacobian determinants (named after Jacobi). Consider the two functions
If wc get all the four partial derivatives
fix 2
and arrange them into a square matrix in a prescribed order, culled a Jacobian matrix and denoted by./, and then take its determinant, the result will be what is known as a Jacobian determinant (or & Jacobian, for short), denoted by |./,:
For economy of space, this Jacobian is sometimes also expressed as
Sixuxj)
More generally, if wc have n differentiate functions in n variables, not necessarily linear,
where the symbol /" denotes the /ith function (and not the function raised to the flth power), we can derive a total of«2 partial derivatives. Adopting the notation fj = dy1 fdxj. we can write the Jacobian w\ =
Byn/dxtl n n r A Jacobian test for the existence of functional dependence among a set of n functions is provided by the following theorem; The Jacobian \ J\ defined in (7.27) will be identically zero for all values of x{l..., xn if and only if the n functions /1in (7.26) arc functionally (linearly or nonlinearly) dependent. As an example, for the two functions in (7.24) the Jacobian as given in (7,25) has the value That is, the Jacobian vanishes for all values of X] and xi. Therefore, according to the theorem, the two functions in (7.24) must be dependent. You can verify that y2 is simply y} squared; thus they are indeed functionally dependent here /io/ilinearly dependent. Let us now consider the special case of linear functions. We have earlier shown that the rows of the coefficient matrix A of a linear-equation system are linearly dependent if and only if the determinant \A\ = 0, This result can now be interpreted as a special application of the Jacobian criterion of functional dependence. Take the left side of each equation in (7.28) as a separate function of the n variables x\t ..., xn, and denote these functions by yu -..> yn ■ The partial derivatives of these functions will turn out to be = a\u ty\/dx2 = anf etc., so that we may write, in general, 'dyi/Sxj = a^. In view of this, the elements of the Jacobian of these n functions will be precisely the elements of the coefficient matrix A, already arranged in the correct order. That is, we have \ J\ ^ \A\> and thus the Jacobian criterion of functional dependence among V],,.,, yn— or, what amounts to the same thing, linear dependence among the rows of the coefficient matrix A—is equivalent to the criterion \A \ — 0 in the present linear case. We have discussed the Jacobian in the context of a system of n functions in n variables. It should be pointed out, however, that the Jacobian in (7.27) is defined even if each function in (7.26) contains more than n variables, say, n + 2 variables: In such a case, if wc hold any two of the variables (say, ^-h ^"d x„+2) constant, or treat them as parameters, we will again have n functions in exactly n variables and can form a Jacobian. Moreover, by holding a different pair of the* variables constant, we can form a different Jacobian. Such a situation will indeed be encountered in Chap. 8 in connection with the discussion of the implicit-function theorem. |
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