32 for all values of x, y, z; then, since in this case INFINITESIMAL CALCULUS we have dF = dx dF du du dx dF = 0, dF dz dv dx dw dx dFdudF dv_ dF dw du dy + dv dy dw dy dF du dF dv dF dw^ du dz dv dz dw dz * 0, dF dw Consequently, eliminating dF dF du dv we get du du du dx dy dz dv dv dv dx dy dw dw = 0. This is an extension of the theorem that when a function of a single variable is constant its derived function is zero. The converse of the preceding theorem can be established, viz., if J = 0, then the functions u, v, w are no longer independent of each other. . , These results are readily extended to any number of variables ; thus, whenever the functions u u u. 2 , . . . u n are connected by a re lation, rf (i. u "")=. Q; and conversely. n ( T T SC i 99. Again, if u, v, w, instead of being given explicitly in terms of x, y, z, be given implicitly, i.e., if they are connected- with them by three equations of the form F^x, y, z, u, v, w) = , F 2 (z, y, z, u, v, w) = Q, F 3 (x, y, z, u, v, w} = , we have, adopting the same notation as before, by C, we have by 97 dfa dfa dfa dx dy dz dfa dfa dfa C.. dx dy dz B d<p 3 d<p% d^>.j dx dy dz and similarly dfa dfa dfa du dv dw dfa dfa, dfa C du dv dw A Clef) ft Ct <z>o Ct <Po du dv dw Hence dfa dfa dfa dfa dfa d(f> 1 du du du dx dy dz du dv dw dx dy dz d(/> 2 d(p% dtp^ dfa dfa. dfa. dv dv dv dx dy dz du dv dw dx dy dz d(f> 3 dfai dfa dfa dfa dfa dw dw dw dx dy dz du dv dw dx dij dz dF! dx -j dv and similar equations for the increments d%x . . . d 3 x, &c. . . . , as also others derived from the functions F 2 , F 3 . . . Hence, as before, dF l dF] dF_i dx dy dz d 3 x d 3 y d 3 z du dv dx dy dz dF 3 dF 3 dFg dx dy dz dFj dFi dFi du dv dw dF dF 9 dF 2 du dv dw dF 3 dFj dFj du dv dw This result, when generalized, may be written as follows : _ dFi ! dx d(x lt x. 2 . . . x n ) (-!)" dx n dx, dx n du n dF dF du n 100. "We shall next consider the generalization of the element- dF(w) dF(u) du ary theorem _LJ = LJ . dx du dx If we suppose fa , fa , fa to represent functions of u, v, w, while u, v, w are functions of a*, y, z, then, adopting the same notation as before, and representing the determinant d. 2 fa d. 2 fa d.,fa d 3 fa d 3 fa d 3 fa Consequently, the Jacobian of fa, fa, fa with respect to x, ?/, z is equal to their Jacobian with respect to u, v, w multiplied by the Jacobian of u, v, iv with respect to x, y, z. This is the required generalization in the case of three variables. 101. Again, if u, v, w be functions of x, y, z, we may regard x, y, z as functions of u, v, w ; and it follows immediately that the Jacobian of u, v, w with respect to x, y, z is the reciprocal of the Jacobian of x, y, z with respect to u, v, w; i.e. = 1 This, when extended to n variables, is the generalization of the theorem that the derived function of y with respect to x is the inverse of that of x with respect to y. The preceding demonstrations are readily extended to any num ber of variables. When generalized for n variables, the results are written in abridged notation thus d(fa , fa , . . . fa) ^ d(fa , fa, . . . fa] x d(i , - du du du dx dx dx dx dy dz du dv dw dv dx dv dy dv ~d7 dy_ du dy dv dy dw dw dw dw dz dz dz dx dy ~dz da dv dw d(X l , - r~ . . . X n ) d(U 1 1 . 102. Again, the Jacobian of any system can be expressed as a mono mial. This result can be established as follows : Reverting to our original discussion, it is readily seen that of 2?). quantities, x l , x 2 , x 3 . . . x n , % , u. 2 , u 3 , . . . u n , connected by n equations, when any n are chosen at pleasure the others are capable of determination. Consequently, if n-l of them be supposed to remain invariable, all the others may be regarded as changing simultaneously, and the ratios of their infinitely small increments are determined. Hence we may suppose our n systems of simultaneous increments attributed as in the following table : djX 2 o a
d a U n The first line indicates that the first system of increments attri buted to a?i , a?2 . . . ar are such that 2 , u 3 . . . u n do not change ; in the second line we suppose that increments of the second system are such that X 1 , u 3 . . . u n do not change ; and so on. Again, since for these values the determinants A, B, reduce to their diagonal terms, we have, in this case, by 97, __ jj 22 33 n n
Also, by what has been stated above, the ratios djWj rfgM 2 d 3 U d,,Un_ d jit" j Ct o-^ 2 ^ 3^3 ^ *t*^ n can each be determined in this case from the given equations.
