Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 165equivalent to z'Ax = 0 for all x, where z = col (zl, . . . , z,), and hence to z'A = 0 orA'z = 0, (2.166)If A has rank r, then as in Section 1.17 the solutions z of (2.166) form a subspaceW of dimension n - r of Vn. We choose a basis zl , . . . , zn-, for W and obtain n - rindependent linear dependencies:where z, = col (z,, , . . . , zan), and all other linear dependencies are linear combinationsof these. If n 5 m and A has maximal rank n, then there are no such relationsand the n functions fi are functionally independent. If m < n, then r 5 m < n andthe dependencies exist. In all cases A maps Vm onto an r-dimensional subspace ofVn and the mapping is one-to-one precisely when r = m.General case.The results for the linear case extend in a natural way to the generalcase of n functions f,(xl,. . . , xm) (i = 1,. . . , n), defined and differentiable indomain D. Functional dependence of fl, . . . , f, in D is understood to mean theexistence of h functions F,(ul, . . . , u,) (j = 1, . . . , h), defined and differentiablein a domain D, of En, such that the composite functionsFJ(f~(x~, ...),..., fnbi,. ..I), J = 1,. .., h, (2.168)are defined in D and are identically 0 in D. Here it is assumed that the Jacobianmatrix (aF,/auk) has maximum rank h in D,, so that h 5 n.When these conditions hold, differentiation yields the equationsWe denote the Jacobian matrix (afk/axe) by A. Equation (2.169) then asserts thatA'z, = 0 at each point x of D, where A is evaluated at x and z[, is the jth rowof (aF,/auk), evaluated at the corresponding point u of D,. Since the last matrix isassumed to have rank h, zl , . . . , zh are linearly independent; accordingly, h 5 n - r,where r is the rank of A. Therefore, r 5 n - h:THEOREMFunctional dependence:F,(f~(xl, ..., xm), ..., fn(x1, ... 9 xm)) = 0 in D (2.170)for j = 1, . . . , h, implies that the Jacobian matrix A = (af, lax;) has rank r 5 n - hat each point of D.To obtain a converse, we assume that A has constant rank r < n in D. Thenone can prove that, in an appropriate neighborhood Dl of each point x0 of D, thefunctions fl, ..., fn are functionally dependent, so that they satisfy relations ofform (2.170) in Dl with h = n - r, where Fl, ..., Fh are as above. In fact, one canshow that, if r 5 n, then in terms of appropriate curvilinear coordinates 61, . . . , emin Dl and ql, ..., qn in a neighborhood of the corresponding point u0 = f(xO), themapping u = f(x) becomes simply the linear mapping