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Implicit Function Theorem
Definition: Given F_1(x,y,z,u,v,w) = 0; F_2(x,y,z,u,v,w) = 0; F_3(x,y,z,u,v,w) = 0, if the determinantof the Jacobian |J F(u,v,w)|=|partial(F_1,F_2,F_3))/(partial(u,v,w))|!=0, then u, v, and w can be solved for in terms of x, y, and z and partial derivatives of u, v, w with respect to x, y, and z can be found by differentiating implicitly. More generally, let A be an open set in R^(n+k) and let f:A -> R^n be a C^r function. Write f in the form f(x,y), where x and y are elements of R^k and R^n. Suppose that (a, b) is a point in A such that f(a,b)=0 and the determinant of the nxn matrix whose elements are the derivatives of the n component functions of f with respect to the n variables, written as y, evaluated at (a,b), is not equal to zero. The latter may be rewritten as rank(Df(a,b))=n. Then there exists a neighborhood B of a in R^k and a unique C^r function g:B -> t;R^n such that g(a)=b and f(x,g(x))=0 for all x in B.