Theorem : suppose that f(x,x,..x)is a rea-valued function such that f and all of its partial derivatives of order less than or equal to r are defined and contin

Theorem : Suppose that f(x,x,..X)is a rea-valued function such that f and all of its partial derivatives of order less than or equal to r are defined and continuOus on an open subset u of R". Then, at each point p in u, every partial derivative of order r is independent of the order in which the partial derivatives are calculated. Example. Suppose that T(X,y.2)= *y°2 + xe" + y' sin(3x - 52). Then, Theorem implies that 0f ôzÖYÔX čZÖXÖYÔX x ôy©z which are equal to every other 4th order partial derivative that's with respect to x twice, and y and Z once each. ntoot lle + W IColl Toll Eroo: 1800 1