11.31 THEOREM [MEAN VALUE THEOREM ON Rnj. Let V be open in Rn and suppose that f : V ---t Rm is differentiable on V. If x,a E V and L(x;a) ~ V, then for each u E R m, there is acE L(x;

a) such that U· (I(x) - f(a)) =U· (Df(c)(x-a)).

PROOF. Let g(t) =a+t(x-a), t ER,