mean-value theorem for several variables

The mean-value theorem for a function of one real variable may be generalised for functions of arbitrarily many real variables; for the sake of concreteness, we here formulate it for the case of three variables:

Theorem.  If a function  $f(x,y,z)$  is continuously differentiable in an open set of${ℝ}^3$ containing the points  $(x_1,y_1,z_1)$  and  $(x_2,y_2,z_2)$  and the line segment connecting them, then an equation

$$f(x_2,y_2,z_2)-f(x_1,y_1,z_1)=f_x^{\prime }(a,b,c)(x_2-x_1)+f_y^{\prime }(a,b,c)(y_2-y_1)+f_z^{\prime }(a,b,c)(z_2-z_1),$$

where $(a,b,c)$ an interior point of the line segment, is .