In Chap. 28 we discussed applications of the differentiation of a variable. We now do the same with the (partial) differentiation of functions of several variables: We describe the (multidimensional) Newton’s method for determining the roots of vector fields and the Taylor expansion for scalar fields, to locally approximate given scalar fields by a tangent plane or osculating parabola. For this, we do not need to learn anything new in terms of content, but only to gather together knowledge we have already acquired.