ABSTRACT
This chapter introduces the notion of an inner product on a Euclidean space. This allows to define the angle between two vectors and to describe some of the geometrical objects in Euclidean space such as planes and hyperplanes. The chapter shows how these ideas can be used to obtain greater insight into the meaning of the derivative. It shows that a function can have all its partial derivatives at every point and yet fail to be continuous. That function, however, does not have directional derivatives in every direction. The chapter also shows that the function has a continuous derivative if all the partial derivatives are continuous. One interpretation given to the derivative in elementary calculus is that it gives the slope of the tangent line to the graph of the function.
