First Steps of Vector Differential Calculus

This paper treats the fundamentals of the *vector differential calculus* part of *universal geometric calculus.* Geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. In order to make the treatment self-contained, I first compile all important *geometric algebra* relationships, which are necessary for vector differential calculus. Then *differentiation by vectors* is introduced and a host of major vector differential and vector derivative relationships is proven explicitly in a very elementary step by step approach. The paper is thus intended to serve as reference material, giving details, which are usually skipped in more advanced discussions of the subject matter.