Before revision:  This paper presents an entire re-derivation of generalized spray and Finsler geometries focusing on building the ideas from familiar concepts in advanced calculus and the calculus of variations.
After revision:  This paper presents an entire re-derivation of generalized spray and Finsler geometries , critical for the development of our recent work on geometric fabrics, which builds the ideas from familiar concepts in advanced calculus and the calculus of variations.