This is a book of applied mathematical proofs. If you have seen a mathematical result, if you want to know why the result is so, you can look for the proof here. The book's purpose is to convey the essential ideas underlying the derivations of a large number of mathematical results useful in the modeling of physical systems. To this end, the book emphasizes main threads of mathematical argument and the motivation underlying the main threads, deemphasizing formal mathematical rigor. It derives mathematical results from the purely applied perspective of the scientist and the engineer. This book defines applied mathematics to be correct mathematics useful to scientists, engineers and the like; proceeding not from reduced, well defined sets of axioms but rather directly from a nebulous mass of natural arithmetical, geometrical and classical-algebraic idealizations of physical systems; demonstrable but generally lacking the detailed rigor of the professional mathematician.