It is widely conceded that modern physics is "badly posed" mathematically. The corollary to this concession is not so widely appreciated, namely that given sufficient mathematical manipulation, any proposition of physics can be "proven" true. Thus modern physics no longer qualifies as science. Starting from a base of mathematics, this book develops a set of coherent ideas related in a framework of logically consistent statements (definitions and equations) useful for understanding physical observations. This set of ideas and equations is called Theoretical Physics. While related to mathematical ideas and operations, the ideas of Theoretical Physics are distinctly different since they both constrain and expand the mathematics. The process of constraint forbids certain otherwise valid mathematical operations. The process of expansion generates from each mathematical idea of panoply of related but distinct ideas. The text also contains some new mathematical material, notably: division in the vector space, a detailed discussion of gradients, and new types of integrals which to invert gradients directly. The text concludes by showing the ambiguities associated with the Eulerian relationship are a major reason why many physical problems are badly posed mathematically. The inconvenience to modern physics is great inasmuch as it follows that all argumentation based on partial differential equations, which is practically all of modern physics, is suspect. The ultimate purpose of this book is to help physicists reestablish physics as science. This is not a book for frivolous skimming, but rather one requiring close study. The intended reader will have attained a graduate level of education in either mathematics or physics. The serious student will be rewarded with a comprehensive set of ideas, definitions, and equations by which physical observations can be described and understood. This is a book for every mathematician and physicist.