Differentiation and Integration of Compound Functions

This book discusses an elemental manipulation on functions which because it is fundamental has wide applicability. A function of a variable is given: the question is then asked: "How is the function changed when it becomes the function of different variable?" In particular how are the derivatives and integrals of the function with respect to one variable related to those of the other variable.? Clearly the relationship between the variables becomes critical. In mathematics the answer is given by arbitrarily assigning one of the variables as independent. For differentiation the arbitrary assignment are called "chain rules." Thus mathematics provides no inherently correct answer, but only one dependent on an arbitrary external naming of the variables. In theoretical physics, however, the ambiguities of mathematics are intolerable. The notion of independent/dependent variable are discarded and replaced by more restrictive relationship between the variables, restrictions which can be brought back to physical observations. Results are given for real derivatives and their integrals, directional derivatives and their integrals, and sectional derivatives and their integrals. Mathematicians and physicists familiar with this volume will be helped in avoiding many common errors while simultaneously enabling themselves to use compound functions in an enlarged scope of applications. This book forms Appendix 2 and Appendix 3 of Theoretical Physics: The First Problem ISBN, 978−0−9844299−1−2 which explains how to transform mathematics into theoretical physics. It will be read only with difficulty by someone not familiar with the main volume.