A New Approach to Classical Multiple-Variable Calculus (TH)

How to define the vector division has been plagued by people. We redefine the fraction , where represents an arbitrary vector, and extend it to discuss the differentiability and integrability of two or more variables function. In this book we directly exploit the definition of derivatives for two variable functions. According to our research, we discover various forms of the Mean Value(Vector) Theorems and the Fundamental Theorems of Calculus for multiple integrals. This book adds a new understanding to the field of multi-variable calculus and thus helps those who are working in the subject.
We divide our research into three sections in this book. In Chapter 1, by defining the line derivector and plane derivector, we explain the chain rule for differentiation of multi-variables function, differential and the mean vector theorem. In Chapter 2, we illustrate the formula derivation of i) line derivector for a 2-variable function, ii) line derivatives for 2-variable functions, iii) plane derivative for a 2-variable function, iv) Surface derivectors for 3-variable functions, v) Surface derivatives, and vi) Solid derivatives, ect. Finally, we discuss various fundamental theorems of calculus for multiple integrals.

Ying-Keh Wu
Retired Professor of Department of Applied Mathematics, Tunghai UniversityPh. D. in Statistics, Virginia Polytechnic Institute

1 DIFFERENTIATIONS
1.1 Derivatives and derivectors
1.2 The chain rules for differentiation of multi-variables functions
1.3 Differentials
1.4 The mean vector theorems (MVTs)
1.5 Second order line derivector for 2-variable functions
1.6 Taylor theorem in R2
2 INTEGRATIONS
2.1 Definitions of variousmultiple integrals
2.2 The fundamental properties of plane integrals and solid integrals
2.3 Various fundamental theorems of calculus for multiple integrals
2.4 Change of variables in multiple integrals
2.5 Relations among various multiple integrals