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• Print publication year: 2013
• Online publication date: April 2013

# III - VECTORS AND TENSORS

## Summary

The next two chapters contain a detailed discussion of vector and tensor analysis.

Chapter 4 contains the basic concepts of vectors and tensors, including vector and tensor algebra. We begin with a description of vectors as an abstract object having a magnitude and direction, whereas tensors are then defined as operators on vectors. Several algebraic operations are summarized together with their matrix representations. Differential calculus of vector and tensors are then introduced with the aid of gradient operators, resulting in operations such as gradients, divergences, and curls. Next, we discuss the transformations of rectangular coordinates to curvilinear coordinates, such as cylindrical, spherical, and other general orthogonal coordinate systems.

Chapter 5 then focuses on the integral calculus of vectors. Detailed discussions of line, surface, and volume integrations are included in the appendix, including the mechanics of calculations. Instead, the chapter discusses various important integral theorems such as the divergence theorem, the Stokes' theorem, and the general Lieb-nitz formula. An application section is included to show how several physical models, especially those based on conservation laws, can be cast in terms of tensor calculus, which is independent of coordinate systems. The models generated are generally in the form of partial differential equations that are applicable to problems in mechanics, fluid dynamics, general physico-chemical processes, and electromagnetics. The solutions of these models are the subject of Part III and Part IV of the book.