Title: Differentiation of Tensor Functions and Representation Theorems
Abstract: Vector and tensor analysisVector(s)and tensor analysis or calculus is prerequisite for many applications in science and engineering. Examples of which include differential geometryGeometrydifferential, electromagnetic field theory and continuum mechanicsContinuum mechanics. It is an important branch of mathematics that studies differentiation and integration of vectorField(s)vector and tensor fieldsField(s)tensor which will extensively be used in this text. This motivates to devote the first part of this chapter as well as the two upcoming chapters of this book to study the fundamental rules of vectorVector(s)and tensor calculus and tensor calculus and their applications. Particularly, in this section, the standard rules of differentiation for tensor functions are first introduced. Then, their gradients are established by means of a first-orderFirst-order Taylor series Taylor series expansion. At the end, some analytical derivatives that frequently appear in this text are approximated by use of finite difference method. The remaining part of this chapter provides an introduction to representation theorems widely used in various branches of physics and engineering. Examples of which include solid mechanics and tissue engineering. These are examples of applications of theory of algebraic invariants in mechanics of isotropic and anisotropic continuum mediums.
Publication Year: 2023
Publication Date: 2023-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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