In Chapter 3 we saw that any vector space gives rise to other vector spaces such as the dual space and the space of all linear operators on . In this chapter we will consider a more general class of spaces constructed from a vector space , known as tensor spaces, of which these are particular cases. In keeping with modern mathematical practice, tensors and their basic operations will be defined invariantly, but we will also relate it to the ‘old-fashioned’ multicomponented formulation that is often better suited to applications in physics [1].
There are two significantly different approaches to tensor theory. Firstly, the method of Section 7.1 defines the tensor product of two vector spaces as a factor space of a free vector space [2]. While somewhat abstract in character, this is an essentially constructive procedure. In particular, it can be used to gain a deeper understanding ofassociative algebras, and supplements the material of Chapter 6. Furthermore, it applies to infinite dimensional vector spaces. The second method defines tensors as multilinear maps [3–5]. Readers may find this second approach the easier to understand, and there will be no significant loss in comprehension if they move immediately to Section 7.2. For finite dimensional vector spaces the two methods are equivalent [6].
Contents
- 7.1 Free vector spaces and tensor spaces
- 7.2 Multilinear maps and tensors
- 7.3 Basis representation of tensors
- 7.4 Operations on tensors
Concept index
Terms by section. Links lead to the brown underlined definitions in the Chinese reading text.
7.1 Free vector spaces and tensor spaces
- free vector space自由向量空间
- tensor product space张量积空间
- tensor product张量积
- tensor张量
- contravariant tensor逆变张量
- covariant tensor协变张量
- free associative algebra自由结合代数
- universal property泛性质
7.2 Multilinear maps and tensors
- multilinear map多重线性映射
- contravariant degree逆变次数
- covariant degree协变次数
- tensor components张量分量
- metric tensor度量张量
- inverse metric tensor逆度量张量
- mixed tensor混合张量