1. Multilinear Functions
In the case of r variables we sometimes use the more specific term r-linear and defining relation is
$$ f(v_1, \cdots, av_i+\bar{a} \bar{v_i}, \cdots, v_r)=af(v_1, \cdots, v_i, \cdots, v_r)+\bar{a} f(v_1, \cdots, \bar{v_i}, \cdots, v_r) $$
Suppose that $ \tau \in V^\* $ and $ \theta \in W^\* $; that is, $ \tau $ and $ \theta $ are linear real-valued functions on $ V $ and $ W $, respectively. Then we obtain a bilinear real-valued funciton $ \tau \otimes \theta : V\times W \to \mathbb{R} $ by the formula $$ \tau \otimes \theta (v, w)=(\tau v)(\theta w). $$ This bilinear function is called the **tensor product** of $ \tau $ and $ \theta $, and we read it $ "\tau $ tensor $ \theta" $.2. Tensor Space
The scalar-valued Multilinear funcitons with variables all in either $ V $ or $ V^* $ are called tensors over $ V $ and the vector spaces they form are called the tensor spaces over $ V $. The numbers of variables from $ V^* $ and $ V $ are called the type numbers or degrees of the tensor, with the number of variables from $ V^* $ are called the contravaiant degree, the number of $ V $ the covariant degree. Thus for a multilinear function on $ V^*\times V\times V $ the type is $ (1, 2) $.
The space of multilinear funciton on $ V^*\times V\times V $ is denoted
In fact, V(finite-dimensional) may be considered to be the same as $ V^{**} $ the linear funcitons on $ V^* $, and $ V^* $ consists of linear funcitons on $ V $. In general, tensors of type (r,s) form a vector space denoted by $ T^r_s=V\otimes \cdots \otimes V\otimes V^*\otimes \cdots \otimes V^* $ and consist of multilinear functions on $ V^*\times \cdots \times V^*\times V\times \cdots \times V $.
A tensor of type $ (0, 0) $is defined to be a scalar,so $ T_0^0 =\mathbb{R} $. A tensor of type $ (1, 0) $sometimes called a contravaiant vector and one of type $ (0, 1) $ a covariant vector. A vector of type $ (r, 0) $ is sometimes called a contravaiant tensor and one of type $ (0, s) $ is sometimes called a covariant tensor.
3. Algebra of Tensors
As part of the vector space structure, we have that tensors of the same type can be added and multiplied by scalars. The tensor product of tensor $ A $ of type $ (r, s) $ and tensor $ B $ of type $ (t, u) $ is a tensor $ A\otimes B $ of type $ (r+t, s+u) $ defined, as a function on $ (V*){r+t}\times V^{s+u} $, by
The associative law and the distributive laws for tensor product are true and easily verified. That is,
But the tensor product is not generally commutative(for instance, if $ v, w \in V $ are linealy independent, show that $ v\otimes w \neq w\otimes v $).
参考文献
[1]Richard L. Bishop, Samuel I. Goldberg, Tensor Analysis on Manifolds, Dover Publicaitons,Inc. New York, 1980: chapter 2.