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Tensors are generalizations of vectors. These manifestate as

  1. product of vector spaces
  2. product of linear maps
  3. multi-indexed arrangements of numbers or scalar functions

In the first case we take two vector spaces V, W and construct a new one V\otimes W by mean of their basis: if V has as basic vectors b_1,b_2,... and W has c_1,c_2,... then

V\otimes W

is the vector space generated by the symbols b_1\otimes c_1,b_1\otimes c_2,... which are all the linear combinations of pairs b_i\otimes c_j.

In another hand, if we want to construct a bilinear map B:V\times W\to F then we pick up a pair of covectors f:V\to F and g:W\to F then we manufacture

B=f\otimes g:V\times W\to F defined via B(v,w)=f\otimes g(v,w)=f(v)g(w)

It is also know that for finite dimensional vector spaces the tensor of rank one are the elements of V and its dual space V^*

In the multi-indexed versions tensors are expressions like g_{ij}, \delta_i^j, \varepsilon_{ijkl}, {{\Gamma^i}_j}_k, {R^i}_{jkl},... which actualy are the components of tensors and are in nature scalars or scalar functions. In the multi-indexed version tensor are arrays of quantities that obeys certain rules of transformations when we change systems of coordinates.

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Exterior links

tensors in wikipedia

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