San José
State University Department of Economics 

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A tensor is an entity in an ndimensional space whose representations in different coordinate systems for that space are multidimensional arrays which are related to each other in a specific way. Suppose X=(x^{1},…,^{n}) and Y=(y^{1},…,y^{n}) are two coordinate systems for the space and they are related by an invertible transformation Y=T(X). Let P and Q be sequences of indices. The representation of a tensor in a particular coordinate system is an array of elements of the form
where the indices of Q are called the covariant indices and those of P are called the contravariant indices. The distinction between the two types of indices is explained below.
If a_{Q}^{P} and b_{S}^{R} are the representations in the X coordinate system and the Y coordinate system, respectively, then the relationship between the elements of the representations is
where M is the product of all terms of the form (∂x^{i}/∂y^{α}) for i being an index in Q and α an index in S and N is the product of all terms of the form (∂y^{β}/∂x^{j}) for β being an index in R and j being an index in P. The length of the sequences P and R are the same and that of Q and S are also the same.
Let a_{Q}^{P} and c_{Q}^{P} be the representation of two tensors in coordinate system X and b_{S}^{R} and d_{S}^{R} their representations in coordinate system Y. Then f_{Q}^{P}=(a_{Q}^{P}+c_{Q}^{P}) is the representation of a tensor in coordinate system X.
Proof:
Therefore f_{Q}^{P} transforms as a tensor. Thus the sum of two tensors of the same structure is a tensor of that same structure. The difference of two tensors is the sum of one tensor with the a tensor of the additive inverses of the elements of another tensor. Therefore the difference of two tensors of the same structure is a tensor of the same structure.
The outerproduct of tensors is the element by element product of the components of tensors. Thus the product of a_{Q}^{P} with c_{U}^{T} results in an array of the structure f_{QU}^{PT}. Remarkably enough this outer product of tensors is a tensor.
Proof:
Again let a_{Q}^{P} and c_{U}^{T} be the representations of two tensors in coordinate system X and b_{S}^{R} and d_{W}^{V} their representations in coordinate system Y.
The outer product is an array of the form f_{QU}^{PT} and these components of the outer product of tensors transform as a tensor.
The outer product of tensors corresponds to the Kronecker product of matrices. It exists but is little used. The useful product for matrices is the one in which there is summation over one index. This corresponds to what is called the contraction of tensors.
Consider a tensor of the form a_{Qj}^{Pi}; i.e., the contravariant indices are the sequence P with index i adjoined and the covariant indices are the sequence Q with the index j adjoined. This is for coordinate system X. For coordinate system Y the array of components is b_{Sβ}^{Rα} where
Terms of the form
Thus the array a_{Qi}^{Pi}, where index j is made identical to i and the terms summed over i, transform as a tensor and can be denoted as a_{Q}^{P}.
The inner product is best introduced in the form of matrix multiplication. Consider a matrix A whose typical element is a_{ij} and a column vector X whose elements are x_{j} for j=1,…,n. Then the matrix product Y=AX is
This is equivalent to taking the outer product of the tensors A and X and then contracting on the index of X.
Consider an array of the form A(P,Qi) where P and Qi are sequences of indices and suppose the inner product of A(P,Qi) with an arbitrary contravariant tensor of rank one (a vector) λ^{i} transforms as a tensor of form C_{Q}^{P} then the array A(P,Qi) is a tensor of type A_{Qi}^{P}.
where μ^{α} is the transform of λ^{i}.
M is the product of terms of the form (∂x^{i}/∂y^{α}) where i belongs to P and α belongs to
S. On the other hand N is the product of terms of the form (∂y^{β}/∂x^{j}) where β belongs to R and
j belongs to Q.
This is the transformation rule for a tensor of the form A_{Qi}^{P}.
(To be continued.)
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