San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The Addition and Quotient Theorems in Tensor Analysis

A tensor is an entity in an n-dimensional 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=(x1,…,n) and Y=(y1,…,yn) 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

#### aQP

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 aQP and bSR are the representations in the X coordinate system and the Y coordinate system, respectively, then the relationship between the elements of the representations is

#### bSR = aQPMN

where M is the product of all terms of the form (∂xi/∂yα) for i being an index in Q and α an index in S and N is the product of all terms of the form (∂yβ/∂xj) 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.

### The Addition Theorem for Tensors

Let aQP and cQP be the representation of two tensors in coordinate system X and bSR and dSR their representations in coordinate system Y. Then fQP=(aQP+cQP) is the representation of a tensor in coordinate system X.

Proof:

• #### bSR = aQPMN and dSR = cQPMN therefore (bSR+dSR) = (aQP+cQP)MN

Therefore fQP 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.

### Outer Multiplication Theorem for Tensors

The outer-product of tensors is the element by element product of the components of tensors. Thus the product of aQP with cUT results in an array of the structure fQUPT. Remarkably enough this outer product of tensors is a tensor.

Proof:

• Again let aQP and cUT be the representations of two tensors in coordinate system X and bSR and dWV their representations in coordinate system Y.

• #### bSR = aQPM1N1and dWV = cUTM2N2therefore (bSR*dWV) = (aQP*cUT)M1N1 M2N2which is the same as (bSR*dWV) = (aQP*cUT)(M1M2)(N1N2)

The outer product is an array of the form fQUPT 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.

### The Contraction Theorem for Tensors

Consider a tensor of the form aQjPi; 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 where

#### bSβRα = aQjPiM(∂xj/∂yα)N(∂yα/∂xi)

Terms of the form

#### (∂xi/∂yα)(∂yα/∂xi) reduce to δβαwhich is 0 if α≠β and 1 if α=β.

Thus the array aQiPi, where index j is made identical to i and the terms summed over i, transform as a tensor and can be denoted as aQP.

### The Inner Product Theorem for Tensors

The inner product is best introduced in the form of matrix multiplication. Consider a matrix A whose typical element is aij and a column vector X whose elements are xj for j=1,…,n. Then the matrix product Y=AX is

#### yi = Σjaijxj

This is equivalent to taking the outer product of the tensors A and X and then contracting on the index of X.

### The Quotient Theorem for Tensors

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) λi transforms as a tensor of form CQP then the array A(P,Qi) is a tensor of type AQiP.

Proof:
• Consider the coordinate system for the array to be X=(x1,…,xn) and consider transformations to the coordinate system Y=(y1,…,yn). For the inner product A(P,Qi)λi to transform as a tensor means that there is an array B(R,Sα) and vector μα such that

#### B(R,Sα)μα = A(P,Qi)λiMN

where μα is the transform of λi.

M is the product of terms of the form (∂xi/∂yα) where i belongs to P and α belongs to S. On the other hand N is the product of terms of the form (∂yβ/∂xj) where β belongs to R and j belongs to Q.

• Since there exist the reverse transformation from Y to X there is a transformation of μ to λ; i.e.,

#### λi = μβ(∂xi/∂yβ)

• When λi is replaced by μβ(∂xi/∂yβ) in the previous equation and the RHS subtracted from the LHS and the result factored then the following is obtained

#### [B(R,Sα) − A(P,Qi)MN(∂xi/∂yβ)]μα = 0

• Since μ is arbitrary as a result of being the transform of an arbitrary vector λ the expression within the brackets must be identically zero. Thus

#### B(R,Sα) = A(P,Qi)MN(∂xi/∂yβ)

This is the transformation rule for a tensor of the form AQiP.

(To be continued.)