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

a_Q^P
 

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

b_S^R = a_Q^PMN
 

where M is the product of all terms of the form (∂xⁱ/∂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.

The Addition Theorem for Tensors
 

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:

• 
  b_S^R = a_Q^PMN
  and
  d_S^R = c_Q^PMN
  therefore
  (b_S^R+d_S^R) = (a_Q^P+c_Q^P)MN
   

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.

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 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.

• 
  b_S^R = a_Q^PM₁N₁
  and
  d_W^V = c_U^TM₂N₂
  therefore
  (b_S^R*d_W^V) = (a_Q^P*c_U^T)M₁N₁ M₂N₂
  which is the same as
  (b_S^R*d_W^V) = (a_Q^P*c_U^T)(M₁M₂)(N₁N₂)
   

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.

The Contraction Theorem for 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

b_Sβ^Rα = a_Qj^PiM(∂x^j/∂y^α)N(∂y^α/∂xⁱ)
 

Terms of the form

(∂xⁱ/∂y^α)(∂y^α/∂xⁱ)
reduce to
δ_β^α
which is 0 if α≠β and 1 if α=β.
 

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 Theorem for Tensors
 

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

y_i = Σ_ja_ijx_j
 

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 C_Q^P then the array A(P,Qi) is a tensor of type A_Qi^P.

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 μ^α such that

  
  B(R,Sα)μ^α = A(P,Qi)λⁱMN
   

  where μ^α is the transform of λⁱ.

  M is the product of terms of the form (∂xⁱ/∂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.
   

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

  
  λⁱ = μ^β(∂xⁱ/∂y^β)
   

• When λⁱ is replaced by μ^β(∂xⁱ/∂y^β) 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/∂y^β)]μ^α = 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/∂y^β)
   

  This is the transformation rule for a tensor of the form A_Qi^P.

(To be continued.)