Section 3.2 Tensor Algebra
Like vector algebra, tensor also obeys the laws of algebra. ,
- Addition: when two similar tensors are added the third tensor of same type will yield.\begin{equation*} A_{q}^{mp} + B_{q}^{mp} = C_{q}^{mp} \end{equation*}
- Multiplication (outer product or direct product): when two tensors of any rank will be multiplied as outer product the third tensor has the rank multiple of their ranks. such as \(A_{q}^{pr} \bigotimes B_{s}^{m} = C_{qs}^{prm}\text{.}\) The result of the direct product is a tensor whose rank is the sum of the ranks of the tensors being multiplied. The division of a tensor of rank greater than zero by another tensor of rank greater than zero is not defined.
- Contraction: The process of reducing the number of indices of a mixed tensor by one. for example: to contract the components of \(A_{klm}^{ij}\) into \(B_{lm}^{j}\) we have to set \(i=j\text{,}\) then \(A_{klm}^{ij}= B_{lm}^{j}\text{,}\) again, if \(j=l\text{,}\) then \(B_{lm}^{j} = C_{m}\text{,}\) a tensor of rank one.
- Inner multiplication: The inner product between two tensors results from first forming the direct product, and then setting the two nearest indices (with one index coming from each tensor) equal to one another and performing the sum according to the summation convention, e.g. If we multiply \(A_{q}^{mp}\) and \(B_{st}^{r}\) then the outer product is \(A_{q}^{mp}B_{st}^{r}\text{.}\) Now set \(q=r\) we get the inner product as \(A_{r}^{mp}B_{st}^{r}\text{,}\) again if \(q=r\) and \(p=s\text{,}\) then another inner product \(A_{r}^{mp}B_{pt}^{r}\) is obtained.
- Quotient Law: This law is useful to determine whether the given quantity is a tensor or not. Suppose it is not known whether X is a tensor or not but its inner product with an arbitrary tensor is a tensor then X itself is also a tensor. This is called a quotient law. The quotient rule can generally be written as \(KA = B\) where \(A\) and \(B\) are tensors of known rank and \(K\) is an unknown quantity. The quotient rule gives the rank of \(K\text{,}\) e.g., in \(\vec{L}=I\vec{\omega}\text{,}\) where \(\vec{l}\) and \(\vec{\omega}\) are known vectors, then from quotient rule we come to know that \(I\) is a second rank tensor. Some well know quotient rules are\begin{align*} K_{i}A_{i} \amp =B;\\ K_{ij}A_{j} \amp =B_{i};\\ K_{ij}A_{jk} \amp =B_{ik}; \\ K_{ijkl}A_{ij} \amp =B_{kl}; \\ K_{ij}A_{k} \amp =B_{ijk}. \end{align*}
