Tensors, Part I

In this post I will be going through index notation, the basic properties of tensors and two crucial functions that will allow us to greatly simplify calculations and express quantities in a concise manner. The index represenation of a tensor consists in

Kronecker's Delta

$$\delta_{ij}= \begin{cases} 1, & \text{if } i=j \\ 0, & \text{if } i\neq j \end{cases}$$

It has the important property of index substitution

$$a_i \delta_{ij} =a_j$$

when the summation runs over all $i$'s. We can also play with higher-order tensors and several Kronecker symbols.

$$b_{ijk} \delta_{jp}\delta_{kq}=b_{ipq} \\ c_{ijkl} \delta_{jp} \delta_{pq} = c_{iqkl}$$

From the definition it is also clear that it satisfies a symmetry property and a dimensional property, since a Kronecker delta is defined to act on a particular space of dimension $n$, generally equal to $3$.

$$\delta_{ij}=\delta_{ji} \\ \delta_{ij} \delta_{ji}=\delta_{ii}=n$$

When dealing with the canonical basis in $\mathbb R^n$, inner products among basis vectors coincide with the Kronecker delta

$$\langle e_i,e_j \rangle=\delta_{ij}$$

Levi-Civita Symbol

$$\varepsilon_{ijk}= \begin{cases} 1, & \text{if $i,j,k$ are different and in cyclic order} \\ -1, & \text{if $i,j,k$ are different not in cyclic order} \\ 0, & \text{if at least two of $i,j,k$ are equal} \end{cases}$$

Vector Algebra computations

$$\langle u,v \rangle=\langle u_i e_i, v_j e_j \rangle=u_i v_j \langle e_i,e_j \rangle=u_i v_j \delta_{ij}=u_i v_i$$