- The Intuitive Nature of Einstein Notation's Tensor Index Notation Using the Schouten Convention

: Explanation of the Schouten convention that enhances the intuitiveness of tensor notation

~ The Schouten convention refers to the notation of attaching prime symbols to indices.

1. Vector as a Contravariant Vector #

In a vector space $V$, when the matrix that transforms the basis is $A$, we have $$e_i = A_i^{j’} e_{j’} = A e_{i’} \quad (e_{i’} = A_{i’}^j e_j = A^{-1} e_i)$$

(Note that both $A$ and $A^{-1}$ can be represented using the $A$ index notation.)

For a vector $v = v^i e_i$: $$v = v^i A_i^{j’} e_{j’} = (A_i^{j’} v^i) e_{j’} = v^{j’} e_{j’}$$

That is, $[v]$ changes to $[v]’ = A [v]$, meaning the basis representation of $v$ changes. The term “contravariant vector” comes from the fact that it transforms in the opposite way to the basis transformation.

2. Dual Vector as a Covariant Vector #

On the other hand, a dual vector $f$, which is a vector in the dual space $V^*$, is called a covariant vector. The basis transformation relationship expressed using the matrix $A$ index is: $$e^i = A^i_{j’} e^{j’}$$

In matrix form, this becomes: $$e^i = (A^{-1})^T e^{i’}$$

Therefore, $[f]$ becomes $[f]’ = (A^{-1})^T [f]$.
(Here, $T$ denotes transpose, and $[\ ]$ refers to the representation of a vector according to its basis.)

The change in the basis representation of $f$ is related to the transpose of $A^{-1}$. The term “covariant” was coined because it transforms in the same way as the basis transformation of the original vector space.

However, when the basis transforms from orthonormal to orthonormal in a real vector space, $A$ becomes an orthogonal matrix, satisfying $A^{-1} = A^T$.

3. Additional Note #

The purpose of this article is to show that there is a consistent rule in tensor index notation using Einstein convention, and that this notation can be naturally converted to matrix notation.

For reference, the matrix of a dual space corresponds to the transpose of the matrix of the vector space. That is,

When $A : V \to W$, $w = Av$, then $A^T : W^* \to V^*$, $\tilde{v} = A^T \tilde{w}$.