mathematics-physics-wiki/docs/en/mathematics/linear-algebra/tensors/tensor-transformations.md

Tensor transformations

We have a n \in \mathbb{N} finite dimensional vector space V such that \dim V = n, with a basis \{\mathbf{e}_i\}_{i=1}^n, a corresponding dual space V^* with a basis \{\mathbf{\hat e}^i\}_{i=1}^n and a pseudo inner product \bm{g} on V.

Let us introduce a different basis \{\mathbf{f}_i\}_{i=1}^n of V with a corresponding dual basis \{\mathbf{\hat f}^i\}_{i=1}^n of V^* which are related to the former basis \{\mathbf{e}_i\}_{i=1}^n by

\mathbf{f}_j = A^i_j \mathbf{e}_i,

so that \mathbf{\hat e}^i = A^i_j \mathbf{\hat f}^j.

Transformation of tensors

Recall from the section of tensor-formalism that a holor depends on the chosen basis, but the corresponding tensor itself does not. This implies that holors transform in a particular way under a change of basis, which is characteristic for tensors.

Theorem 1: let \mathbf{T} \in \mathscr{T}^p_q(V) be a tensor with p=q=1 without loss of generality and B = A^{-1}. Then \mathbf{T} may be decomposed into

\begin{align*} \mathbf{T} &= T^i_j \mathbf{e}_i \otimes \mathbf{\hat e}^j, \ &= \overline T^i_j \mathbf{f}_i \otimes \mathbf{\hat f}^j, \end{align*}

with the holors related by

\overline T^i_j = B^i_k A^j_l T^k_l.

??? note "Proof:"

Will be added later.

The homogeneous nature of the tensor transformation implies that a holor equation of the form T^i_j = 0 holds relative to any basis if it holds relative to a particular one.

Transformation of volume forms

Lemma 1: let (V, \bm{\mu}) be a vector space with an oriented volume form with

\begin{align*} \bm{\mu} &= \mu_{i_1 \dots i_n} \mathbf{\hat e}^{i_1} \otimes \cdots \otimes \mathbf{\hat e}^{i_n}, \ &= \overline \mu_{i_1 \dots i_n} \mathbf{\hat f}^{i_1} \otimes \cdots \otimes \mathbf{\hat f}^{i_n}, \end{align*}

then we have

\overline \mu_{j_1 \dots j_n} = A^{i_1}{j_1} \cdots A^{i_n}{j_n} \mu_{i_1 \dots i_n} = \mu_{j_1 \dots j_n} \det (A).

??? note "Proof:"

Will be added later.

Then \det(A) is the volume scaling factor of the transformation with A. So that if \bm{\mu}(\mathbf{e}_1, \dots, \mathbf{e}_n) = 1, then \bm{\mu}(\mathbf{f}_1, \dots, \mathbf{f}_n) = \det(A).

Theorem 2: let (V, \bm{\mu}) be a vector space with an oriented volume form with

\begin{align*} \bm{\mu} &= \mu_{i_1 \dots i_n} \mathbf{\hat e}^{i_1} \otimes \cdots \otimes \mathbf{\hat e}^{i_n}, \ &= \overline \mu_{i_1 \dots i_n} \mathbf{\hat f}^{i_1} \otimes \cdots \otimes \mathbf{\hat f}^{i_n}, \end{align*}

and if we define

\overline \mu_{i_1 \dots i_n} \overset{\text{def}}{=} \frac{1}{\det (A)} A^{j_1}{i_1} \cdots A^{j_n}{i_n} \mu_{j_1 \dots j_n},

then \mu_{i_1 \dots i_n} = \overline \mu_{i_1 \dots i_n} = [i_1, \dots, i_n] is an invariant holor.

??? note "Proof:"

Will be added later.

Transformation of Levi-Civita form

Theorem 3: let \bm{\epsilon} \in \bigwedge_n(V) be the Levi-Civita tensor with

\begin{align*} \bm{\epsilon} &= \epsilon_{i_1 \dots i_n} \mathbf{\hat e}^{i_1} \otimes \cdots \otimes \mathbf{\hat e}^{i_n}, \ &= \overline \epsilon_{i_1 \dots i_n} \mathbf{\hat f}^{i_1} \otimes \cdots \otimes \mathbf{\hat f}^{i_n}, \end{align*}

then \epsilon_{i_1 \dots i_n} = \overline \epsilon_{i_1 \dots i_n} is an invariant holor.

??? note "Proof:"

Will be added later.