port from mathematics-physics notes

docs/mathematics/linear-algebra/tensors/tensor-transformations.md

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+# 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.