From 7d7fea70c91fa42a1e2b75352965039c4e8ccc39 Mon Sep 17 00:00:00 2001 From: luc Date: Sat, 20 Sep 2025 21:44:37 +0200 Subject: [PATCH] linear-algebra: update --- .../linear-algebra/dual-vector-spaces.md | 6 +-- .../tensors/tensor-formalism.md | 22 +++++----- .../tensors/tensor-symmetries.md | 44 +++++++++++++++++-- 3 files changed, 54 insertions(+), 18 deletions(-) diff --git a/docs/mathematics/linear-algebra/dual-vector-spaces.md b/docs/mathematics/linear-algebra/dual-vector-spaces.md index c46fc45..5b7dda0 100644 --- a/docs/mathematics/linear-algebra/dual-vector-spaces.md +++ b/docs/mathematics/linear-algebra/dual-vector-spaces.md @@ -1,8 +1,8 @@ # Dual vector spaces -We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n.$ In the following sections we make use of the Einstein summation convention introduced and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}$. +We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$ and a field $F$. In the following sections we make use of the Einstein summation convention introduced. -> *Definition 1*: let $\mathbf{\hat f}: V \to \mathbb{K}$ be a **covector** or **linear functional** on $V$ if for all $\mathbf{v}_{1,2} \in V$ and $\lambda, \mu \in \mathbb{K}$ we have +> *Definition 1*: let $\mathbf{\hat f}: V \to F$ be a **covector** or **linear functional** on $V$ if for all $\mathbf{v}_{1,2} \in V$ and $\lambda, \mu \in F$ we have > > $$ > \mathbf{\hat f}(\lambda \mathbf{v}_1 + \mu \mathbf{v}_2) = \lambda \mathbf{\hat f}(\mathbf{v}_1) + \mu \mathbf{\hat f}(\mathbf{v}_2). @@ -10,7 +10,7 @@ We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim Throughout this section covectors will be denoted by hats to increase clarity. -> *Definition 2*: let the the dual space $V^* \overset{\text{def}} = \mathscr{L}(V, \mathbb{K})$ denote the vector space of covectors on the vector space $V$. +> *Definition 2*: let the the dual space $V^* \overset{\text{def}} = \mathscr{L}(V, F)$ denote the vector space of covectors on the vector space $V$. Each basis $\{\mathbf{e}_i\}$ of $V$ therefore induces a basis $\{\mathbf{\hat e}^i\}$ of $V^*$ by diff --git a/docs/mathematics/linear-algebra/tensors/tensor-formalism.md b/docs/mathematics/linear-algebra/tensors/tensor-formalism.md index 6a2723d..5b246d0 100644 --- a/docs/mathematics/linear-algebra/tensors/tensor-formalism.md +++ b/docs/mathematics/linear-algebra/tensors/tensor-formalism.md @@ -1,13 +1,13 @@ # Tensor formalism -We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$ and a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}.$ In the following sections we make use of the Einstein summation convention and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}$. +We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim V = n$, with a basis $\{\mathbf{e}_i\}_{i=1}^n$, a corresponding dual space $V^*$ with a basis $\{\mathbf{\hat e}^i\}$ and a field $F$. In the following sections we make use of the Einstein summation convention. ## Definition > *Definition 1*: a **tensor** is a multilinear mapping of the type > > $$ -> \mathbf{T}: \underbrace{V^* \times \dots \times V^*}_p \times \underbrace{V \times \dots \times V}_q \to \mathbb{K}, +> \mathbf{T}: \underbrace{V^* \times \dots \times V^*}_p \times \underbrace{V \times \dots \times V}_q \to F, > $$ > > with $p, q \in \mathbb{N}$. Tensors are collectively denoted as @@ -16,7 +16,7 @@ We have a $n \in \mathbb{N}$ finite dimensional vector space $V$ such that $\dim > \mathbf{T} \in \underbrace{V \otimes \dots \otimes V}_p \otimes \underbrace{V^* \otimes \dots \otimes V^*}_q = \mathscr{T}_q^p(V), > $$ > -> with $\mathscr{T}_0^0(V) = \mathbb{K}$. +> with $\mathscr{T}_0^0(V) = F$. We refer to $\mathbf{T} \in \mathscr{T}_q^p(V)$ as a $(p, q)$-tensor; a mixed tensor of **contravariant rank** $p$ and **covariant rank** $q.$ It may be observed that we have $\dim \mathscr{T}_q^p (V) = n^{p+q}$ with $\dim V = n \in \mathbb{N}$. @@ -45,7 +45,7 @@ $$ ## Outer product -> *Definition 3*: the outer product $f \otimes g: X \times Y \to \mathbb{K}$ of two scalar functions $f: X \to \mathbb{K}$ and $g: Y \to \mathbb{K}$ is defined as +> *Definition 3*: the outer product $f \otimes g: X \times Y \to F$ of two scalar functions $f: X \to F$ and $g: Y \to F$ is defined as > > $$ > (f \otimes g)(x,y) = f(x) g(y), @@ -69,7 +69,7 @@ The following statements are given with $p=q=r=s=1$ without loss of generality. From this definition the subsequent theorem follows naturally. -> *Theorem 1*: let $\mathbf{T} \in \mathscr{T}_q^p(V)$ be a tensor, then there exists **holors** $T_j^i \in \mathbb{K}$ such that +> *Theorem 1*: let $\mathbf{T} \in \mathscr{T}_q^p(V)$ be a tensor, then there exists **holors** $T_j^i \in F$ such that > > $$ > \mathbf{T} = T^i_j \mathbf{e}_i \otimes \mathbf{\hat e}^j, @@ -90,11 +90,11 @@ From this definition the subsequent theorem follows naturally. \end{align*} $$ -For $\mathbf{T} \in \mathscr{T}^0_q(V)$ it follows that there exists holors $T_i \in \mathbb{K}$ such that $\mathbf{T} = T_i \mathbf{\hat e}^i$ with $T_i = \mathbf{T}(\mathbf{e}_i)$, are referred to as the **covariant components** of $\mathbf{T}$ relative to a basis $\{\mathbf{e}_i\}$. +For $\mathbf{T} \in \mathscr{T}^0_q(V)$ it follows that there exists holors $T_i \in F$ such that $\mathbf{T} = T_i \mathbf{\hat e}^i$ with $T_i = \mathbf{T}(\mathbf{e}_i)$, are referred to as the **covariant components** of $\mathbf{T}$ relative to a basis $\{\mathbf{e}_i\}$. -For $\mathbf{T} \in \mathscr{T}^p_0(V)$ it follows that there exists holors $T^i \in \mathbb{K}$ such that $\mathbf{T} = T^i \mathbf{e}_i$ with $T^i = \mathbf{T}(\mathbf{\hat e}^i)$, are referred to as the **contravariant components** of $\mathbf{T}$ relative to a basis $\{\mathbf{e}_i\}$. +For $\mathbf{T} \in \mathscr{T}^p_0(V)$ it follows that there exists holors $T^i \in F$ such that $\mathbf{T} = T^i \mathbf{e}_i$ with $T^i = \mathbf{T}(\mathbf{\hat e}^i)$, are referred to as the **contravariant components** of $\mathbf{T}$ relative to a basis $\{\mathbf{e}_i\}$. -If $\mathbf{T} \in \mathscr{T}^p_q(V)$, it follows that there exists holors $T^i_j \in \mathbb{K}$ are coined the **mixed components** of $\mathbf{T}$ relative to a basis $\{\mathbf{e}_i\}$. +If $\mathbf{T} \in \mathscr{T}^p_q(V)$, it follows that there exists holors $T^i_j \in F$ are coined the **mixed components** of $\mathbf{T}$ relative to a basis $\{\mathbf{e}_i\}$. By definition tensors are basis independent. Holors are basis dependent. @@ -135,10 +135,10 @@ We have from theorem 2 that the outer product of two tensors yields another tens ## Inner product -> *Definition 5*: an **inner product** on $V$ is a bilinear mapping $\bm{g}: V \times V \to \mathbb{K}$ which satisfies +> *Definition 5*: an **inner product** on $V$ is a bilinear mapping $\bm{g}: V \times V \to F$ which satisfies > > 1. for all $\mathbf{u}, \mathbf{v} \in V: \; \bm{g}(\mathbf{u}, \mathbf{v}) = \overline{\bm{g}}(\mathbf{v}, \mathbf{u}),$ -> 2. for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\lambda, \mu \in \mathbb{K}: \;\bm{g}(\mathbf{u}, \lambda \mathbf{v} + \mu \mathbf{w}) = \lambda \bm{g}(\mathbf{u}, \mathbf{v}) + \mu \bm{g}(\mathbf{u}, \mathbf{w}),$ +> 2. for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\lambda, \mu \in F: \;\bm{g}(\mathbf{u}, \lambda \mathbf{v} + \mu \mathbf{w}) = \lambda \bm{g}(\mathbf{u}, \mathbf{v}) + \mu \bm{g}(\mathbf{u}, \mathbf{w}),$ > 3. for all $\mathbf{u} \in V\backslash \{\mathbf{0}\}: \bm{g}(\mathbf{u},\mathbf{u}) > 0,$ > 4. for $\mathbf{u} = \mathbf{0} \iff \bm{g}(\mathbf{u},\mathbf{u}) = 0.$ @@ -209,7 +209,7 @@ $$ \mathbf{k}(\mathbf{g}(\mathbf{u}), \mathbf{v}) = \text{g}_{ij} u^i v^k\mathbf{k}(\mathbf{\hat e}^j, \mathbf{e}_k) = \text{g}_{ij} u^i v^k \delta^j_k = \text{g}_{ij} u^i v^j. $$ - Since $u^i, v^j \in \mathbb{K}$ are arbitrary it follows that $\text{g}_{ij} = g_{ij}$. + Since $u^i, v^j \in F$ are arbitrary it follows that $\text{g}_{ij} = g_{ij}$. Consequently, the inverse $\mathbf{g}^{-1}: V^* \to V$ has the property $\mathbf{g}^{-1}(\mathbf{\hat u}) = G^{-1} \mathbf{\hat u}$ for all $\mathbf{\hat u} \in V^*$. The bijective linear map $\mathbf{g}$ is commonly known as the **metric** and $\mathbf{g}^{-1}$ as the **dual metric**. diff --git a/docs/mathematics/linear-algebra/tensors/tensor-symmetries.md b/docs/mathematics/linear-algebra/tensors/tensor-symmetries.md index c555cde..2b24de2 100644 --- a/docs/mathematics/linear-algebra/tensors/tensor-symmetries.md +++ b/docs/mathematics/linear-algebra/tensors/tensor-symmetries.md @@ -1,6 +1,6 @@ # Tensor symmetries -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 $\{\mathbf{\hat e}^i\}$ and a pseudo inner product $\bm{g}$ on $V.$ +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 $\{\mathbf{\hat e}^i\}$, a field F and a pseudo inner product $\bm{g}$ on $V$. ## Symmetric tensors @@ -53,7 +53,7 @@ Alternatively one may write $\bigwedge_q(V) = V^* \otimes_a \cdots \otimes_a V^* It follows from the definitions of symmetric and antisymmetric tensors that for $0$-tensors we have $$ - {\bigvee}_0(V) = {\bigvee}^0(V) = {\bigwedge}_0(V) = {\bigwedge}^0(V) = \mathbb{K}. + {\bigvee}_0(V) = {\bigvee}^0(V) = {\bigwedge}_0(V) = {\bigwedge}^0(V) = F. $$ Furthermore, for $1$-tensors we have @@ -188,6 +188,42 @@ An interesting result of the definition of the symmetric and antisymmetric produ ??? note "*Proof*:" - Will be added later. + We have for the symmetric case -In some literature theorem 2 is used as definition for the symmetric and antisymmetric product from which the relation with the symmetrisation maps can be proven. Either method is valid, however it has been chosen that defining the products in terms of the symmetrisation maps is more general. \ No newline at end of file + $$ + \begin{align*} + \mathbf{\hat u}_1 \vee \mathbf{\hat u}_2 &= 2 \mathscr{S}(\mathbf{\hat u}_1 \otimes \mathbf{\hat u}_2),\\ + &= u^1_{(i} u^2_{j)} \mathbf{\hat e}_i \otimes \mathbf{\hat e}^j,\\ + &= u^1_i u^2_j \mathbf{\hat e}^i \otimes \mathbf{\hat e}^j + u^2_i u^1_j \mathbf{\hat e}^i \otimes \mathbf{\hat e}^j, + \end{align*} + $$ + + such that + + $$ + \begin{align*} + \mathbf{\hat u}_1 \vee \mathbf{\hat u}_2 (\mathbf{v}_1, \mathbf{v}_2) &= u^1_i u^2_j v_1^i v_2^j + u^2_i u^1_j v_1^i v_2^j,\\ + &= \mathrm{perm}\big(\mathbf{k}(\mathbf{\hat u}_i, \mathbf{v}_j)\big). + \end{align*} + $$ + + And we have for the antisymmetric case + + $$ + \begin{align*} + \mathbf{\hat u}_1 \wedge \mathbf{\hat u}_2 &= 2 \mathscr{A}(\mathbf{\hat u}_1 \otimes \mathbf{\hat u}_2),\\ + &= u^1_{[i} u^2_{j]} \mathbf{\hat e}_i \otimes \mathbf{\hat e}^j,\\ + &= u^1_i u^2_j \mathbf{\hat e}^i \otimes \mathbf{\hat e}^j - u^2_i u^1_j \mathbf{\hat e}^i \otimes \mathbf{\hat e}^j, + \end{align*} + $$ + + such that + + $$ + \begin{align*} + \mathbf{\hat u}_1 \wedge \mathbf{\hat u}_2 (\mathbf{v}_1, \mathbf{v}_2) &= u^1_i u^2_j v_1^i v_2^j - u^2_i u^1_j v_1^i v_2^j,\\ + &= \det\big(\mathbf{k}(\mathbf{\hat u}_i, \mathbf{v}_j)\big). + \end{align*} + $$ + +In some literature theorem 2 is used as definition for the symmetric and antisymmetric product from which the relation with the symmetrisation maps can be proven. Either method is valid, however it has been chosen that defining the products in terms of the symmetrisation maps is more general.