diff --git a/docs/en/mathematics/index.md b/docs/en/mathematics/index.md index 806d996..d2ee104 100755 --- a/docs/en/mathematics/index.md +++ b/docs/en/mathematics/index.md @@ -19,3 +19,4 @@ The mathematics sections of this wiki are based on various books and lectures. A * The section of calculus is based on the lectures of Luc Habets and the book Calculus by Robert Adams. * The sections of linear algebra and complex numbers in number theory are based on the lectures and lecture notes of Rik Kaasschieter and the book Linear Algebra by Steven Leon. * The sections of multivariable calculus and ordinary differential equations are based on the lectures and lecture notes of Georg Prokert and the book Calculus by Robert Adams. +* The sections of dual vector spaces, tensors and differential geometry are based on the lectures and lecture notes of Luc Florack. \ No newline at end of file diff --git a/docs/en/mathematics/linear-algebra/dual-vector-spaces.md b/docs/en/mathematics/linear-algebra/dual-vector-spaces.md index 27977ae..16d2a0f 100644 --- a/docs/en/mathematics/linear-algebra/dual-vector-spaces.md +++ b/docs/en/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 in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) 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 $\{\mathbf{e}_i\}_{i=1}^n.$ In the following sections we make use of the Einstein summation convention introduced in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}$. -> *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 $\lambda, \mu \in \mathbb{K}$ +> *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 > > $$ > \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 $V$. +> *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$. Each basis $\{\mathbf{e}_i\}$ of $V$ therefore induces a basis $\{\mathbf{\hat e}^i\}$ of $V^*$ by diff --git a/docs/en/mathematics/linear-algebra/inner-product-spaces.md b/docs/en/mathematics/linear-algebra/inner-product-spaces.md index 4febcc9..bd69cf8 100644 --- a/docs/en/mathematics/linear-algebra/inner-product-spaces.md +++ b/docs/en/mathematics/linear-algebra/inner-product-spaces.md @@ -4,7 +4,7 @@ An introduction of length in a vector space may be formulated in terms of an inner product space. -> *Definition 1*: an **inner product** $V$ is an operation on $V$ that assigns, to each pair of vectors $\mathbf{x},\mathbf{y} \in V$ a real number $\langle \mathbf{x},\mathbf{y}\rangle$ satisfying the following conditions +> *Definition 1*: an **inner product** on $V$ is an operation on $V$ that assigns, to each pair of vectors $\mathbf{x},\mathbf{y} \in V$, a real number $\langle \mathbf{x},\mathbf{y}\rangle$ satisfying the following conditions > > 1. $\langle \mathbf{x},\mathbf{x}\rangle > 0, \text{ for } \mathbf{x} \in V\backslash\{\mathbf{0}\} \text{ and } \langle \mathbf{x},\mathbf{x}\rangle = 0, \; \text{for } \mathbf{x} = \mathbf{0}$, > 2. $\langle \mathbf{x},\mathbf{y}\rangle = \overline{\langle \mathbf{y},\mathbf{x}\rangle}, \; \forall \mathbf{x}, \mathbf{y} \in V$, diff --git a/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md b/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md index e69de29..c47e97e 100644 --- a/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md +++ b/docs/en/mathematics/linear-algebra/tensors/tensor-formalism.md @@ -0,0 +1,241 @@ +# 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 introduced in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) and $\mathbb{K} = \mathbb{R} \lor\mathbb{K} = \mathbb{C}.$ + +## 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}, +> $$ +> +> with $p, q \in \mathbb{N}$. Tensors are collectively denoted as +> +> $$ +> \mathbf{T} = \underbrace{V \otimes \dots \otimes V}_p \otimes \underbrace{V^* \otimes \dots \otimes V^*}_q = \mathscr{T}_q^p(V), +> $$ +> +> with $\mathscr{T}_0^0(V) = \mathbb{K}$. + +We refer to $\mathbf{T} \in \mathscr{T}_q^p(V)$ as a $(p, q)$-tensor; a mixed tensor of **contravariant rank** $p$ and **covariant rank** $q.$ It may be observed that we have $\dim \mathscr{T}_q^p (V) = n^{p+q}$ with $\dim V = n \in \mathbb{N}$. + +It follows from definition 1 and by virtue of the isomorphism between $V^{**}$ and $V$ that $\mathbf{T} \in \mathscr{T}_1^0(V) = V^*$ is a covector and $\mathbf{T} \in \mathscr{T}_0^1(V) = V$ is a vector. + +## Kronecker tensor + +> *Definition 2*: let $\mathbf{k} \in \mathscr{T}_1^1(V)$ be the **Kronecker tensor** be defined such that +> +> $$ +> \mathbf{k}(\mathbf{\hat e}^i, \mathbf{e}_j) = \delta^i_j, +> $$ +> +> with $\delta_j^i$ the Kronecker symbol. + +Let $\mathbf{\hat u} = u_i \mathbf{\hat e}^i \in V^*$ and $\mathbf{v} = v^j \mathbf{e}_j \in V$ then the tensor properties and the definition of the Kronecker tensor imply that + +$$ +\begin{align*} + \mathbf{k}(\mathbf{\hat u}, \mathbf{v}) &= \mathbf{k}(u_i \mathbf{\hat e}^i, v^j \mathbf{e}_j), \\ + &= u_i v^j \mathbf{k}(\mathbf{\hat e}^i, \mathbf{e}_j), \\ + &= u_i v^j \delta^i_j, \\ + &= u_i v^i. +\end{align*} +$$ + +## Outer product + +> *Definition 3*: the outer product $f \otimes g: X \times Y \to \mathbb{K}$ of two scalar functions $f: X \to \mathbb{K}$ and $g: Y \to \mathbb{K}$ is defined as +> +> $$ +> (f \otimes g)(x,y) = f(x) g(y), +> $$ +> +> for all $(x,y) \in X \times Y$. + +The outer product is associative, distributive with respect to addition and scalar multiplication, but not commutative. + +Note that although the same symbol is used for the outer product and the denotion of a tensor space, these are not equivalent. But are closely related. + +For the following statements we take $p=q=r=s=1$ without loss of generality. + +> *Definition 4*: the mixed $(p, q)$-tensor $\mathbf{e}_i \otimes \mathbf{\hat e}^j \in \mathscr{T}_q^p(V)$ is defined as +> +> $$ +> (\mathbf{e}_i \otimes \mathbf{\hat e}^j)(\mathbf{\hat u}, \mathbf{v}) = \mathbf{k}(\mathbf{\hat u}, \mathbf{e}_i) \mathbf{k}(\mathbf{\hat e}^j, \mathbf{v}), +> $$ +> +> for all $(\mathbf{\hat u}, \mathbf{v}) \in V^* \times V$. + +From this definition the subsequent theorem follows naturally. + +> *Theorem 1*: let $\mathbf{T} \in \mathscr{T}_q^p(V)$ be a tensor, then there exists **holors** $T_j^i \in \mathbb{K}$ such that +> +> $$ +> \mathbf{T} = T^i_j \mathbf{e}_i \otimes \mathbf{\hat e}^j, +> $$ +> +> with $T^i_j = \mathbf{T}(\mathbf{\hat e}^i, \mathbf{e}_j)$. + +??? note "*Proof*:" + + Let $\mathbf{T} \in \mathscr{T}_q^p(V)$ such that + + $$ + \begin{align*} + \mathbf{T}(\mathbf{\hat e}^i, \mathbf{e}_j) &= T^k_l (\mathbf{e}_k \otimes \mathbf{\hat e}^l)(\mathbf{\hat e}^i, \mathbf{e}_j), \\ + &= T^k_l \mathbf{k}(\mathbf{\hat e}^i, \mathbf{e}_k) \mathbf{k}(\mathbf{\hat e}^l,\mathbf{e}_j), \\ + &= T^k_l \delta^i_k \delta^l_j, \\ + &= T^i_j. + \end{align*} + $$ + +For $\mathbf{T} \in \mathscr{T}^0_q(V)$ it follows that there exists holors $T_i \in \mathbb{K}$ such that $\mathbf{T} = T_i \mathbf{\hat e}^i$ with $T_i = \mathbf{T}(\mathbf{e}_i)$, are referred to as the **covariant components** of $\mathbf{T}$ relative to a basis $\{\mathbf{e}_i\}$. + +For $\mathbf{T} \in \mathscr{T}^p_0(V)$ it follows that there exists holors $T^i \in \mathbb{K}$ such that $\mathbf{T} = T^i \mathbf{e}_i$ with $T^i = \mathbf{T}(\mathbf{\hat e}^i)$, are referred to as the **contravariant components** of $\mathbf{T}$ relative to a basis $\{\mathbf{e}_i\}$. + +If $\mathbf{T} \in \mathscr{T}^p_q(V)$, it follows that there exists holors $T^i_j \in \mathbb{K}$ are coined the **mixed components** of $\mathbf{T}$ relative to a basis $\{\mathbf{e}_i\}$. + +By definition tensors are basis independent. Holors are basis dependent. + +> *Theorem 2*: let $\mathbf{S} \in \mathscr{T}^p_q(V)$ and $\mathbf{T} \in \mathscr{T}^r_s(V)$ be tensors with +> +> $$ +> \mathbf{S} = S^i_j \mathbf{e}_i \otimes \mathbf{\hat e}^j \quad \land \quad \mathbf{T} = T^r_s \mathbf{e}_r \otimes \mathbf{\hat e}^s, +> $$ +> +> then the outer product of $\mathbf{S}$ and $\mathbf{T}$ is given by +> +> $$ +> \mathbf{S} \otimes \mathbf{T} = S^i_j T^k_l \mathbf{e}_i \otimes \mathbf{e}_k \otimes \mathbf{\hat e}^j \otimes \mathbf{\hat e}^l, +> $$ +> +> with $\mathbf{S} \otimes \mathbf{T} \in \mathscr{T}^{p+r}_{q+s}(V)$. + +??? note "*Proof*:" + + Let $\mathbf{S} \in \mathscr{T}^p_q(V)$ and $\mathbf{T} \in \mathscr{T}^r_s(V)$ with + + $$ + \mathbf{S} = S^i_j \mathbf{e}_i \otimes \mathbf{\hat e}^j \quad \land \quad \mathbf{T} = T^r_s \mathbf{e}_r \otimes \mathbf{\hat e}^s, + $$ + + then + + $$ + \begin{align*} + \mathbf{S} \otimes \mathbf{T} &= S^i_j (\mathbf{e}_i \otimes \mathbf{\hat e}^j) \otimes T^r_s (\mathbf{e}_r \otimes \mathbf{\hat e}^s), \\ + &= S^i_j T^r_s \mathbf{e}_i \otimes \mathbf{e}_r \otimes \mathbf{\hat e}^j \otimes \mathbf{\hat e}^s. + \end{align*} + $$ + + Which maps two vectors and two covectors, therefore $\mathbf{S} \otimes \mathbf{T} \in \mathscr{T}^{p+r}_{q+s}(V)$. + +We have from theorem 2 that the outer product of two tensors yields another tensor, with ranks adding up. + +## Inner product + +> *Definition 5*: a **pseudo inner product** on $V$ is a nondegenerate bilinear mapping $\bm{g}: V \times V \to \mathbb{K}$ which satisfies +> +> 1. for all $\mathbf{u} \in V \backslash \{\mathbf{0}\} \exists \mathbf{v} \in V: \; \bm{g}(\mathbf{u},\mathbf{v}) \neq 0$, +> 2. for all $\mathbf{u}, \mathbf{v} \in V: \; \bm{g}(\mathbf{u}, \mathbf{v}) = \overline{\bm{g}}(\mathbf{v}, \mathbf{u})$, +> 3. for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $\lambda, \mu \in \mathbb{K}: \;\bm{g}(\mathbf{u}, \lambda \mathbf{v} + \mu \mathbf{w}) = \lambda \bm{g}(\mathbf{u}, \mathbf{v}) + \mu \bm{g}(\mathbf{u}, \mathbf{w}).$ + +It may be observed that $\bm{g} \in \mathscr{T}_2^0$. Unlike the Kronecker tensor, the existance of an inner product is never implied. + +> *Definition 6*: let $G$ be the Gram matrix with its components $G \overset{\text{def}}= (g_{ij})$ defined as +> +> $$ +> g_{ij} = \bm{g}(\mathbf{e}_i, \mathbf{e}_j). +> $$ + +For $\mathbf{u} = u^i \mathbf{e}_i, \mathbf{v} = v^j \mathbf{e}_j \in V$ we then have + +$$ +\begin{align*} + \bm{g}(\mathbf{u}, \mathbf{v}) &= \bm{g}(u^i \mathbf{e}_i, v^j \mathbf{e}_j), \\ + &= u^i v^j \bm{g}(\mathbf{e}_i, \mathbf{e}_j), \\ + &\overset{\text{def}}= u^i v^j g_{ij}. +\end{align*} +$$ + +> *Proposition 1*: the Gram matrix $G$ is symmetric and nonsingular such that +> +> $$ +> g^{ik} g_{kj} = \delta^i_j, +> $$ +> +> with $G^{-1} \overset{\text{def}}= (g^{ij})$. + +??? note "*Proof*:" + + Let $G$ be the Gram matrix, symmetry of $G$ follows from defintion 5. Suppose that $G$ is singular, then there exists $\mathbf{u} = u^i \mathbf{e}_i \in V \backslash \{\mathbf{0}\}$ such that $G \mathbf{u} = \mathbf{0} \implies u^i g_{ij} = 0$, as a result we find that + + $$ + \forall \mathbf{v} = v^j \mathbf{e}_j \in V: 0 = u^i g_{ij} v^j = u^i \bm{g}(\mathbf{e}_i, \mathbf{e}_j) v^j = \bm{g}(u^i \mathbf{e}_i, v^j \mathbf{e}_j) = \bm{g}(\mathbf{u}, \mathbf{v}), + $$ + + which contradicts the non-degeneracy of the pseudo inner product in definition 5. + +> *Theorem 3*: there exists a bijective linear map $\mathbf{g}: V \to V^*$ with inverse $\mathbf{g}^{-1}$ such that +> +> 1. $\forall \mathbf{u}, \mathbf{v} \in V: \; \bm{g}(\mathbf{u}, \mathbf{v}) = \mathbf{k}(\mathbf{g}(\mathbf{u}), \mathbf{v})$, +> 2. $\forall \mathbf{\hat u} \in V^*, \mathbf{v} \in V: \; \bm{g}(\mathbf{g}^{-1}(\mathbf{\hat u}), \mathbf{v}) = \mathbf{k}(\mathbf{\hat u}, \mathbf{v})$, +> +> with $\mathbf{g}(\mathbf{v}) = G \mathbf{v}$ for all $\mathbf{v} \in V$. + +??? note "*Proof*:" + + Let $\mathbf{u} \in V$ and let $\mathbf{\hat u} \in V^*$, suppose $\mathbf{\hat u}: \mathbf{v} \mapsto \bm{g}(\mathbf{u}, \mathbf{v})$ then we may define $\mathbf{g}: V \to V^*: \mathbf{u} \mapsto \mathbf{g}(\mathbf{u}) \overset{\text{def}} = \mathbf{\hat u}$. + + Let $\mathbf{v} \in V \backslash \{\mathbf{0}\}: \mathbf{g}(\mathbf{v}) = \mathbf{0}$, then + + $$ + 0 = \mathbf{k}(\mathbf{g}(\mathbf{v}), \mathbf{w}) \overset{\text{def}} = \bm{g}(\mathbf{v}, \mathbf{w}), + $$ + + for all $\mathbf{w} \in V$, which contradicts the non-degeneracy of the pseude inner product in definition 5. Hence $\mathbf{g}$ is injective, since $\dim V$ is finite $\mathbf{g}$ is also bijective. + + Let $\mathbf{u} = u^i \mathbf{e}_i, \mathbf{v} = v^j \mathbf{e}_j \in V$ and define $\mathbf{g}(\mathbf{e}_i) = \text{g}_{ij} \mathbf{\hat e}^j$ such that + + $$ + \mathbf{k}(\mathbf{g}(\mathbf{u}), \mathbf{v}) \overset{\text{def}} = \bm{g}(\mathbf{u}, \mathbf{v}) = g_{ij} u^i v^j, + $$ + + but also + + $$ + \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. + $$ + + Since $u^i, v^j \in \mathbb{K}$ are arbitrary it follows that $\text{g}_{ij} = g_{ij}$. + +Consequently the inverse $\mathbf{g}^{-1}: V^* \to V$ has the property $\mathbf{g}^{-1}(\mathbf{\hat u}) = G^{-1} \mathbf{\hat u}$ for all $\mathbf{\hat u} \in V^*$. The bijective linear map $\mathbf{g}$ is commonly known as the **metric** and $\mathbf{g}^{-1}$ as the **dual metric**. + +It follows from theorem 3 that for $\mathbf{u} = u^i \mathbf{e}_i \in V$ and $\mathbf{\hat u} = u_i \mathbf{\hat e}^i \in V^*$ we have + +$$ + \mathbf{g}(\mathbf{u}) = g_{ij} u^i \mathbf{\hat e}^j = u_j \mathbf{\hat e}^j = \mathbf{\hat u}, +$$ + +with $u_j = g_{ij} u^i$ and + +$$ + \mathbf{g}^{-1}(\mathbf{\hat u}) = g^{ij} u_i \mathbf{e}_j = u^j \mathbf{e}_j = \mathbf{u}, +$$ + +with $u^j = g^{ij} u_i$. + +> *Definition 7*: the basis $\{\mathbf{e}_i\}$ of $V$ induces a **reciprocal basis** $\{\mathbf{g}^{-1}(\mathbf{\hat e}^i)\}$ of $V$ given by +> +> $$ +> \mathbf{g}^{-1}(\mathbf{\hat e}^i) = g^{ij} \mathbf{e}_j. +> $$ +> +> Likewise the basis $\{\mathbf{\hat e}^i\}$ of $V^*$ induces a **reciprocal dual basis** $\{\mathbf{g}(\mathbf{e}_i)\}$ of $V^*$ given by +> +> $$ +> \mathbf{g}(\mathbf{e}^i) = g_{ij} \mathbf{\hat e}^j. +> $$ + +Sofar, a vector space $V$ and its associated dual space $V^*$ have been introduced as a priori independent entities. An inner product provides us with an explicit mechanism to construct a bijective linear mapping associated with each vector by virtue of the metric. \ No newline at end of file