diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index e6ffb85..ce895dd 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -97,6 +97,8 @@ nav: - 'Tensor symmetries': mathematics/linear-algebra/tensors/tensor-symmetries.md - 'Volume forms': mathematics/linear-algebra/tensors/volume-forms.md - 'Tensor transformations': mathematics/linear-algebra/tensors/tensor-transformations.md + - 'Topology': + - 'Fiber bundles': mathematics/topology/fiber-bundles.md - 'Calculus': - 'Limits': mathematics/calculus/limits.md - 'Continuity': mathematics/calculus/continuity.md @@ -125,6 +127,8 @@ nav: - 'The Laplace transform': mathematics/ordinary-differential-equations/laplace-transform.md - 'Differential geometry': - 'Differential manifolds': mathematics/differential-geometry/differential-manifolds.md + - 'Tangent spaces': mathematics/differential-geometry/tangent-spaces.md + - 'Transformations': mathematics/differential-geometry/transformations.md - 'Linear connections': mathematics/differential-geometry/linear-connections.md - 'Derivatives': mathematics/differential-geometry/derivatives.md - 'Torsion': mathematics/differential-geometry/torsion.md diff --git a/docs/en/mathematics/differential-geometry/differential-manifolds.md b/docs/en/mathematics/differential-geometry/differential-manifolds.md index c921915..3f71cea 100644 --- a/docs/en/mathematics/differential-geometry/differential-manifolds.md +++ b/docs/en/mathematics/differential-geometry/differential-manifolds.md @@ -1,6 +1,6 @@ # Differential manifolds -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 $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}.$ +In the following sections of differential geometry we make use of the Einstein summation convention introduced in [vector analysis](/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates/) and $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}.$ ## Definition @@ -16,9 +16,9 @@ Differential geometry is concerned with *differential manifolds*, smooth continu The last axiom ensures that any chart is tacitly assumed to be already contained in the atlas. -## Transformations +## Coordinate transformations -> *Definition 2*: let $p,q \in \mathrm{M}$ be points on the differential manifold and let $\psi: \mathscr{D}(\psi) \to\mathrm{M}: p \mapsto \psi(p) \overset{\text{def}}{=} q$ be a **transformation** on the manifold, we define two diffeomorphisms +> *Definition 2*: let $p,q \in \mathrm{M}$ be points on the differential manifold and let $\psi: \mathscr{D}(\psi) \to\mathrm{M}: p \mapsto \psi(p) \overset{\text{def}}{=} q$ be a **transformation** from $p$ to $q$ on the manifold, we define two diffeomorphisms > > $$ > \phi_\alpha: \mathscr{D}(\phi_\alpha) \to \mathscr{R}(\phi_\alpha): p \mapsto \phi_\alpha(p) \overset{\text{def}}{=} x, @@ -38,191 +38,4 @@ The last axiom ensures that any chart is tacitly assumed to be already contained To clarify the definitions, a passive transformation corresponds only to a descriptive transformation. Whereas an active transformation corresponds to a transformation on the manifold $M$. -A passive transformation may also be given directly by $\phi_\beta \circ \phi_\alpha: x \mapsto y$ since $\psi = \mathrm{id}$ in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms. - -## Fiber bundles - -(This subsection should probably be moved to a more general setting of manifolds.) - -> *Definition 3*: a **fiber** $V_x$ at a point $x \in \mathrm{M}$ on a manifold is a finite dimensional vector space. With the collection of fibers $V_x$ for all $x \in \mathrm{M}$ define the **fiber bundle** as -> -> $$ -> V = \bigcup_{x \in \mathrm{M}} V_x. -> $$ - -Then by definition we have the projection map $\pi$ given by - -$$ - \pi: V \to\mathrm{M}: (x,\mathbf{v}) \mapsto \pi(x, \mathbf{v}) \overset{\text{def}}{=} x, -$$ - -and its inverse - -$$ - \pi^{-1}:\mathrm{M} \to V: x \mapsto \pi(x) \overset{\text{def}}{=} V_x. -$$ - -Similarly, a dual fiber $V_x^*$ may be defined for $x \in \mathrm{M}$, with its fiber bundle defined by - -$$ - V^* = \bigcup_{x \in \mathrm{M}} V_x^*. -$$ - -> *Definition 4*: a **tensor fiber** $\mathscr{B}_x$ at a point $x \in \mathrm{M}$ on a manifold is defined as -> -> $$ -> \mathscr{B}_x = \bigcup_{p,q \in \mathbb{N}} \mathscr{T}^p_q(V_x). -> $$ -> -> With the collection of tensor fibers $\mathscr{B}_x$ for all $x \in \mathrm{M}$ define the **tensor fiber bundle** as -> -> $$ -> \mathscr{B} = \bigcup_{x \in \mathrm{M}} \mathscr{B}_x. -> $$ - -Then for a point $x \in \mathrm{M}$ we have a tensor $\mathbf{T} \in \mathscr{B}_x$ such that - -$$ - \mathbf{T} = T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k, -$$ - -with $T^{ij}_k \in \mathbb{K}$ holors of $\mathbf{T}$. Furthermore, $\{\mathbf{e}_i\}_{i=1}^n$ a basis of $V_x$ and $\{\mathbf{\hat e}^i\}_{i=1}^n$ a basis of $V_x^*$. - -> *Definition 5*: a tensor field $\mathbf{T}$ on a manifold $M$ is a [section]() -> -> $$ -> \mathbf{T} \in \Gamma(\mathrm{M}, \mathscr{B}), -> $$ -> -> of the tensor fiber bundle $\mathscr{B}$. - -Therefore, a tensor field assigns a tensor fiber (or tensor) to each point on a section of the manifold. These tensors may vary smoothly along the section of the manifold. - -## Tangent bundles - -> *Definition 6*: let $f \in C^{\infty}(\mathrm{M})$ with $C^{\infty}$ the class of [smooth functions]() and $M$ a differential manifold. A derivation of $f$ at $x \in \mathrm{M}$ is defined as a linear map $\mathbf{v}_x: C^\infty(\mathrm{M}) \to \mathbb{K}$ that satisfies -> -> $$ -> \forall f,g \in C^{\infty}(\mathrm{M}): \mathbf{v}_x(f g) = (\mathbf{v}_xf) g + f (\mathbf{v}_x g). -> $$ -> -> Let $\mathrm{T}_x\mathrm{M}$ be the set of all derivations at $x$ such that $\mathbf{v}_x \in \mathrm{T}_x\mathrm{M}$. With $\mathrm{T}_x\mathrm{M}$ denoted as the **tangent space** at $x$. - -We may think of the tangent space at a point $x \in \mathrm{M}$ as a space attached to $x$ on the differential manifold $M$. - -> *Theorem 1*: let $M$ be a differential manifold and let $x \in \mathrm{M}$, the tangent space $\mathrm{T}_x\mathrm{M}$ is a vector space. - -??? note "*Proof*:" - - Will be added later. - -Thus, the tangent space is a vector space attached to $x \in \mathrm{M}$ on the differential manifold. It follows that its vectors have interesting properties. - -> *Theorem 2*: let $M$ be a differential manifold, let $x \in \mathrm{M}$ and let $\mathbf{v}_x \in \mathrm{T}_x\mathrm{M}$, then we have that -> -> $$ -> \forall f \in C^{\infty}(\mathrm{M}): \mathbf{v}_x f = v^i \partial_i f(x), -> $$ -> -> such that $\mathbf{v}_x = v^i \partial_i \in \mathrm{T}_x\mathrm{M}$ is denoted as a **tangent vector** in the tangent space $\mathrm{T}_x\mathrm{M}$. - - -??? note "*Proof*:" - - Will be added later. - -Theorem 2 adds the notion of tangent vectors to the explanation of the tangent space. The tangent space at a point on the manifold thus represents the space of tangent vectors. - -> *Proposition 1*: let $M$ be a differential manifold of $\dim\mathrm{M} = n \in \mathbb{N}$. The tangent space $\mathrm{T}_x\mathrm{M}$ has dimension $n$ such that -> -> $$ -> \forall x \in \mathrm{M}: \dim \mathrm{T}_x\mathrm{M} = \dim\mathrm{M} -> $$ -> -> and is span by the vector basis $\{\partial_i\}_{i=1}^n$. - -??? note "*Proof*:" - - Will be added later. - -Proposition 1 states that the tangent space is of the same dimension as the manifold and its basis are partial derivative operators. In the context of the [covariant basis](), this definition of the basis leaves out the coordinate map, but is in fact equivalent to the covariant basis. - -As a last step in the explanation, we may think of the 2 dimensional surface of a sphere, which may define a differential manifold $M$. The tangent space at a point $x \in \mathrm{M}$ on the surface of the sphere may then be compared to the tangent plane to the sphere attached at point $x \in \mathrm{M}$. The catch is that the 3 dimensional space necessary to understand this construction exists only in our imagination and not in the mathematical construct. - -> *Definition 7*: let $M$ be a differential manifold, the collection of tangent spaces $\mathrm{T}_x\mathrm{M}$ for all $x \in \mathrm{M}$ define the **tangent bundle** as -> -> $$ -> \mathrm{TM} = \bigcup_{x \in \mathrm{M}} \mathrm{T}_x\mathrm{M}. -> $$ - -In particular, we may think of the tangent bundle $\mathrm{TM}$ as a subspace $\mathrm{TM} \subset V$ of the vector bundle $V$ for a differential manifold. With the special properties given in theorem 2 and proposition 1. - -The connection of each tangent vector to its base point may be formalised with the projection map $\pi$ which in this case is given by - -$$ - \pi: \mathrm{TM} \to\mathrm{M}: (x, \mathbf{v}) \mapsto \pi(x, \mathbf{v}) \overset{\text{def}}{=} x, -$$ - -and its inverse - -$$ - \pi^{-1}:\mathrm{M} \to \mathrm{TM}: x \mapsto \pi^{-1}(x) \overset{\text{def}}{=} \mathrm{T}_x\mathrm{M}. -$$ - -> *Definition 8*: a vector field $\mathbf{v}$ on a differential manifold $M$ is a section -> -> $$ -> \mathbf{v} \in \Gamma(\mathrm{TM}), -> $$ -> -> of the tangent bundle $\mathrm{TM}$. - -## Cotangent bundles - -> *Definition 9*: let $M$ be a differential manifold and $\mathrm{T}_x\mathrm{M}$ the tangent space at $x \in \mathrm{M}$. We define the **cotangent space** $\mathrm{T}_x^*\mathrm{M}$ as the dual space of $\mathrm{T}_x\mathrm{M}$ -> -> $$ -> \mathrm{T}_x^*\mathrm{M} = (\mathrm{T}_x\mathrm{M})^*. -> $$ -> -> Then every element $\bm{\omega}_x \in \mathrm{T}_x^*\mathrm{M}$ is a linear map $\bm{\omega}_x: \mathrm{T}_x\mathrm{M} \to \mathbb{K}$ denoted as the **cotangent vector**. - -This definition is a logical consequence of the notion of the [dual vector space](). It then also follows that the dual cotangent space is isomorphic to the tangent space at a point $x \in \mathrm{M}$. - -> *Theorem 3*: let $\mathrm{M}$ be a differential manifold of $\dim \mathrm{M} = n \in \mathbb{N}$, then we have that for every $x \in \mathrm{M}$ the basis $\{dx^i\}_{i=1}^n$ of $\mathrm{T}_x^*\mathrm{M}$ is uniquely determined by -> -> $$ -> dx^i(\partial_j) = \delta^i_j, -> $$ -> -> for each basis $\{\partial_j\}_{j=1}^n$ in $\mathrm{T}_x\mathrm{M}$. - -??? note "*Proof*:" - - The proof follows directly from theorem 1 in [dual vector spaces](). - -The choice of $dx^i$ can be explained by taking the differential $df = \partial_i f dx^i \in \mathrm{T}_x^*\mathrm{M}$ with $f \in C^\infty(\mathrm{M})$. Then if we take - -$$ - \mathbf{k}_x(df, \mathbf{v}) = \mathbf{k}(\partial_i f dx^i, v^j \partial_j) = v^j \partial_i f \mathbf{k}(dx^i, \partial_j) = v^j \partial_i f \delta^i_j = v^i \partial_i f = \mathbf{v} f, -$$ - -with $\mathbf{k}_x: \mathrm{T}_x^*\mathrm{M} \times \mathrm{T}_x\mathrm{M} \to \mathbb{K}$ the Kronecker tensor at $x \in \mathrm{M}$. Which shows that defining the basis of the cotangent space as differentials corresponds with respect to the basis of the tangent space. - -So, a cotangent vector $\bm{\omega}_x \in \mathrm{T}_x^*\mathrm{M}$ may be decomposed into - -$$ - \bm{\omega}_x = \omega_i dx^i. -$$ - -## Push forward and pull back - -> *Definition 10*: let $\mathrm{M}, \mathrm{N}$ be two differential manifolds with $\dim \mathrm{N} \geq \dim \mathrm{M}$ and let $\psi: \mathrm{M} \to \mathrm{N}$ be the diffeomorphism between the manifolds. Then we define the **pull back** $\psi^*$ and **push forward** $\psi_*$ operators, such that for $\mathbf{v} \in \mathrm{T}_x \mathrm{M}$ and $\bm{\omega} \in \mathrm{T}_{\psi(x)}^* \mathrm{M}$ we have -> -> $$ -> \mathbf{k}_x(\psi^* \bm{\omega}, \mathbf{v}) = \mathbf{k}_{\psi(x)}(\bm{\omega}, \psi_* \mathbf{v}), -> $$ -> -> for all $x \in \mathrm{M}$. - -Which indicates the proper separation between the elements of both spaces. \ No newline at end of file +A passive transformation may also be given directly by $\phi_\beta \circ \phi_\alpha: x \mapsto y$ since $\psi = \mathrm{id}$ in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms. \ No newline at end of file diff --git a/docs/en/mathematics/differential-geometry/linear-connections.md b/docs/en/mathematics/differential-geometry/linear-connections.md index e69de29..0af3b84 100644 --- a/docs/en/mathematics/differential-geometry/linear-connections.md +++ b/docs/en/mathematics/differential-geometry/linear-connections.md @@ -0,0 +1,18 @@ +# Linear connections + +Let $\mathrm{M}$ be a differential manifold with $\dim \mathrm{M} = n \in \mathbb{N}$ used throughout the section. Let $\mathrm{TM}$ and $\mathrm{T^*M}$ denote the tangent and cotangent bundle. + +> *Definition 1*: a **linear connection** on the fiber bundle $\mathscr{B}$ is a map +> +> $$ +> \nabla: \Gamma(\mathrm{TM}) \times \Gamma(\mathscr{B}) \to \Gamma(\mathscr{B}): (\mathbf{v}, \mathbf{T}) \mapsto \nabla_\mathbf{v} \mathbf{T}, +> $$ +> +> satisfying the following properties, if $f,g \in C^\infty(\mathrm{M})$, $\mathbf{v} \in \mathrm{TM}$ and $\mathbf{T}, \mathbf{S} \in \mathscr{B}$ then +> +> 1. $\nabla_{f\mathbf{v}} \mathbf{T} = f \nabla_\mathbf{v} \mathbf{T}$ +> 2. $\nabla_\mathbf{v} (f \mathbf{T} + g \mathbf{S}) = (\nabla_\mathbf{v} f) \mathbf{T} + f \nabla_\mathbf{v} \mathbf{T} + (\nabla_\mathbf{v} g) \mathbf{S} + g \nabla_{\mathbf{v}} \mathbf{S}$, +> 3. $\nabla_\mathbf{v} f = \mathbf{v} f = \mathbf{k}(df, \mathbf{v})$. + +From property 3 it becomes clear that $\nabla_\mathbf{v}$ is an analogue of a directional derivative. + diff --git a/docs/en/mathematics/differential-geometry/tangent-spaces.md b/docs/en/mathematics/differential-geometry/tangent-spaces.md new file mode 100644 index 0000000..41b37cc --- /dev/null +++ b/docs/en/mathematics/differential-geometry/tangent-spaces.md @@ -0,0 +1,136 @@ +# Tangent spaces + +Let $\mathrm{M}$ be a differential manifold with $\dim \mathrm{M} = n \in \mathbb{N}$ used throughout the section. + +## Definition + +> *Definition 1*: let $f \in C^{\infty}(\mathrm{M})$ with $C^{\infty}$ the class of [smooth functions]() and $M$ a differential manifold. A derivation of $f$ at $x \in \mathrm{M}$ is defined as a linear map $\mathbf{v}_x: C^\infty(\mathrm{M}) \to \mathbb{K}$ that satisfies +> +> $$ +> \forall f,g \in C^{\infty}(\mathrm{M}): \mathbf{v}_x(f g) = (\mathbf{v}_xf) g + f (\mathbf{v}_x g). +> $$ +> +> Let $\mathrm{T}_x\mathrm{M}$ be the set of all derivations at $x$ such that $\mathbf{v}_x \in \mathrm{T}_x\mathrm{M}$. With $\mathrm{T}_x\mathrm{M}$ denoted as the **tangent space** at $x$. + +We may think of the tangent space at a point $x \in \mathrm{M}$ as a space attached to $x$ on the differential manifold $M$. + +## Properties of tangent spaces + +> *Theorem 1*: let $M$ be a differential manifold and let $x \in \mathrm{M}$, the tangent space $\mathrm{T}_x\mathrm{M}$ is a vector space. + +??? note "*Proof*:" + + Will be added later. + +Thus, the tangent space is a vector space attached to $x \in \mathrm{M}$ on the differential manifold. It follows that its vectors have interesting properties. + +> *Theorem 2*: let $M$ be a differential manifold, let $x \in \mathrm{M}$ and let $\mathbf{v}_x \in \mathrm{T}_x\mathrm{M}$, then we have that +> +> $$ +> \forall f \in C^{\infty}(\mathrm{M}): \mathbf{v}_x f = v^i \partial_i f(x), +> $$ +> +> such that $\mathbf{v}_x = v^i \partial_i \in \mathrm{T}_x\mathrm{M}$ is denoted as a **tangent vector** in the tangent space $\mathrm{T}_x\mathrm{M}$. + + +??? note "*Proof*:" + + Will be added later. + +Theorem 2 adds the notion of tangent vectors to the explanation of the tangent space. The tangent space at a point on the manifold thus represents the space of tangent vectors. + +> *Proposition 1*: let $M$ be a differential manifold of $\dim\mathrm{M} = n \in \mathbb{N}$. The tangent space $\mathrm{T}_x\mathrm{M}$ has dimension $n$ such that +> +> $$ +> \forall x \in \mathrm{M}: \dim \mathrm{T}_x\mathrm{M} = \dim\mathrm{M} +> $$ +> +> and is span by the vector basis $\{\partial_i\}_{i=1}^n$. + +??? note "*Proof*:" + + Will be added later. + +Proposition 1 states that the tangent space is of the same dimension as the manifold and its basis are partial derivative operators. In the context of the [covariant basis](), this definition of the basis leaves out the coordinate map, but is in fact equivalent to the covariant basis. + +As a last step in the explanation, we may think of the 2 dimensional surface of a sphere, which may define a differential manifold $M$. The tangent space at a point $x \in \mathrm{M}$ on the surface of the sphere may then be compared to the tangent plane to the sphere attached at point $x \in \mathrm{M}$. The catch is that the 3 dimensional space necessary to understand this construction exists only in our imagination and not in the mathematical construct. + +## Tangent bundle + +> *Definition 2*: let $M$ be a differential manifold, the collection of tangent spaces $\mathrm{T}_x\mathrm{M}$ for all $x \in \mathrm{M}$ define the **tangent bundle** as +> +> $$ +> \mathrm{TM} = \bigcup_{x \in \mathrm{M}} \mathrm{T}_x\mathrm{M}. +> $$ + +In particular, we may think of the tangent bundle $\mathrm{TM}$ as a subspace $\mathrm{TM} \subset V$ of the fiber bundle $V$ for a differential manifold. With the special properties given in theorem 2 and proposition 1. + +The connection of each tangent vector to its base point may be formalised with the projection map $\pi$ which in this case is given by + +$$ + \pi: \mathrm{TM} \to\mathrm{M}: (x, \mathbf{v}) \mapsto \pi(x, \mathbf{v}) \overset{\text{def}}{=} x, +$$ + +and its inverse + +$$ + \pi^{-1}:\mathrm{M} \to \mathrm{TM}: x \mapsto \pi^{-1}(x) \overset{\text{def}}{=} \mathrm{T}_x\mathrm{M}. +$$ + +> *Definition 3*: a vector field $\mathbf{v}$ on a differential manifold $M$ is a section +> +> $$ +> \mathbf{v} \in \Gamma(\mathrm{TM}), +> $$ +> +> of the tangent bundle $\mathrm{TM}$. + +## Cotangent spaces + +> *Definition 4*: let $M$ be a differential manifold and $\mathrm{T}_x\mathrm{M}$ the tangent space at $x \in \mathrm{M}$. We define the **cotangent space** $\mathrm{T}_x^*\mathrm{M}$ as the dual space of $\mathrm{T}_x\mathrm{M}$ +> +> $$ +> \mathrm{T}_x^*\mathrm{M} = (\mathrm{T}_x\mathrm{M})^*. +> $$ +> +> Then every element $\bm{\omega}_x \in \mathrm{T}_x^*\mathrm{M}$ is a linear map $\bm{\omega}_x: \mathrm{T}_x\mathrm{M} \to \mathbb{K}$ denoted as the **cotangent vector**. + +This definition is a logical consequence of the notion of the [dual vector space](). It then also follows that the dual cotangent space is isomorphic to the tangent space at a point $x \in \mathrm{M}$. + +> *Theorem 3*: let $\mathrm{M}$ be a differential manifold of $\dim \mathrm{M} = n \in \mathbb{N}$, then we have that for every $x \in \mathrm{M}$ the basis $\{dx^i\}_{i=1}^n$ of $\mathrm{T}_x^*\mathrm{M}$ is uniquely determined by +> +> $$ +> dx^i(\partial_j) = \delta^i_j, +> $$ +> +> for each basis $\{\partial_j\}_{j=1}^n$ in $\mathrm{T}_x\mathrm{M}$. + +??? note "*Proof*:" + + The proof follows directly from theorem 1 in [dual vector spaces](). + +The choice of $dx^i$ can be explained by taking the differential $df = \partial_i f dx^i \in \mathrm{T}_x^*\mathrm{M}$ with $f \in C^\infty(\mathrm{M})$. Then if we take + +$$ + \mathbf{k}_x(df, \mathbf{v}) = \mathbf{k}(\partial_i f dx^i, v^j \partial_j) = v^j \partial_i f \mathbf{k}(dx^i, \partial_j) = v^j \partial_i f \delta^i_j = v^i \partial_i f = \mathbf{v} f, +$$ + +with $\mathbf{k}_x: \mathrm{T}_x^*\mathrm{M} \times \mathrm{T}_x\mathrm{M} \to \mathbb{K}$ the Kronecker tensor at $x \in \mathrm{M}$. Which shows that defining the basis of the cotangent space as differentials corresponds with respect to the basis of the tangent space. + +So, a cotangent vector $\bm{\omega}_x \in \mathrm{T}_x^*\mathrm{M}$ may be decomposed into + +$$ + \bm{\omega}_x = \omega_i dx^i. +$$ + +In the context of the [contravariant basis](), this definition of the basis leaves out the coordinate map, but is in fact equivalent to the contravariant basis. + +## Cotangent bundle + +> *Definition 5*: let $M$ be a differential manifold, the collection of cotangent spaces $\mathrm{T}_x^*\mathrm{M}$ for all $x \in \mathrm{M}$ define the **cotangent bundle** as +> +> $$ +> \mathrm{T^*M} = \bigcup_{x \in \mathrm{M}} \mathrm{T}_x^*\mathrm{M}. +> $$ + +Thus, we may think of the cotangent bundle $\mathrm{T^*M}$ as a subspace $\mathrm{T^*M} \subset V^*$ of the dual fiber bundle $V^*$ for a differential manifold. \ No newline at end of file diff --git a/docs/en/mathematics/differential-geometry/transformations.md b/docs/en/mathematics/differential-geometry/transformations.md new file mode 100644 index 0000000..a605e47 --- /dev/null +++ b/docs/en/mathematics/differential-geometry/transformations.md @@ -0,0 +1,35 @@ +# Transformations + +Let $\mathrm{M}$ be a differential manifold with $\dim \mathrm{M} = n \in \mathbb{N}$ used throughout the section. Let $\mathrm{TM}$ and $\mathrm{T^*M}$ denote the tangent and cotangent bundle. + +## Push forward and pull back + +> *Definition 1*: let $\mathrm{M}, \mathrm{N}$ be two differential manifolds with $\dim \mathrm{N} \geq \dim \mathrm{M}$ and let $\psi: \mathrm{M} \to \mathrm{N}$ be the diffeomorphism between the manifolds. Then we define the **pull back** $\psi^*$ and **push forward** $\psi_*$ operators, such that for $\mathbf{v} \in \mathrm{T}_x \mathrm{M}$ and $\bm{\omega} \in \mathrm{T}_{\psi(x)}^* \mathrm{M}$ we have +> +> $$ +> \mathbf{k}_x(\psi^* \bm{\omega}, \mathbf{v}) = \mathbf{k}_{\psi(x)}(\bm{\omega}, \psi_* \mathbf{v}), +> $$ +> +> for all $x \in \mathrm{M}$. + +Which indicates the proper separation between the elements of both spaces. + +## Basis transformation + +Let $\psi: \mathscr{D}(\mathrm{M}) \to \mathrm{M}: x \mapsto \psi(x) \overset{\text{def}}{=} \overline{x}$ be an active coordinate transformation from a point $x$ to a point $\overline{x}$ on $\mathrm{M}$. Then we have a basis $\{\partial_i\}_{i=1}^n \subset \mathrm{T}_x\mathrm{M}$ for the tangent space $\mathrm{T}_x\mathrm{M}$ at $x$ and a basis $\{\overline{\partial_i}\}_{i=1}^n \subset \mathrm{T}_{\overline{x}}\mathrm{M}$ for the tangent space $\mathrm{T}_{\overline{x}}\mathrm{M}$ at $\overline{x}$. Which are related by + +$$ + \partial_i = J^j_i \overline{\partial_j} = \partial_i \psi^j(x) \overline{\partial_j}, +$$ + +with $J^j_i = \partial_i \psi^j(x)$ the [Jacobian]() at $x \in \mathrm{M}$. For it to make sense, it helps to change notation to + +$$ + \frac{\partial}{\partial x_i} = \frac{\partial \overline{x}^j}{\partial x_i} \frac{\partial}{\partial \overline{x}_j} = \frac{\partial \psi^j}{\partial x_i} \frac{\partial}{\partial \overline{x}_j}. +$$ + +Similarly, we have a basis $\{dx^i\}_{i=1}^n \subset \mathrm{T}_x^*\mathrm{M}$ for the cotangent space $\mathrm{T}_x\mathrm{M}$ at $x$ and a basis $\{d\overline{x}^i\}_{i=1}^n \subset \mathrm{T}_{\overline{x}}^*\mathrm{M}$ for the cotangent space $\mathrm{T}_{\overline{x}}^*\mathrm{M}$ at $\overline{x}$. Which are related by + +$$ + d\overline{x}^i = J^i_j dx^j = \partial_j \psi^i(x) dx^j. +$$ \ No newline at end of file diff --git a/docs/en/mathematics/topology/fiber-bundles.md b/docs/en/mathematics/topology/fiber-bundles.md new file mode 100644 index 0000000..1ee96e2 --- /dev/null +++ b/docs/en/mathematics/topology/fiber-bundles.md @@ -0,0 +1,57 @@ +# Fiber bundles + +Let $X$ be a manifold over a field $F$. + +> *Definition 1*: a **fiber** $V_x$ at a point $x \in X$ on a manifold is a finite dimensional vector space. With the collection of fibers $V_x$ for all $x \in X$ define the **fiber bundle** as +> +> $$ +> V = \bigcup_{x \in X} V_x. +> $$ + +Then by definition we have the projection map $\pi$ given by + +$$ + \pi: V \to X: (x,\mathbf{v}) \mapsto \pi(x, \mathbf{v}) \overset{\text{def}}{=} x, +$$ + +and its inverse + +$$ + \pi^{-1}: X \to V: x \mapsto \pi(x) \overset{\text{def}}{=} V_x. +$$ + +Similarly, a dual fiber $V_x^*$ may be defined for $x \in X$, with its fiber bundle defined by + +$$ + V^* = \bigcup_{x \in X} V_x^*. +$$ + +> *Definition 2*: a **tensor fiber** $\mathscr{B}_x$ at a point $x \in X$ on a manifold is defined as +> +> $$ +> \mathscr{B}_x = \bigcup_{p,q \in \mathbb{N}} \mathscr{T}^p_q(V_x). +> $$ +> +> With the collection of tensor fibers $\mathscr{B}_x$ for all $x \in X$ define the **tensor fiber bundle** as +> +> $$ +> \mathscr{B} = \bigcup_{x \in X} \mathscr{B}_x. +> $$ + +Then for a point $x \in X$ we have a tensor $\mathbf{T} \in \mathscr{B}_x$ such that + +$$ + \mathbf{T} = T^{ij}_k \mathbf{e}_i \otimes \mathbf{e}_j \otimes \mathbf{\hat e}^k, +$$ + +with $T^{ij}_k \in \mathbb{K}$ holors of $\mathbf{T}$. Furthermore, we have a basis $\{\mathbf{e}_i\}_{i=1}^n$ of $V_x$ and a basis $\{\mathbf{\hat e}^i\}_{i=1}^n$ of $V_x^*$. + +> *Definition 3*: a tensor field $\mathbf{T}$ on a manifold $X$ is a [section]() +> +> $$ +> \mathbf{T} \in \Gamma(X, \mathscr{B}), +> $$ +> +> of the tensor fiber bundle $\mathscr{B}$. + +Therefore, a tensor field assigns a tensor fiber (or tensor) to each point on a section of the manifold. These tensors may vary smoothly along the section of the manifold. \ No newline at end of file