diff --git a/.vscode/keybindings.json b/.vscode/keybindings.json index b3729ab..38c9935 100644 --- a/.vscode/keybindings.json +++ b/.vscode/keybindings.json @@ -1,7 +1,23 @@ [ + { + "key": "altgr+4", + "command": "editor.action.insertSnippet", + "args": { + "snippet": "$ $" + }, + "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'" + }, { "key": "ctrl+1", "command": "editor.action.insertSnippet", + "args": { + "snippet": "\\mathbf{}" + }, + "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'" + }, + { + "key": "ctrl+shift+1", + "command": "editor.action.insertSnippet", "args": { "snippet": "\\boldsymbol{}" }, @@ -32,11 +48,19 @@ "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'" }, { - "key": "ctrl+|", + "key": "ctrl+5", "command": "editor.action.insertSnippet", "args": { "snippet": "
\nProof\n\n
" }, "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'" + }, + { + "key": "ctrl+6", + "command": "editor.action.insertSnippet", + "args": { + "snippet": "```math\n\n```" + }, + "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'" } ] \ No newline at end of file diff --git a/README.md b/README.md index a0639aa..daf4534 100644 --- a/README.md +++ b/README.md @@ -1,4 +1,9 @@ ## Run Typescript with Node - Add `"type": "module"` to `package.json` -- Run .ts files with `tsx script.ts` \ No newline at end of file +- Run .ts files with `tsx script.ts` + +## VS Code regex replacements + +\$\$((.|\n)*?)\$\$ +```math\n$1\n``` \ No newline at end of file diff --git a/content/notes/math/differential_geometry.mdx b/content/notes/math/differential_geometry.mdx index 1d68d4d..a0c0512 100644 --- a/content/notes/math/differential_geometry.mdx +++ b/content/notes/math/differential_geometry.mdx @@ -1735,7 +1735,7 @@ $$ (\nabla_X^0 + \nabla_X^1)(f\mathbf{s}) = 2(Xf)\mathbf{s} + f(\nabla_X^0 + \nabla_X^1)\mathbf{s} $$ -Because of the extra factor of $2$ the sum of two connections does not satisfy the Leibniz rule and so is not a connection. However, if we multiply by \nabla_X^0 (f\mathbf{s})$ by $1-t$ and $\nabla_X^1 (f\mathbf{s})$ by $t$, then $(1-t)\nabla_X^0 + t\nabla_X^1$ satisfies the Leibniz rule. +Because of the extra factor of $2$ the sum of two connections does not satisfy the Leibniz rule and so is not a connection. However, if we multiply by $\nabla_X^0 (f\mathbf{s})$ by $1-t$ and $\nabla_X^1 (f\mathbf{s})$ by $t$, then $(1-t)\nabla_X^0 + t\nabla_X^1$ satisfies the Leibniz rule. Any finite linear combination $\sum t_i \nabla^i$ of connections $\nabla^i$ is a connection provided that the coefficients add up to $1$, i.e. $\sum_i t_i = 1$. Such a linear combination is called *convex*. @@ -1920,7 +1920,7 @@ Because $\nabla_{\bar{X}} \bar{\mathbf{s}}$ is a local operator in $\bar{X}$ and ### Connections at a point -Let $\mathcal{F} = C^\infty (M)$ be the ring of smooth functions on a smooth manifold $M$. Suppose $\nabla$ is a connection on a vector bundle $E$ over $M$. For $X\in\mathfrak{X}(M)$ and $\mathbf{s}\in\Gamma(E)$, since $\nabla_X \mathbf{s}$ is $\mathcal{F}$-linear in $X$, it is a point operator in $X$ that can be defined pointwise in $X: There is a unique maps, also denoted by $\nabla$, +Let $\mathcal{F} = C^\infty (M)$ be the ring of smooth functions on a smooth manifold $M$. Suppose $\nabla$ is a connection on a vector bundle $E$ over $M$. For $X\in\mathfrak{X}(M)$ and $\mathbf{s}\in\Gamma(E)$, since $\nabla_X \mathbf{s}$ is $\mathcal{F}$-linear in $X$, it is a point operator in $X$ that can be defined pointwise in $X$: There is a unique maps, also denoted by $\nabla$, $$ \nabla: T_\mathbf{p}M \times\Gamma(E)\to E_\mathbf{p} @@ -1950,13 +1950,13 @@ $$ \nabla^U :\mathfrak{X}(U)\times\Gamma(U,E) \to\Gamma(U,E) $$ -Suppose $U$ is a trivializing open set for $E$ and $\Set{\mathbf{e}_i}_{i=1}^r$ is a frame for $E$ over $U$ and let $X\in\mathfrak{X}(U)$ be a smooth vector field on $U$. On $U$, since any section $\mathbf{s}\in\Gamma(U,E)$ is a linear combination $\mathbf{s} = \sum_{i=1}^r a^j \mathbf{e}_j$, the section $\nabla_X \mathbf{s}$ cab be computed from $\nabla_X \mathbf{e}_j$ by linearity and the Leibniz rule. As a section of $E$ over $U$, $\nabla_X \mathbf{e}_j$ is a linear combination of the $\mathbf{e}_i$ with coefficients $\omega_j^i$ depending on $X$ +Suppose $U$ is a trivializing open set for $E$ and $\Set{\mathbf{e}_i}_{i=1}^r$ is a frame for $E$ over $U$ and let $X\in\mathfrak{X}(U)$ be a smooth vector field on $U$. On $U$, since any section $\mathbf{s}\in\Gamma(U,E)$ is a linear combination $\mathbf{s} = \sum_{i=1}^r a^j \mathbf{e}_j$, the section $\nabla_X \mathbf{s}$ can be computed from $\nabla_X \mathbf{e}_j$ by linearity and the Leibniz rule. As a section of $E$ over $U$, $\nabla_X \mathbf{e}_j$ is a linear combination of the $\mathbf{e}_i$ with coefficients $\omega_j^i$ depending on $X$ $$ \nabla_X \mathbf{e}_j = \sum_{i=1}^r \omega_j^i (X) \mathbf{e}_i $$ -The $\mathcal{F}$-linearity of $\nabla_X \mathbf{e}_j$ in $X$ implies that $\omega_j^i$ is $\mathcal{F}$-linear in $X$ and so $\omega_j^i$ is a $1$-form on $U$. The $1$-forms $\omega_j^i$ on $U$ are called the *connection forms*, and dthe matrix $\omega = [\omega_j^i]$ is called the *connection matrix*, of the connection $\nabla$ relative to the frame $\Set{\mathbf{e}_i}_{i=1}^r$ on $U$. +The $\mathcal{F}$-linearity of $\nabla_X \mathbf{e}_j$ in $X$ implies that $\omega_j^i$ is $\mathcal{F}$-linear in $X$ and so $\omega_j^i$ is a $1$-form on $U$. The $1$-forms $\omega_j^i$ on $U$ are called the *connection forms*, and the matrix $\omega = [\omega_j^i]$ is called the *connection matrix*, of the connection $\nabla$ relative to the frame $\Set{\mathbf{e}_i}_{i=1}^r$ on $U$. Similarly, for $X,Y\in\mathfrak{X}(U)$, the section $R(X,Y)\mathbf{e}_j$ is a linear combination of $\mathbf{e}_i$ @@ -2023,7 +2023,7 @@ $$ \end{align*} $$ -Compaing this with the definition of the curvature form $\Omega_j^i$ gives +Comparing this with the definition of the curvature form $\Omega_j^i$ gives $$ \Omega_j^i = \mathrm{d}\omega_j^i + \sum_k \omega_k^i \wedge \omega_j^k @@ -2271,6 +2271,359 @@ $$ ## Gauss's Theorema Egregium using forms +### The Gauss curvature equation + +Consider a smooth surface $M$ in $\R^3$ and a point $\mathbf{p}\in M$. Let $U$ be an open neighbourhood of $\mathbf{p}$ on which there is an orthonormal frame $\Set{\mathbf{e}_1,\mathbf{e}_2}$. This is always possible by the Gram-Schmidth process, which turns any frame into an orthonormal frame. If $\mathbf{e}_3 = \mathbf{e}_1 \times\mathbf{e}_2$ is a unit normal vector field, then $B = \Set{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3}$ is an orthonormal frame for the vector bundle $T\R^3|_U$ over $U$. + +For the connection $\mathrm{D}$ on the bundle $T\R^3|_M$, let $[\omega_j^i]$ be the connection matrix of $1$-forms relative to the orthonormal frame $B$ over $U$. Since $\mathrm{D}$ is compatible with the metric and the frame $B$ is orthonormal, the matrix $[\omega_j^i]$ is skew-symmetric. Thus, for $X\in\mathfrak{X}(M)$ + +$$ +\begin{align*} + \mathrm{D}_X \mathbf{e}_1 =& -\omega_2^1 (X)\mathbf{e}_2 -\omega_3^1 (X)\mathbf{e}_3 \\ + \mathrm{D}_X \mathbf{e}_2 =& -\omega_2^1 (X)\mathbf{e}_1 -\omega_3^2 (X)\mathbf{e}_3 \\ + \mathrm{D}_X \mathbf{e}_3 =& -\omega_3^1 (X)\mathbf{e}_1 -\omega_3^2 (X)\mathbf{e}_2 +\end{align*} +$$ + +Let $\nabla$ be the Levi-Civita connection on the surface $M$. Recall that for $X,Y\in\mathfrak{X}(M)$, the directional derivative $\mathrm{D}_X Y$ need not be tangent to the surface $M$, and $\nabla_X Y$ is simply the tangential component of $\mathrm{D}_X Y$ + +$$ +\begin{align*} + \nabla_X \mathbf{e}_1 =& -\omega_2^1 (X)\mathbf{e}_2 \\ + \nabla_X \mathbf{e}_2 =& \omega_2^1 (X)\mathbf{e}_1 +\end{align*} +$$ + +Thus the connection matrix of the Levi-Civita connection $\nabla$ on $M$ is + +$$ + \omega = \begin{bmatrix} 0 & \omega_2^1 \\ -\omega_2^1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \omega_2^1 +$$ + +Since + +$$ +\begin{align*} + \omega\wedge\omega =& \begin{bmatrix} 0 & \omega_2^1 \\ -\omega_2^1 & 0 \end{bmatrix} \wedge \begin{bmatrix} 0 & \omega_2^1 \\ -\omega_2^1 & 0 \end{bmatrix} \\ + =& \begin{bmatrix} -\omega_2^1 \wedge \omega_2^1 & 0 \\ 0 & -\omega_2^1 \wedge \omega_2^1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} +\end{align*} +$$ + +the curvature matrix of $\nabla$ is + +$$ +\begin{align*} + \Omega =& \mathrm{d}\omega + \omega\wedge\omega = \mathrm{d}\omega \\ + =& \begin{bmatrix} 0 & \mathrm{d}\omega_2^1 \\ -\mathrm{d}\omega_2^1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \mathrm{d}\omega_2^1 +\end{align*} +$$ + +So the curvature matrix of $\nabla$ is completely described by $\Omega_2^1 = \mathrm{d}\omega_2^1$. Set the unit normal vector field $N$ on $U$ to be $N = -\mathbf{e}_3$. The shape operator becomes + +$$ +\begin{align*} + L(X) =& -\mathrm{D}_X N = \mathrm{D}_X \mathbf{e}_3,\; X\in\mathfrak{X}(M) \\ + =& \omega_3^1 (X)\mathbf{e}_1 + \omega_3^2 (X)\mathbf{e}_2 +\end{align*} +$$ + + +Let $B = \Set{\mathbf{e}_1,\mathbf{e}_2}$ be an orthonormal frame of vector fields on an oriented open subset $U$ of a surface $M$ in $\R^3$, and let $\mathbf{e}_3$ be a unit normal vector field on $U$. Relative to $B$, the curvature form $\Omega_2^1$ of the Levi-Civita connection on $M$ is related to the connection forms of the direction derivative $\mathrm{D}$ on the bundle $T\R^3|_M$ by the Gauss curvature equation + +$$ + \omega_2^1 = \omega_3^1 \wedge \omega_3^2 +$$ + +
+Proof + +Let $\tilde{\Omega_j^i}$ be the curvature forms of the connection $\mathrm{D}$ on $T\R^3|_M$. Because the curvature tensor of $\mathrm{D}$ is zero, the second equation for $\tilde{\Omega}$ gives + +$$ + \tilde{\Omega}_j^i = \mathrm{d}\omega_j^i + \sum_{k=1}^3 \omega_k^i \wedge\omega_j^k = 0 +$$ + +In particular + +$$ + \mathrm{d}\omega_2^1 + \omega_1^1 \wedge\omega_2^1 + \omega_2^1 \wedge \omega_2^2 + \omega_3^1 \wedge \omega_2^3 = 0 +$$ + +Since $\omega_1^1 = \omega_2^2 = 0$, this reduces to + +$$ + \mathrm{d}\omega_2^1 + \omega_3^1 \wedge\omega_2^3 = 0 +$$ + +Since the matrix $[\omega_j^i]$ is skew-symmetric and $\Omega_2^1 = \mathrm{d}\omega_2^1$ + +$$ + \mathrm{d}\omega_2^1 = \omega_3^1 \wedge \omega_3^2 = \Omega_2^1 +$$ +
+ + + +For a smooth surface in $\R^3$, if $\Set{\mathbf{e}_1,\mathbf{e}_2}$ is an orthonormal frame over an oriented subset $U$ of the surface and $\mathbf{e}_3$ is a unit normal vector field on $U$, then the Gaussian curvature $K$ on $U$ is given by + +$$ + K = \det\begin{align*} \omega_3^1 (\mathbf{e}_1) & \omega_3^1 (\mathbf{e}_2) \\ \omega_3^2 (\mathbf{e}_1) & \omega_3^2 (\mathbf{e}_2) \end{align*} = (\omega_3^1 \wedge \omega_3^2)(\mathbf{e}_1,\mathbf{e}_2) +$$ + +
+Proof + +From $L(X) = \omega_3^1 (X)\mathbf{e}_1 + \omega_3^2 (X)\mathbf{e}_2$ for $X\in\mathfrak{X}(U)$ we have + +$$ +\begin{align*} + L(\mathbf{e}_1) =& \omega_3^1 (\mathbf{e}_1)\mathbf{e}_1 + \omega_3^2 (\mathbf{e}_1)\mathbf{e}_2 \\ + L(\mathbf{e}_2) =& \omega_3^1 (\mathbf{e}_2)\mathbf{e}_1 + \omega_3^2 (\mathbf{e}_2)\mathbf{e}_2 +\end{align*} +$$ + +So the matrix of $L$ relative to the frame $\Set{\mathbf{e}_1,\mathbf{e}_2}$ is + +$$ + L = \begin{bmatrix} \omega_3^1 (\mathbf{e}_1) & \omega_3^1 (\mathbf{e}_2) \\ \omega_3^2 (\mathbf{e}_1) & \omega_3^2 \end{bmatrix} +$$ + +Thus + +$$ +\begin{align*} + K =& \det(L) = \begin{vmatrix} \omega_3^1 (\mathbf{e}_1) & \omega_3^1 (\mathbf{e}_2) \\ \omega_3^2 (\mathbf{e}_1) & \omega_3^2 (\mathbf{e}_2) \end{vmatrix} \\ + =& \omega_3^1 (\mathbf{e}_1)\omega_3^2 (\mathbf{e}_2) - \omega_3^1 (\mathbf{e}_2)\omega_3^2 (\mathbf{e}_1) \\ + =& (\omega_3^1 \wedge \omega_3^2)(\mathbf{e}_1,\mathbf{e}_2) +\end{align*} +$$ +
+ + + + +For a smooth surface in $\R^3$, if $\Set{\mathbf{e}_1,\mathbf{e}_2}$ is an orthonormal frame over an open subset $U$ of the surface with dual frame $\Set{\varepsilon^1,\varepsilon^2}$, then the Gaussian curvature $K$ on $U$ is given by + +$$ + K = \Omega_2^1 (\mathbf{e}_1,\mathbf{e}_2) +$$ + +or + +$$ + \mathrm{d}\omega_2^1 = K\varepsilon^1 \wedge\varepsilon^2 +$$ + +
+Proof + +The first formula follows immediately from the previous theorem and the Gauss curvature equation. As for the second formula, since + +$$ + K = K(\varepsilon^1 \wedge\varepsilon^2)(\mathbf{e}_1,\mathbf{e}_2) = \omega_2^1 (\mathbf{e}_1,\mathbf{e}_2) +$$ + +and a $2$-form on $U$ is completely determined by its value on $\mathbf{e}_1,\mathbf{e_2}$ we have $\Omega_2^1 = K\varepsilon^1 \wedge\varepsilon^2$ and since $\Omega_2^1 = \mathrm{d}\omega_2^1$ we get + +$$ + \mathrm{d}\omega_2^1 = K\varepsilon^1 \wedge\varepsilon^2 +$$ +
+ + +Since $\Omega_2^1$ depends only on the metric and not on the embedding of the surface in $\R^3$, then $K = \Omega_2^1 (\mathbf{e}_1,\mathbf{e}_2)$ shows that the same i true of the Gaussian curvature. This formula is therefore a definition of the Gaussian curvature of an abstract Riemannian $2$-manifold. It is consistent with the Theorema Egregium for vector fields, for by the definition of the curvature matrix, if $\Set{\mathbf{e}_1,\mathbf{e}_2}$ is an orthonormal frame on an open subset $U$ of $M$, then + +$$ + R(X,Y)\mathbf{e}_2 = \Omega_2^1(X,Y)\mathbf{e}_1,\; \forall X,Y\in\mathfrak{X}(U) +$$ + +so that + +$$ + \langle R(\mathbf{e}_1,\mathbf{e}_2)\mathbf{e}_2,\mathbf{e}_1\rangle = \langle\Omega_2^1 (\mathbf{e}_1,\mathbf{e}_2)\mathbf{e}_1,\mathbf{e}_2\rangle = \Omega_2^1 (\mathbf{e}_1,\mathbf{e}_2) +$$ + + +The *Gaussian curvature* $K$ at a point $\mathbf{p}$ of a Riemannian 2-manifold $M$ is defined to be + +$$ + K_\mathbf{p} = \langle R_\mathbf{p}(\mathbf{u},\mathbf{v})\mathbf{v},\mathbf{u}\rangle +$$ + +for any orthonormal basis $\Set{\mathbf{u},\mathbf{v}}$ for the tangent plane $T_\mathbf{p}M$. + + +### Skew-symmetries of the curvature tensor + +Recall that the curvature tensor of an affine connection $\nabla$ on a manifold $M$ is defined to be + +$$ +\begin{gather*} + R:\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M) \to\mathfrak{X}(M) \\ + R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]}Z +\end{gather*} +$$ + +We showed that $R(X,Y)Z$ is $\mathcal{F}$-linear in every argument. It is immediate from the definition that the curvature tensor $R(X,Y)Z$ is skew-symmetric in $X$ and $Y$. + + +If an affine connection $\nabla$ on a Riemannian manifold $M$ is compatible with the metric, then for vector fields $X,Y,Z,W\in\mathfrak{X}(M)$, the tensor $\langle R(X,Y)Z,W\rangle$ is skew-symmetric in $Z$ and $W$. + +
+Proof + +Because $\langle R(X,Y)Z,W\rangle$ is a tensor, it is enough to check its skew-symmetry locally, e.g. on an open set $U$ with an orthonormal frame $\Set{\mathbf{e}_i}_{i=1}^n$ (assured by the Gram-Schmidt process). Then + +$$ + \langle R(X,Y)\mathbf{e}_j,\mathbf{e}_i\rangle = \Omega_j^i (X,Y) +$$ + +Since $\nabla$ is compatible with the metric, its curvature matrix $\Omega = [\Omega_j^i]$ relative to an orthonormal frame is skew-symmetric. Thus + +$$ +\begin{align*} + \langle R(X,Y)\mathbf{e}_j,\mathbf{e}_i \rangle =& \omega_j^i (X,Y) = -\Omega_i^j (X,Y) \\ + =& -\langle R(X,Y)\mathbf{e}_1,\mathbf{e}_2 \rangle +\end{align*} +$$ + +On $U$, we can write $Z = \sum_{i=1}^n z^i \mathbf{e}_i$ and $W = \sum_{i=1}^n w^j \mathbf{e}_j$ for some smooth functions $z^i, w^j \in C^\infty (U)$. Then + +$$ +\begin{align*} + \langle R(X,Y)Z,W \rangle =& \sum_{i,j}^n z^i w^j \langle R(X,Y)\mathbf{e}_i, \mathbf{e}_j \rangle \\ + =& -\sum_{i,j}^n z^i w^j \langle R(X,Y)\mathbf{e}_j, \mathbf{e}_i \rangle +\end{align*} +$$ + +We now show that $\langle R(\mathbf{u},\mathbf{v})\mathbf{v},\mathbf{u}\rangle$ is independent of the orthonormal basis $\Set{\mathbf{u},\mathbf{v}}$ for $T_\mathbf{p}M$. Suppse $\Set{\bar{\mathbf{u}},\bar{\mathbf{v}}}$ is another orthonormal basis. Then + +$$ +\begin{align*} + \bar{\mathbf{u}} = a\mathbf{u} + b\mathbf{v} \\ + \bar{\mathbf{v}} = c\mathbf{u} + d\mathbf{v} +\end{align*} +$$ + +for an orthonormal matrix $\mathbf{A} = \left[\begin{smallmatrix}a & b \\ c & d\end{smallmatrix}\right]$ and + +$$ +\begin{align*} + \langle R(\bar{\mathbf{u}},\bar{\mathbf{v}})\bar{\mathbf{v}},\bar{\mathbf{u}}\rangle = \langle \det(\mathbf{A})R(\mathbf{u},\mathbf{v})(c\mathbf{u} + d\mathbf{v}),a\mathbf{u} + d\mathbf{v}\rangle \\ + =& \det(\mathbf{A})^2 \langle R(\mathbf{u},\mathbf{v})\mathbf{v},\mathbf{u}\rangle +\end{align*} +$$ + +by the skew-symmetry of $\langle R(\mathbf{u},\mathbf{v}\mathbf{z},\mathbf{w}\rangle$ in $\mathbf{z}$ and $\mathbf{w}$. Since $\mathbf{A}\in \mathrm{O}(2)$, then $\det(\mathbf{A}) = \pm 1$. Hence + +$$ + \langle R(\bar{\mathbf{u}},\bar{\mathbf{v}})\mathbf{v},\mathbf{u}\rangle = \langle R(\mathbf{u},\mathbf{v})\mathbf{v},\mathbf{u}\rangle +$$ +
+ + +### Sectional curvature + +Let $M$ be a Riemannian manifold and $\mathbf{p}$ a point in $M$. If $P$ is a $2$-dimensional subspace of the tangent space $T_\mathbf{p}M$, then we define the *sectional curvature* of $P$ to be + +$$ + K(P) = \langle R(\mathbf{e}_1,\mathbf{e}_2)\mathbf{e}_2,\mathbf{e}_1\rangle +$$ + +for any orthonormal basis $\Set{\mathbf{e}_1,\mathbf{e}_2}$ of $P$. Just as in the definition of the Gaussian curvature, the right-hand side is independent of the orthonormal basis $\Set{\mathbf{e}_1,\mathbf{e}_2}$. + +If $\Set{\mathbf{u},\mathbf{v}}$ is an arbitrary basis for the $2$-plane $P$, then the sectional curvature of $P$ is given by + +$$ + K(P) = \frac{\langle R(\mathbf{u},\mathbf{v})\mathbf{v},\mathbf{u}\rangle}{\langle\mathbf{u},\mathbf{u}\rangle\langle\mathbf{v},\mathbf{v}\rangle - \langle\mathbf{u},\mathbf{v}\rangle^2} +$$ + +### Poincaré half-plane + + +The Poincaré half-plane is the upper half-plane + +$$ + \mathbb{H}^2 = \Set{(x,y)\in\R^2 | y > 0} +$$ + +with the metric + +$$ + g_{xy} = \frac{\mathrm{d}x \otimes \mathrm{d}x + \mathrm{d}y \otimes \mathrm{d}y}{y^2} +$$ + +With this metric, and orthonormal frame is + +$$ + \mathbf{e}_1 = y\frac{\partial}{\partial x} \quad \mathbf{e}_2 = y\frac{\partial}{\partial y} +$$ + +The dual frame is + +$$ + \varepsilon_1 = \frac{1}{y}\mathrm{d}x \quad \varepsilon_2 = \frac{1}{y}\mathrm{d}y +$$ + +Hence + +$$ + \mathrm{d}\varepsilon^1 = \frac{1}{y^2}\mathrm{d}x \wedge \mathrm{d}y \quad \mathrm{d}\varepsilon^2 = 0 +$$ + +On the Poincaré half-plane the connection form $\omega_2^1$ is a linear combination of $\mathrm{d}x$ and $\mathrm{d}y$, taking the form + +$$ + \omega_2^1 = a\mathrm{d}x + b\mathrm{d}y +$$ + +We will determine the coefficients $a$ and $b$ from the first structural equation + +$$ +\begin{align*} + \mathrm{d}\varepsilon^1 =& -\omega_2^1 \wedge\varepsilon^2 \\ + \mathrm{d}\varepsilon^2 =& -\omega_1^2 \wedge\varepsilon^1 = \omega_2^1 \wedge\varepsilon^1 +\end{align*} +$$ + +Combining the results give + +$$ +\begin{align*} + \frac{1}{y^2}\mathrm{d}x \wedge\mathrm{d}y =& \mathrm{d}\varepsilon^1 \\ + =& -(a\mathrm{d}x + b\mathrm{d}y)\wedge \frac{1}{y}\mathrm{d}y \\ + =& -\frac{a}{y}\mathrm{d}x\wedge\mathrm{d}y +\end{align*} +$$ + +showing that $a = -\frac{1}{y}$. Furthermore + +$$ +\begin{align*} + 0 =& \mathrm{d}\varepsilon^2 = \left(-\frac{1}{y}\mathrm{d}x + b\mathrm{d}y \right)\wedge\frac{1}{y}\mathrm{d}x \\ + =& -\frac{b}{y}\mathrm{d}x\wedge\mathrm{d}y +\end{align*} +$$ + +giving $b = 0$. Thus + +$$ +\begin{align*} + \omega_2^1 =& -\frac{1}{y}\mathrm{d}x \\ + \mathrm{d}\omega_2^1 =& \frac{1}{y^2}\mathrm{d}y\wedge\mathrm{d}x = -\frac{1}{y^2}\mathrm{d}x\wedge\mathrm{d}y +\end{align*} +$$ + +By definition, the Gaussian curvature of the Poincaré half-plane is + +$$ +\begin{align*} + K =& \Omega_2^1 (\mathbf{e}_1,\mathbf{e}_2) = -\frac{1}{y^2} (\mathrm{d}x\wedge\mathrm{d}y)\left(y\frac{\partial}{\partial x}, y\frac{\partial}{\partial y} \right) \\ + =& -(\mathrm{d}x\wedge\mathrm{d}y)\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right) = -1 +\end{align*} +$$ + + # Geodesics # Partial derivative (basis vector) diff --git a/content/notes/math/functional_analysis.mdx b/content/notes/math/functional_analysis.mdx index 62cc46f..d445794 100644 --- a/content/notes/math/functional_analysis.mdx +++ b/content/notes/math/functional_analysis.mdx @@ -23,7 +23,7 @@ Let $(X, d)$ be a metric space. The open ball about a point $x\in X$ with radius $$ B_\epsilon (x) := \Set{ y \in X | d(x, y) < \epsilon } -$$ +$$ diff --git a/content/notes/math/logic.mdx b/content/notes/math/logic.mdx index d9a2070..6e46a46 100644 --- a/content/notes/math/logic.mdx +++ b/content/notes/math/logic.mdx @@ -90,7 +90,7 @@ Arguments in subjective logic are called subjected opinions, commonly shortened $$ \omega_X^A = (b_X, u_X, a_X) -$$ +$$ where - $b_X:\mathcal{R}(\Omega)\to[0,1]$ is a belief mass distribution over the possible state values of $X$, defined on the reduced power set $\mathcal{R}(\Omega) := \mathcal{P}(\Omega)\setminus\Set{\Omega, \emptyset}$ diff --git a/content/notes/math/real_analysis.mdx b/content/notes/math/real_analysis.mdx index 99f1860..66a3026 100644 --- a/content/notes/math/real_analysis.mdx +++ b/content/notes/math/real_analysis.mdx @@ -729,7 +729,7 @@ Let $f, g: I \subseteq \R \to \R$ be differentiable at $x_0 \in I$. Then $f \cdo $$ (f \cdot g)'\left(x_0\right) = f\left(x_0\right) g'\left(x_0\right) + f'\left(x_0\right) g\left(x_0\right) -$$ +$$
Proof diff --git a/content/notes/math/statistics.mdx b/content/notes/math/statistics.mdx index 61484ce..ef6d25b 100644 --- a/content/notes/math/statistics.mdx +++ b/content/notes/math/statistics.mdx @@ -898,7 +898,7 @@ Suppose $C(\mathbf{x})$ is a $1 - \alpha$ level confidence set for $\theta$. Con $$ H_0:\theta = \theta_0 \textrm{ versus }H_1:\theta\neq\theta_0 -$$ +$$ The following test has significance level $\alpha$ for the hypothesis: Reject $H_0$ if and only if $\theta_0\notin C(\mathbf{x})$. diff --git a/content/notes/math/stochastic_processes.mdx b/content/notes/math/stochastic_processes.mdx index 7346283..96d7e2f 100644 --- a/content/notes/math/stochastic_processes.mdx +++ b/content/notes/math/stochastic_processes.mdx @@ -3306,7 +3306,7 @@ Recall that the positive part of $x\in\R$ is $$ x^+ = x\vee 0 = \begin{cases} x,\quad x > 0 \\ 0,\quad x \leq 0 \end{cases} -$$ +$$ Suppose $\mathbf{X}$ is a sub-martingale. For $t\in T$, diff --git a/content/notes/math/tensor_analysis.mdx b/content/notes/math/tensor_analysis.mdx index 64f0f9d..7ba5207 100644 --- a/content/notes/math/tensor_analysis.mdx +++ b/content/notes/math/tensor_analysis.mdx @@ -2242,7 +2242,7 @@ $$ &= \left[ \partial_a \left( \Gamma_{bc}^i \right) - \partial_b \left( \Gamma_{ac}^d \right) + \Gamma_{bc}^i \Gamma_{ai}^d - \Gamma_{ac}^j \Gamma_{bj}^d \right] \boldsymbol{\partial}_d \\ &= R_{cab}^d \boldsymbol{\partial}_d \end{align*} -$$ +$$ ## Symmetries diff --git a/content/notes/physics/electromagnetism.mdx b/content/notes/physics/electromagnetism.mdx index 4b8bec4..1915013 100644 --- a/content/notes/physics/electromagnetism.mdx +++ b/content/notes/physics/electromagnetism.mdx @@ -2033,7 +2033,7 @@ $$ k = \frac{\omega}{c} \\ E_x = cB_y \quad E_y = - cB_x \end{gather*} -$$ +$$ implying that the waves travel at speed $c$ and are nondispersive. Assuming vacuum in the transmission line, the field amplitudes correspond to the static fields for an infinite line charge and and infinite straigth current diff --git a/content/notes/physics/quantum_mechanics.mdx b/content/notes/physics/quantum_mechanics.mdx index b059259..91a1060 100644 --- a/content/notes/physics/quantum_mechanics.mdx +++ b/content/notes/physics/quantum_mechanics.mdx @@ -54,7 +54,7 @@ The Hermitian conjugate of an operator $\hat{A}$ is defined as $$ \langle \varphi | \hat{A} \psi\rangle = \langle \hat{A}^\dagger \varphi | \psi \rangle -$$ +$$ or alternatively diff --git a/content/notes/physics/spacetime_physics.mdx b/content/notes/physics/spacetime_physics.mdx index 8a1e648..2a7b152 100644 --- a/content/notes/physics/spacetime_physics.mdx +++ b/content/notes/physics/spacetime_physics.mdx @@ -8,9 +8,9 @@ showToc: true | Form | Multivector expansion | Terms | | --- | :-: | --- | -| Graded | $M = \alpha + v + \mathbf{F} + wI + \beta I$ | $$\begin{aligned} \alpha&:\textrm{ scalar} \\ v&:\textrm{ vector} \\ \mathbf{F}&:\textrm{ bivector} \\ wI&:\textrm{ pseudovector} \\ \beta I&:\textrm{ pseudoscalar} \end{aligned}$$ | -| Complex | $M = \zeta + z + \mathbf{F}$ | $$\begin{aligned} \zeta = \alpha + \beta I&:\textrm{ complex scalar} \\ z = v + wI&:\textrm{ complex vector} \\ \mathbf{F} = \mathbf{f}e^{\varphi I}&:\textrm{ complex bivector} \end{aligned}$$ | -| Relative | $M = \left[ \left(\alpha + \vec{E} \right) + \left( \delta + \mathbf{p} \right)\gamma_0 \right] + \left[\left( \beta + \vec{B} \right) + \left( \omega + \mathbf{a} \right)\gamma_0 \right]I$ | $$\begin{aligned} \alpha&:\textrm{ proper scalar} \\ \vec{E}&:\textrm{ relative polar 3-vector part of }\mathbf{F} \\ \delta&:\textrm{ relative scalar part of }v \\ \mathbf{p}&:\textrm{ relative polar 3-vector part of }v \\ \beta I&:\textrm{ proper pseudoscalar} \\ \vec{B}&:\textrm{ relative axial 3-vector part of }\mathbf{F} \\ \omega I&:\textrm{ relative pseudoscalar part of }wI \\ \mathbf{a}&:\textrm{ relative axial 3-vector part of }wI \end{aligned}$$ | +| Graded | $M = \alpha + v + \mathbf{F} + wI + \beta I$ | $$\begin{aligned} \alpha&:\textrm{ scalar} \\ v&:\textrm{ vector} \\ \mathbf{F}&:\textrm{ bivector} \\ wI&:\textrm{ pseudovector} \\ \beta I&:\textrm{ pseudoscalar} \end{aligned}$$| +| Complex | $M = \zeta + z + \mathbf{F}$ | $$\begin{aligned} \zeta = \alpha + \beta I&:\textrm{ complex scalar} \\ z = v + wI&:\textrm{ complex vector} \\ \mathbf{F} = \mathbf{f}e^{\varphi I}&:\textrm{ complex bivector} \end{aligned}$$| +| Relative | $M = \left[ \left(\alpha + \vec{E} \right) + \left( \delta + \mathbf{p} \right)\gamma_0 \right] + \left[\left( \beta + \vec{B} \right) + \left( \omega + \mathbf{a} \right)\gamma_0 \right]I$ | $$\begin{aligned} \alpha&:\textrm{ proper scalar} \\ \vec{E}&:\textrm{ relative polar 3-vector part of }\mathbf{F} \\ \delta&:\textrm{ relative scalar part of }v \\ \mathbf{p}&:\textrm{ relative polar 3-vector part of }v \\ \beta I&:\textrm{ proper pseudoscalar} \\ \vec{B}&:\textrm{ relative axial 3-vector part of }\mathbf{F} \\ \omega I&:\textrm{ relative pseudoscalar part of }wI \\ \mathbf{a}&:\textrm{ relative axial 3-vector part of }wI \end{aligned}$$| ## Spacetime The principle of invariant light speed, which toghether with the principle of relativity form the postulates of special relativity, predicates that the scalar time $t$ and vector spatial coordinates $\mathbf{x} := \left(x_1, x_2, x_3\right)$ form an invariant interval $(ct)^2 - \mathbf{x}^2$ under coordinate transformations. This motivates the concept of spacetime in which physical quantities are described by 4-vectors $x = (ct, \mathbf{x}) = (ct, x_1, x_2, x_3)\in\R^{1,3}$ combining the scaled time $ct$ with the spatial component $\mathbf{x}$. For compatibility with the principle of invariant light speed, the squared norm of a 4-vector $x\in\R^{1,3}$ is defined to equal the invariant interval, i.e. $|x|^2 = (ct)^2 - \mathbf{x}^2$. This invariant interval can be generalized into a bilinear form, $\eta: \R^{1,3}\to\R^{1,3}\to\mathbf{R}$ with $\eta = \mathrm{diag}(1, -1, -1, -1)$, called the Minkowski metric (inner product). For two 4-vectors $u = (u_0, u_1, u_2, u_3), v = (v_0, v_1, v_2, v_3) \in\R^{1,3}$ this metric has the form @@ -137,7 +137,7 @@ where $R := mn$ defines a rotor satisfying $R\tilde{R} = \tilde{R} = 1$. | Grade | Orthonormal basis | Blade type | Geometry | | :-: | :-: | --- | --- | | 4 | $\color{red}{\gamma_{0123}} = I$ | pseudoscalars | 4-volumes | -| 3 | $$\begin{aligned}\color{red}{\gamma_{123}} &= \gamma_0 I \\ \color{blue}{\gamma_{230}} &= \gamma_1 I \\ \color{blue}{\gamma_{310}} &= \gamma_2 I \\ \color{blue}{\gamma_{120}} &= \gamma_3 I\end{aligned}$$ | pseudovectors | 3-volumes | +| 3 | $$\begin{aligned}\color{red}{\gamma_{123}} &= \gamma_0 I \\ \color{blue}{\gamma_{230}} &= \gamma_1 I \\ \color{blue}{\gamma_{310}} &= \gamma_2 I \\ \color{blue}{\gamma_{120}} &= \gamma_3 I\end{aligned}$$| pseudovectors | 3-volumes | | 2 | $\color{red}{\gamma_{10}}\;\color{red}{\gamma_{20}}\;\color{red}{\gamma_{30}}\;\color{blue}{\gamma_{23}}\;\color{blue}{\gamma_{31}}\;\color{blue}{\gamma_{12}}$ | bivectors | planes | | 1 | $\color{blue}{\gamma_{0}}\;\color{red}{\gamma_{1}}\;\color{red}{\gamma_{2}}\;\color{red}{\gamma_{3}}$ | vectors | lines | | 0 | $\color{blue}{1}$ | scalars | points | @@ -254,7 +254,7 @@ Any trivector $\mathfrak{F}$ can be written as a product of a vector and the pse $$ \mathfrak{F} = f_0\gamma_{123} + f_1\gamma_{230} + f_2\gamma_{310} + f_3\gamma_{120} = \left(\sum_{\mu=0}^3 f_\mu \gamma_\mu\right) I = fI -$$ +$$ where $f$ is a vector. The expanded form of a multivector can thus be reformulated as diff --git a/content/notes/physics/special_relativity.mdx b/content/notes/physics/special_relativity.mdx index b839fd2..bf6cde3 100644 --- a/content/notes/physics/special_relativity.mdx +++ b/content/notes/physics/special_relativity.mdx @@ -221,7 +221,7 @@ where $\beta = \frac{v}{c}$ and $$ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{\sqrt{1 - \beta^2}} -$$ +$$ is the Lorentz factor. In matrix form, the Lorentz boost is written as