Small fixes to sample notebook.

Author: kpeeters

contrib/structure_equations_and_bianchi.cnb

@@ -1,5 +1,5 @@
 {
-	"cell_id" : 14643089922803689914,
+	"cell_id" : 2082837756733559988,
 	"cells" : 
 	[
 		{
@@ -9,14 +9,14 @@
 			"cells" : 
 			[
 				{
-					"cell_id" : 18043264383205729330,
+					"cell_id" : 6179245722104386386,
 					"cell_origin" : "client",
 					"cell_type" : "latex_view",
-					"source" : "\\part*{Cartan structural equations and Bianchi identity}\n\n\\author{Oscar Castillo-Felisola}\n\n\\section*{Theoretical background}\n\nLet $M$ be a manifold, and $g$ a (semi)Riemannian metric defined on $M$. Then the line element for the metric $g$ is\n\\begin{equation*}\n  \\mathrm{d}{s}^2(g) = g_{\\mu\\nu} \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu.\n\\end{equation*}\nNonetheless, the information about the metric structure of the manifold can be translated to the language of frames,\n\\begin{equation*}\n  \\begin{split}\n    \\mathrm{d}{s}^2(g)\n    & = g_{\\mu\\nu} \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu \\\\\n    & = \\eta_{ab} \\; e^{a}_{\\mu}(x) \\, e^{b}_{\\nu}(x) \\; \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu \\\\\n    & = \\eta_{ab} \\; \\mathrm{e}^{a} \\otimes \\mathrm{e}^{b}.\n  \\end{split}\n\\end{equation*}\nTherefore, the vielbein 1-form, $\\mathrm{e}^{a} \\equiv e^{a}_{\\mu}(x) \\mathrm{d}{x}^\\mu$, encodes the information of the metric tensor.\nIn order to complete the structure, one needs information about the transport of geometrical objects lying on bundles\nbased on $M$. That information is encoded on the spin connection 1-form, $\\omega^{a}{}_{b}$. Using these quantities one finds\nthe generalisation of the structure equations of Cartan,\n\\begin{align}\n  \\mathrm{d}{\\mathrm{e}^{a}} + \\omega^{a}{}_{b} \\wedge \\mathrm{e}^{b} & = \\mathrm{T}^{a},\n  \\label{firstSE}\\\\\n  \\mathrm{d}{\\omega^{a}{}_{c}} + \\omega^{a}{}_{b} \\wedge \\omega^{b}{}_{c} & = \\mathrm{R}^{a}{}_{c}\n  \\label{secondSE}.\n\\end{align}\nThe torsion 2-form, $\\mathrm{T}^{a}$, and the curvature 2-form, $\\mathrm{R}^{a}{}_{c}$, measure the impossibility of endowing $M$ with\nan Euclidean structure.\n\n\\section*{Manipulation of the structural equations}\n\n\\subsection*{Definitions}"
+					"source" : "\\section*{Cartan structural equations and Bianchi identity}\n\n\\author{Oscar Castillo-Felisola}\n\n\\section*{Theoretical background}\n\nLet $M$ be a manifold, and $g$ a (semi)Riemannian metric defined on $M$. Then the line element for the metric $g$ is\n\\begin{equation*}\n  \\mathrm{d}{s}^2(g) = g_{\\mu\\nu} \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu.\n\\end{equation*}\nNonetheless, the information about the metric structure of the manifold can be translated to the language of frames,\n\\begin{equation*}\n  \\begin{split}\n    \\mathrm{d}{s}^2(g)\n    & = g_{\\mu\\nu} \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu \\\\\n    & = \\eta_{ab} \\; e^{a}_{\\mu}(x) \\, e^{b}_{\\nu}(x) \\; \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu \\\\\n    & = \\eta_{ab} \\; \\mathrm{e}^{a} \\otimes \\mathrm{e}^{b}.\n  \\end{split}\n\\end{equation*}\nTherefore, the vielbein 1-form, $\\mathrm{e}^{a} \\equiv e^{a}_{\\mu}(x) \\mathrm{d}{x}^\\mu$, encodes the information of the metric tensor.\nIn order to complete the structure, one needs information about the transport of geometrical objects lying on bundles\nbased on $M$. That information is encoded on the spin connection 1-form, $\\omega^{a}{}_{b}$. Using these quantities one finds\nthe generalisation of the structure equations of Cartan,\n\\begin{align}\n  \\mathrm{d}{\\mathrm{e}^{a}} + \\omega^{a}{}_{b} \\wedge \\mathrm{e}^{b} & = \\mathrm{T}^{a},\n  \\label{firstSE}\\\\\n  \\mathrm{d}{\\omega^{a}{}_{c}} + \\omega^{a}{}_{b} \\wedge \\omega^{b}{}_{c} & = \\mathrm{R}^{a}{}_{c}\n  \\label{secondSE}.\n\\end{align}\nThe torsion 2-form, $\\mathrm{T}^{a}$, and the curvature 2-form, $\\mathrm{R}^{a}{}_{c}$, measure the impossibility of endowing $M$ with\nan Euclidean structure.\n\n\\section*{Manipulation of the structural equations}\n\n\\subsection*{Definitions}"
 				}
 			],
 			"hidden" : true,
-			"source" : "\\part*{Cartan structural equations and Bianchi identity}\n\n\\author{Oscar Castillo-Felisola}\n\n\\section*{Theoretical background}\n\nLet $M$ be a manifold, and $g$ a (semi)Riemannian metric defined on $M$. Then the line element for the metric $g$ is\n\\begin{equation*}\n  \\mathrm{d}{s}^2(g) = g_{\\mu\\nu} \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu.\n\\end{equation*}\nNonetheless, the information about the metric structure of the manifold can be translated to the language of frames,\n\\begin{equation*}\n  \\begin{split}\n    \\mathrm{d}{s}^2(g)\n    & = g_{\\mu\\nu} \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu \\\\\n    & = \\eta_{ab} \\; e^{a}_{\\mu}(x) \\, e^{b}_{\\nu}(x) \\; \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu \\\\\n    & = \\eta_{ab} \\; \\mathrm{e}^{a} \\otimes \\mathrm{e}^{b}.\n  \\end{split}\n\\end{equation*}\nTherefore, the vielbein 1-form, $\\mathrm{e}^{a} \\equiv e^{a}_{\\mu}(x) \\mathrm{d}{x}^\\mu$, encodes the information of the metric tensor.\nIn order to complete the structure, one needs information about the transport of geometrical objects lying on bundles\nbased on $M$. That information is encoded on the spin connection 1-form, $\\omega^{a}{}_{b}$. Using these quantities one finds\nthe generalisation of the structure equations of Cartan,\n\\begin{align}\n  \\mathrm{d}{\\mathrm{e}^{a}} + \\omega^{a}{}_{b} \\wedge \\mathrm{e}^{b} & = \\mathrm{T}^{a},\n  \\label{firstSE}\\\\\n  \\mathrm{d}{\\omega^{a}{}_{c}} + \\omega^{a}{}_{b} \\wedge \\omega^{b}{}_{c} & = \\mathrm{R}^{a}{}_{c}\n  \\label{secondSE}.\n\\end{align}\nThe torsion 2-form, $\\mathrm{T}^{a}$, and the curvature 2-form, $\\mathrm{R}^{a}{}_{c}$, measure the impossibility of endowing $M$ with\nan Euclidean structure.\n\n\\section*{Manipulation of the structural equations}\n\n\\subsection*{Definitions}"
+			"source" : "\\section*{Cartan structural equations and Bianchi identity}\n\n\\author{Oscar Castillo-Felisola}\n\n\\section*{Theoretical background}\n\nLet $M$ be a manifold, and $g$ a (semi)Riemannian metric defined on $M$. Then the line element for the metric $g$ is\n\\begin{equation*}\n  \\mathrm{d}{s}^2(g) = g_{\\mu\\nu} \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu.\n\\end{equation*}\nNonetheless, the information about the metric structure of the manifold can be translated to the language of frames,\n\\begin{equation*}\n  \\begin{split}\n    \\mathrm{d}{s}^2(g)\n    & = g_{\\mu\\nu} \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu \\\\\n    & = \\eta_{ab} \\; e^{a}_{\\mu}(x) \\, e^{b}_{\\nu}(x) \\; \\mathrm{d}{x}^\\mu \\otimes \\mathrm{d}{x}^\\nu \\\\\n    & = \\eta_{ab} \\; \\mathrm{e}^{a} \\otimes \\mathrm{e}^{b}.\n  \\end{split}\n\\end{equation*}\nTherefore, the vielbein 1-form, $\\mathrm{e}^{a} \\equiv e^{a}_{\\mu}(x) \\mathrm{d}{x}^\\mu$, encodes the information of the metric tensor.\nIn order to complete the structure, one needs information about the transport of geometrical objects lying on bundles\nbased on $M$. That information is encoded on the spin connection 1-form, $\\omega^{a}{}_{b}$. Using these quantities one finds\nthe generalisation of the structure equations of Cartan,\n\\begin{align}\n  \\mathrm{d}{\\mathrm{e}^{a}} + \\omega^{a}{}_{b} \\wedge \\mathrm{e}^{b} & = \\mathrm{T}^{a},\n  \\label{firstSE}\\\\\n  \\mathrm{d}{\\omega^{a}{}_{c}} + \\omega^{a}{}_{b} \\wedge \\omega^{b}{}_{c} & = \\mathrm{R}^{a}{}_{c}\n  \\label{secondSE}.\n\\end{align}\nThe torsion 2-form, $\\mathrm{T}^{a}$, and the curvature 2-form, $\\mathrm{R}^{a}{}_{c}$, measure the impossibility of endowing $M$ with\nan Euclidean structure.\n\n\\section*{Manipulation of the structural equations}\n\n\\subsection*{Definitions}"
 		},
 		{
 			"cell_id" : 13528271774984453456,

web2/cadabra2/source/user_notebooks.html

@@ -21,10 +21,10 @@ <h2>Notebooks</h2>
   <a href="notebooks/einstein_equations.html">read online</a>
   <a href="notebooks/einstein_equations.cnb">download notebook</a>
 </div>
-<!-- <div class="view_options">
-  <span>Structure equations and Bianchi identities:</span>
+<div class="view_options">
+  <span>Structure equations and Bianchi ids:</span>
   <a href="notebooks/structure_equations_and_bianchi.html">read online</a>
   <a href="notebooks/structure_equations_and_bianchi.cnb">download notebook</a>
-</div> -->
-  
+</div>
+
 {%- endblock %}