{
	"cell_id" : 17942456533940163234,
	"cells" : 
	[
		{
			"cell_id" : 11340042467900394662,
			"cell_origin" : "client",
			"cell_type" : "latex",
			"cells" : 
			[
				{
					"cell_id" : 9417804978734591005,
					"cell_origin" : "client",
					"cell_type" : "latex_view",
					"source" : "\\section*{Differential forms}\n\nCadabra can handle the calculus of differential forms. Differential forms are declared by attaching\nthe property \\prop{DifferentialForm} to an object, as in the example below, where we declare\nfour forms, with degree 0 to 3, and one form with symbolic degree $p$."
				}
			],
			"hidden" : true,
			"source" : "\\section*{Differential forms}\n\nCadabra can handle the calculus of differential forms. Differential forms are declared by attaching\nthe property \\prop{DifferentialForm} to an object, as in the example below, where we declare\nfour forms, with degree 0 to 3, and one form with symbolic degree $p$."
		},
		{
			"cell_id" : 8887426986920168478,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775809,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property DifferentialForm to~}A^{(0)}.\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775810,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property DifferentialForm to~}A^{(1)}.\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775811,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property DifferentialForm to~}A^{(2)}.\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775812,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property DifferentialForm to~}A^{(3)}.\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775813,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property DifferentialForm to~}B^{(p)}.\\end{dmath*}"
				}
			],
			"source" : "A0::LaTeXForm(\"A^{(0)}\").\nA1::LaTeXForm(\"A^{(1)}\").\nA2::LaTeXForm(\"A^{(2)}\").\nA3::LaTeXForm(\"A^{(3)}\").\nBp::LaTeXForm(\"B^{(p)}\").\nA0::DifferentialForm(degree=0);\nA1::DifferentialForm(degree=1);\nA2::DifferentialForm(degree=2);\nA3::DifferentialForm(degree=3);\nBp::DifferentialForm(degree=p);"
		},
		{
			"cell_id" : 9245929567340263576,
			"cell_origin" : "client",
			"cell_type" : "latex",
			"cells" : 
			[
				{
					"cell_id" : 10731088970142001207,
					"cell_origin" : "client",
					"cell_type" : "latex_view",
					"source" : "The exterior product (or wedge product) of forms is denoted with a \\verb|^| symbol (make sure to put a space\nbefore and after this symbol to avoid confusion with a superscript)."
				}
			],
			"hidden" : true,
			"source" : "The exterior product (or wedge product) of forms is denoted with a \\verb|^| symbol (make sure to put a space\nbefore and after this symbol to avoid confusion with a superscript)."
		},
		{
			"cell_id" : 4953780234914878080,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775815,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775816,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "A1 ^ A2 + A2 ^ A1"
						}
					],
					"source" : "\\begin{dmath*}{}A^{(1)}\\wedge A^{(2)}+A^{(2)}\\wedge A^{(1)}\\end{dmath*}"
				}
			],
			"source" : "ex:=A1 ^ A2 + A2 ^ A1;"
		},
		{
			"cell_id" : 1319382506597754390,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775818,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775819,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "2A1 ^ A2"
						}
					],
					"source" : "\\begin{dmath*}{}2A^{(1)}\\wedge A^{(2)}\\end{dmath*}"
				}
			],
			"source" : "sort_product(_);"
		},
		{
			"cell_id" : 9265232820241736682,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775821,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775822,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "A1 ^ A2 ^ A1"
						}
					],
					"source" : "\\begin{dmath*}{}A^{(1)}\\wedge A^{(2)}\\wedge A^{(1)}\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775823,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775824,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "0"
						}
					],
					"source" : "\\begin{dmath*}{}0\\end{dmath*}"
				}
			],
			"source" : "ex:=A1 ^ A2 ^ A1;\nsort_product(_);"
		},
		{
			"cell_id" : 4020737267376578434,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775826,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775827,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "A3 ^ A2 ^ A1 ^ A2"
						}
					],
					"source" : "\\begin{dmath*}{}A^{(3)}\\wedge A^{(2)}\\wedge A^{(1)}\\wedge A^{(2)}\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775828,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}-A^{(1)}\\wedge A^{(2)}\\wedge A^{(2)}\\wedge A^{(3)}\\end{dmath*}"
				}
			],
			"source" : "ex:= A3 ^ A2 ^ A1 ^ A2;\nsort_product(_);"
		},
		{
			"cell_id" : 14722099834296306032,
			"cell_origin" : "client",
			"cell_type" : "latex",
			"cells" : 
			[
				{
					"cell_id" : 15780429776499677377,
					"cell_origin" : "client",
					"cell_type" : "latex_view",
					"source" : "An exterior derivative can be declared by using the \\prop{ExteriorDerivative} property. Applying an exterior\nderivative twice produces zero, and it obeys the product rule taking into account the degree of differential forms."
				}
			],
			"hidden" : true,
			"source" : "An exterior derivative can be declared by using the \\prop{ExteriorDerivative} property. Applying an exterior\nderivative twice produces zero, and it obeys the product rule taking into account the degree of differential forms."
		},
		{
			"cell_id" : 15528343336642185683,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775831,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property ExteriorDerivative to~}d{\\#}.\\end{dmath*}"
				}
			],
			"source" : "d{#}::ExteriorDerivative;\nd{#}::LaTeXForm(\"{\\rm d}\")."
		},
		{
			"cell_id" : 1144807404142193736,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775833,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775834,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "-d(A1 ^ A2 ^ A2 ^ A3)"
						}
					],
					"source" : "\\begin{dmath*}{}-{\\rm d}\\left(A^{(1)}\\wedge A^{(2)}\\wedge A^{(2)}\\wedge A^{(3)}\\right)\\end{dmath*}"
				}
			],
			"source" : "ex2:= d{ @(ex) };"
		},
		{
			"cell_id" : 12023467755791252355,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775836,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775837,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "-d(A1) ^ A2 ^ A2 ^ A3 + A1 ^ d(A2) ^ A2 ^ A3 + A1 ^ A2 ^ d(A2) ^ A3 + A1 ^ A2 ^ A2 ^ d(A3)"
						}
					],
					"source" : "\\begin{dmath*}{}-{\\rm d}{A^{(1)}}\\wedge A^{(2)}\\wedge A^{(2)}\\wedge A^{(3)}+A^{(1)}\\wedge {\\rm d}{A^{(2)}}\\wedge A^{(2)}\\wedge A^{(3)}+A^{(1)}\\wedge A^{(2)}\\wedge {\\rm d}{A^{(2)}}\\wedge A^{(3)}+A^{(1)}\\wedge A^{(2)}\\wedge A^{(2)}\\wedge {\\rm d}{A^{(3)}}\\end{dmath*}"
				}
			],
			"source" : "product_rule(_);"
		},
		{
			"cell_id" : 8520254541109020363,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775839,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775840,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "d(A1 ^ A2 ^ d(A3))"
						}
					],
					"source" : "\\begin{dmath*}{}{\\rm d}\\left(A^{(1)}\\wedge A^{(2)}\\wedge {\\rm d}{A^{(3)}}\\right)\\end{dmath*}"
				}
			],
			"source" : "ex:= d{ A1 ^ A2 ^ d{ A3 } };"
		},
		{
			"cell_id" : 9883512770807428270,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775842,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775843,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "d(A1) ^ A2 ^ d(A3)-A1 ^ d(A2) ^ d(A3)"
						}
					],
					"source" : "\\begin{dmath*}{}{\\rm d}{A^{(1)}}\\wedge A^{(2)}\\wedge {\\rm d}{A^{(3)}}-A^{(1)}\\wedge {\\rm d}{A^{(2)}}\\wedge {\\rm d}{A^{(3)}}\\end{dmath*}"
				}
			],
			"source" : "product_rule(_);"
		},
		{
			"cell_id" : 3882037680778395718,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775845,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775846,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "A2 ^ d(A1) ^ d(A3)-A1 ^ d(A2) ^ d(A3)"
						}
					],
					"source" : "\\begin{dmath*}{}A^{(2)}\\wedge {\\rm d}{A^{(1)}}\\wedge {\\rm d}{A^{(3)}}-A^{(1)}\\wedge {\\rm d}{A^{(2)}}\\wedge {\\rm d}{A^{(3)}}\\end{dmath*}"
				}
			],
			"source" : "sort_product(_);"
		},
		{
			"cell_id" : 7388374175146732527,
			"cell_origin" : "client",
			"cell_type" : "latex",
			"cells" : 
			[
				{
					"cell_id" : 7893177832322756865,
					"cell_origin" : "client",
					"cell_type" : "latex_view",
					"source" : "Two forms in a product can only be swapped around if one of their degrees is zero:"
				}
			],
			"hidden" : true,
			"source" : "Two forms in a product can only be swapped around if one of their degrees is zero:"
		},
		{
			"cell_id" : 7517601938794019119,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775848,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775849,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "A2 A0 + A0 A2"
						}
					],
					"source" : "\\begin{dmath*}{}A^{(2)} A^{(0)}+A^{(0)} A^{(2)}\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775850,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775851,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "2A0 A2"
						}
					],
					"source" : "\\begin{dmath*}{}2A^{(0)} A^{(2)}\\end{dmath*}"
				}
			],
			"source" : "ex4:= A2 A0 + A0 A2;\nsort_product(_);"
		},
		{
			"cell_id" : 4180056803145046336,
			"cell_origin" : "client",
			"cell_type" : "latex",
			"cells" : 
			[
				{
					"cell_id" : 4007756470112194266,
					"cell_origin" : "client",
					"cell_type" : "latex_view",
					"source" : "In a wedge product, forms can be swapped around taking into account their degree:"
				}
			],
			"hidden" : true,
			"source" : "In a wedge product, forms can be swapped around taking into account their degree:"
		},
		{
			"cell_id" : 1994676414667850866,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775853,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775854,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "A3 ^ A1-A1 ^ A3 + A2 ^ A2 ^ A0"
						}
					],
					"source" : "\\begin{dmath*}{}A^{(3)}\\wedge A^{(1)}-A^{(1)}\\wedge A^{(3)}+A^{(2)}\\wedge A^{(2)}\\wedge A^{(0)}\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775855,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775856,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "-2A1 ^ A3 + A0 ^ A2 ^ A2"
						}
					],
					"source" : "\\begin{dmath*}{}-2A^{(1)}\\wedge A^{(3)}+A^{(0)}\\wedge A^{(2)}\\wedge A^{(2)}\\end{dmath*}"
				}
			],
			"source" : "ex5:= A3 ^ A1 - A1 ^ A3 + A2 ^ A2 ^ A0;\nsort_product(_);"
		},
		{
			"cell_id" : 3478897637566888729,
			"cell_origin" : "client",
			"cell_type" : "latex",
			"cells" : 
			[
				{
					"cell_id" : 1174853405546436749,
					"cell_origin" : "client",
					"cell_type" : "latex_view",
					"source" : "Differential forms can have tensor indices, e.g.~for vielbeine or spin connections:"
				}
			],
			"hidden" : true,
			"source" : "Differential forms can have tensor indices, e.g.~for vielbeine or spin connections:"
		},
		{
			"cell_id" : 16323106096533524039,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775858,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property Indices(position=free) to~}\\left[a,~\\discretionary{}{}{} b,~\\discretionary{}{}{} c\\right].\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775859,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property DifferentialForm to~}\\left[e^{a},~\\discretionary{}{}{} \\omega^{a}\\,_{b}\\right].\\end{dmath*}"
				}
			],
			"source" : "{a,b,c}::Indices;\n{e^{a}, \\omega^{a}_{b}}::DifferentialForm(degree=1);"
		},
		{
			"cell_id" : 569944791880263975,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775861,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775862,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "d(e^{a}) = -\\omega^{a}_{b} ^ e^{b}"
						}
					],
					"source" : "\\begin{dmath*}{}{\\rm d}{e^{a}} = -\\omega^{a}\\,_{b}\\wedge e^{b}\\end{dmath*}"
				}
			],
			"source" : "ex:=d{e^{a}} = - \\omega^{a}_{b} ^ e^{b};"
		},
		{
			"cell_id" : 15096640937298439073,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775864,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775865,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "0 = -d(\\omega^{a}_{b} ^ e^{b})"
						}
					],
					"source" : "\\begin{dmath*}{}0 = -{\\rm d}\\left(\\omega^{a}\\,_{b}\\wedge e^{b}\\right)\\end{dmath*}"
				}
			],
			"source" : "cv:= d{ @(ex) };"
		},
		{
			"cell_id" : 2802247552303887267,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775867,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775868,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "0 = -d(\\omega^{a}_{b} ^ e^{b})"
						}
					],
					"source" : "\\begin{dmath*}{}0 = -{\\rm d}\\left(\\omega^{a}\\,_{b}\\wedge e^{b}\\right)\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775869,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775870,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "0 = -d(\\omega^{a}_{b}) ^ e^{b} + \\omega^{a}_{b} ^ d(e^{b})"
						}
					],
					"source" : "\\begin{dmath*}{}0 = -{\\rm d}{\\omega^{a}\\,_{b}}\\wedge e^{b}+\\omega^{a}\\,_{b}\\wedge {\\rm d}{e^{b}}\\end{dmath*}"
				}
			],
			"source" : "distribute(cv);\nproduct_rule(cv);"
		},
		{
			"cell_id" : 16406542458279560327,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775872,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775873,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "0 = -d(\\omega^{a}_{b}) ^ e^{b}-\\omega^{a}_{b} ^ \\omega^{b}_{c} ^ e^{c}"
						}
					],
					"source" : "\\begin{dmath*}{}0 = -{\\rm d}{\\omega^{a}\\,_{b}}\\wedge e^{b}-\\omega^{a}\\,_{b}\\wedge \\omega^{b}\\,_{c}\\wedge e^{c}\\end{dmath*}"
				}
			],
			"source" : "substitute(cv, ex);"
		},
		{
			"cell_id" : 10258893899376774313,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775878,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775879,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "0 = -d(\\omega^{a}_{b}) ^ e^{b}-\\omega^{a}_{b} ^ \\omega^{b}_{c} ^ e^{c}"
						}
					],
					"source" : "\\begin{dmath*}{}0 = -{\\rm d}{\\omega^{a}\\,_{b}}\\wedge e^{b}-\\omega^{a}\\,_{b}\\wedge \\omega^{b}\\,_{c}\\wedge e^{c}\\end{dmath*}"
				}
			],
			"source" : "rename_dummies(cv);"
		},
		{
			"cell_id" : 16913818003335919797,
			"cell_origin" : "client",
			"cell_type" : "latex",
			"cells" : 
			[
				{
					"cell_id" : 6957951781041518552,
					"cell_origin" : "client",
					"cell_type" : "latex_view",
					"source" : "Some more random assorted ramblings below, not finished yet!!!"
				}
			],
			"hidden" : true,
			"source" : "Some more random assorted ramblings below, not finished yet!!!"
		},
		{
			"cell_id" : 5914123710050700613,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775875,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775876,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "d(-2A1 ^ A3 + A0 ^ A2 ^ A2)"
						}
					],
					"source" : "\\begin{dmath*}{}{\\rm d}\\left(-2A^{(1)}\\wedge A^{(3)}+A^{(0)}\\wedge A^{(2)}\\wedge A^{(2)}\\right)\\end{dmath*}"
				}
			],
			"source" : "ex4:= d{ @(ex5) };"
		},
		{
			"cell_id" : 6700747750860875507,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775878,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775879,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "-2d(A1 ^ A3) + d(A0 ^ A2 ^ A2)"
						}
					],
					"source" : "\\begin{dmath*}{}-2{\\rm d}\\left(A^{(1)}\\wedge A^{(3)}\\right)+{\\rm d}\\left(A^{(0)}\\wedge A^{(2)}\\wedge A^{(2)}\\right)\\end{dmath*}"
				}
			],
			"source" : "distribute(_);"
		},
		{
			"cell_id" : 15415966661854912494,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775881,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775882,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "-2d(A1) ^ A3 + 2A1 ^ d(A3) + d(A0) ^ A2 ^ A2 + A0 ^ d(A2) ^ A2 + A0 ^ A2 ^ d(A2)"
						}
					],
					"source" : "\\begin{dmath*}{}-2{\\rm d}{A^{(1)}}\\wedge A^{(3)}+2A^{(1)}\\wedge {\\rm d}{A^{(3)}}+{\\rm d}{A^{(0)}}\\wedge A^{(2)}\\wedge A^{(2)}+A^{(0)}\\wedge {\\rm d}{A^{(2)}}\\wedge A^{(2)}+A^{(0)}\\wedge A^{(2)}\\wedge {\\rm d}{A^{(2)}}\\end{dmath*}"
				}
			],
			"source" : "product_rule(_);"
		},
		{
			"cell_id" : 4457834258149327361,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775884,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775885,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "-2A3 ^ d(A1) + 2A1 ^ d(A3) + A2 ^ A2 ^ d(A0) + 2A0 ^ A2 ^ d(A2)"
						}
					],
					"source" : "\\begin{dmath*}{}-2A^{(3)}\\wedge {\\rm d}{A^{(1)}}+2A^{(1)}\\wedge {\\rm d}{A^{(3)}}+A^{(2)}\\wedge A^{(2)}\\wedge {\\rm d}{A^{(0)}}+2A^{(0)}\\wedge A^{(2)}\\wedge {\\rm d}{A^{(2)}}\\end{dmath*}"
				}
			],
			"source" : "sort_product(_);"
		},
		{
			"cell_id" : 4984116761811257240,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775809,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property Indices(position=free) to~}\\left[i,~\\discretionary{}{}{} j,~\\discretionary{}{}{} k,~\\discretionary{}{}{} l\\right].\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775810,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property Integer to~}\\left[i,~\\discretionary{}{}{} j,~\\discretionary{}{}{} k,~\\discretionary{}{}{} l\\right].\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775811,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property EpsilonTensor to~}\\epsilon^{i j k}.\\end{dmath*}"
				},
				{
					"cell_id" : 9223372036854775812,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}\\text{Attached property ExteriorDerivative to~}d{\\#}.\\end{dmath*}"
				}
			],
			"source" : "{i,j,k,l}::Indices(values={1,2,3});\n{i,j,k,l}::Integer(1..3);\n\\epsilon^{i j k}::EpsilonTensor;\nd{#}::ExteriorDerivative;\nd{#}::LaTeXForm(\"{\\rm d}\")."
		},
		{
			"cell_id" : 6955768955919678707,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775814,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775815,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "\\epsilon^{i j k}"
						}
					],
					"source" : "\\begin{dmath*}{}\\epsilon^{i j k}\\end{dmath*}"
				}
			],
			"source" : "ex:=\\epsilon^{i j k};"
		},
		{
			"cell_id" : 146115869353297447,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775817,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775818,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "\\components^{i j k}({{1, 2, 3} = 1, {1, 3, 2} = -1, {2, 1, 3} = -1, {2, 3, 1} = 1, {3, 1, 2} = 1, {3, 2, 1} = -1})"
						}
					],
					"source" : "\\begin{dmath*}{}\\square{}^{i}{}^{j}{}^{k}\\left\\{\\begin{aligned}\\square{}^{1}{}^{2}{}^{3}= & 1\\\\[-.5ex]\n\\square{}^{1}{}^{3}{}^{2}= & -1\\\\[-.5ex]\n\\square{}^{2}{}^{1}{}^{3}= & -1\\\\[-.5ex]\n\\square{}^{2}{}^{3}{}^{1}= & 1\\\\[-.5ex]\n\\square{}^{3}{}^{1}{}^{2}= & 1\\\\[-.5ex]\n\\square{}^{3}{}^{2}{}^{1}= & -1\\\\[-.5ex]\n\\end{aligned}\\right.\n\\end{dmath*}"
				}
			],
			"source" : "evaluate(_, $$);"
		},
		{
			"cell_id" : 6999986821528213344,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775820,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"cells" : 
					[
						{
							"cell_id" : 9223372036854775821,
							"cell_origin" : "server",
							"cell_type" : "input_form",
							"source" : "{\\Sigma^{1} = \\cos(\\psi) d(\\theta) + \\sin(\\psi) \\sin(\\theta) d(\\phi), \\Sigma^{2} = -\\sin(\\psi) d(\\theta) + \\cos(\\psi) \\sin(\\theta) d(\\phi), \\Sigma^{3} = \\cos(\\theta) d(\\phi) + d(\\psi)}"
						}
					],
					"source" : "\\begin{dmath*}{}\\left[\\Sigma^{1} = \\cos{\\psi} {\\rm d}{\\theta}+\\sin{\\psi} \\sin{\\theta} {\\rm d}{\\phi},~\\discretionary{}{}{} \\Sigma^{2} = -\\sin{\\psi} {\\rm d}{\\theta}+\\cos{\\psi} \\sin{\\theta} {\\rm d}{\\phi},~\\discretionary{}{}{} \\Sigma^{3} = \\cos{\\theta} {\\rm d}{\\phi}+{\\rm d}{\\psi}\\right]\\end{dmath*}"
				}
			],
			"source" : "rl:= { \\Sigma^{1} =  \\cos{\\psi} d{\\theta} + \\sin{\\psi} \\sin{\\theta} d{\\phi},\n       \\Sigma^{2} = -\\sin{\\psi} d{\\theta} + \\cos{\\psi} \\sin{\\theta} d{\\phi},\n       \\Sigma^{3} =  \\cos{\\theta} d{\\phi} + d{\\psi} };"
		},
		{
			"cell_id" : 2266077326488286745,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775823,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{} - \\frac{1}{2}\\epsilon^{i j k} \\Sigma^{j}\\wedge \\Sigma^{k}\\end{dmath*}"
				}
			],
			"source" : "tst:= -1/2 \\epsilon^{i j k} \\Sigma^{j} ^ \\Sigma^{k};"
		},
		{
			"cell_id" : 11941205435825729702,
			"cell_origin" : "client",
			"cell_type" : "input",
			"cells" : 
			[
				{
					"cell_id" : 9223372036854775826,
					"cell_origin" : "server",
					"cell_type" : "latex_view",
					"source" : "\\begin{dmath*}{}0\\end{dmath*}"
				}
			],
			"source" : "evaluate(tst, rl);"
		},
		{
			"cell_id" : 8836380534498282216,
			"cell_origin" : "client",
			"cell_type" : "input",
			"source" : ""
		}
	],
	"description" : "Cadabra JSON notebook format",
	"version" : 1
}