This worksheet provides a short introduction to SageManifolds (version 0.8).
It is released under the GNU General Public License version 3.
(c) Eric Gourgoulhon, Michal Bejger (2015)
The corresponding worksheet file can be downloaded from here.
The following assumes that the SageManifolds package is installed on your computer (see the download and installation instructions if not).
As an example let us define a differentiable manifold of dimension 3 over $\mathbb{R}$:
︡be9b3b4a-cd00-4a51-90d5-99fd4b384fc4︡︡{"done":true,"html":"This worksheet provides a short introduction to SageManifolds (version 0.8).
\nIt is released under the GNU General Public License version 3.
\n(c) Eric Gourgoulhon, Michal Bejger (2015)
\nThe corresponding worksheet file can be downloaded from here.
\n\n
The following assumes that the SageManifolds package is installed on your computer (see the download and installation instructions if not).
\nAs an example let us define a differentiable manifold of dimension 3 over $\\mathbb{R}$:
"} ︠9ad61099-879a-49c5-a935-3cf8f4ca4638si︠ M = Manifold(3, 'M', r'\mathcal{M}', start_index=1) ︡28a5d5fe-8267-4bb3-93c1-7241589a4eb8︡︡{"done":true} ︠cacb8712-43e7-4399-b0b3-59b537e1d78ei︠ %htmlIf we ask for M, the LaTeX symbol is returned (provided the worksheet's Typeset box is selected):
︡b828f8cc-a4e8-4f62-8bf1-1bc76dec695c︡︡{"done":true,"html":"If we ask for M, the LaTeX symbol is returned (provided the worksheet's Typeset box is selected):
"} ︠5974d532-7c7a-4994-beb0-daead5c9b1e8s︠ M ︡88a3af8a-3ee6-4c9a-a78b-c4ac235059db︡︡{"html":"If we use the command print instead, we get a short description of the object:
︡074d7b77-0d7b-43fd-8816-f5b4e78297ee︡︡{"done":true,"html":"If we use the command print instead, we get a short description of the object:
"} ︠d6c5c22e-3dc6-4941-9337-2766a2b2e1d6s︠ print M ︡264d1d4d-261b-4991-bbe4-5f714b4ed85b︡︡{"stdout":"3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠e664fbe2-81e5-4b58-af8f-a1ed117a9debi︠ %htmlVia the command type, we get the type of the Python object corresponding to M (here the Python class Manifold_with_category):
︡742bad64-11a6-4f68-a602-fe27c853bde0︡︡{"done":true,"html":"Via the command type, we get the type of the Python object corresponding to M (here the Python class Manifold_with_category):
"} ︠2f3612ac-38d9-47fb-8721-bd1fdfc03005s︠ type(M) ︡187b9b10-1d8a-42c0-b3ba-41c669cbdeaa︡︡{"stdout":"The indices on the manifold are generated by the method irange(), to be used in loops:
︡8e9f87ac-384c-47d5-bf18-6a76853919dd︡︡{"done":true,"html":"The indices on the manifold are generated by the method irange(), to be used in loops:
"} ︠23b74003-0ed6-4be1-9714-73ec362d0b60s︠ for i in M.irange(): print i, ︡2479be7f-4059-4b71-a25a-636d050c80c9︡︡{"stdout":"1 2 3","done":false}︡{"done":true} ︠7743a638-07d0-46da-86f5-0aba39121356i︠ %htmlIf the parameter start_index had not been specified, the default range of the indices would have been $\{0,1,2\}$ instead:
︡cec5d53a-602c-4c8b-ac76-baea199d5093︡︡{"done":true,"html":"If the parameter start_index had not been specified, the default range of the indices would have been $\\{0,1,2\\}$ instead:
"} ︠265ba4f0-33e6-4c9d-b359-048f4d6b28c6s︠ M0 = Manifold(3, 'M', r'\mathcal{M}') for i in M0.irange(): print i, ︡d6cd8a7b-64e7-47d5-81d7-49e588c2581c︡︡{"stdout":"0 1 2","done":false}︡{"done":true} ︠75003083-f337-4b33-a828-727a6c84c97ei︠ %htmlLet us assume that the manifold $\mathcal{M}$ can be covered by a single chart (other cases are discussed below); the chart is declared as follows:
︡395befa7-4e75-40ef-ba5c-c2d19be0d7d3︡︡{"done":true,"html":"Let us assume that the manifold $\\mathcal{M}$ can be covered by a single chart (other cases are discussed below); the chart is declared as follows:
"} ︠6b27b492-2737-4e94-81c7-f956a1cc1478s︠ X.The writing .<x,y,z> in the left-hand side means that the Python variables x, y and z are set to the three coordinates of the chart. This allows one to refer subsequently to the coordinates by their names.
In this example, the function chart() has no arguments, which implies that the coordinate symbols will be 'x', 'y' and 'z' (i.e. exactly the characters set in the <...> operator) and that each coordinate range is $(-\infty,+\infty)$. For other cases, an argument must be passed to chart() to specify the coordinate symbols and range, as well as the LaTeX symbols of the coordinates if the latter are different from the coordinate names (an example will be provided below).
The writing .<x,y,z> in the left-hand side means that the Python variables x, y and z are set to the three coordinates of the chart. This allows one to refer subsequently to the coordinates by their names.
\nIn this example, the function chart() has no arguments, which implies that the coordinate symbols will be 'x', 'y' and 'z' (i.e. exactly the characters set in the <...> operator) and that each coordinate range is $(-\\infty,+\\infty)$. For other cases, an argument must be passed to chart() to specify the coordinate symbols and range, as well as the LaTeX symbols of the coordinates if the latter are different from the coordinate names (an example will be provided below).
\nThe chart is displayed as a pair formed by the open set covered by it (here the whole manifold) and the coordinate names:
︡a100737e-67f4-4e0e-a193-11b9ba664655︡︡{"done":true,"html":"The chart is displayed as a pair formed by the open set covered by it (here the whole manifold) and the coordinate names:
"} ︠2232bcb2-7fce-4cd3-8d61-102ac52bbe56s︠ X ︡9f2bff13-4798-47b3-a00d-0455cf9bf2a1︡︡{"html":"The coordinates can be accessed individually, by means of their indices, following the convention defined by start_index=1 in the manifold's definition:
︡d9bf58bb-dde0-4d54-b921-f587e9715d71︡︡{"done":true,"html":"The coordinates can be accessed individually, by means of their indices, following the convention defined by start_index=1 in the manifold's definition:
"} ︠cc0afe32-d160-47bb-aeda-b0daf1ab9f92s︠ X[1] ︡35384efe-eb9e-4749-9fe0-4332df5005ae︡︡{"html":"The full set of coordinates is obtained by means of the operator [:]:
︡5b2733b0-f4b5-4049-a107-b8350bbe2d3a︡︡{"done":true,"html":"The full set of coordinates is obtained by means of the operator [:]:
"} ︠48db4cae-c81d-4e1d-92a9-ae09f878b498s︠ X[:] ︡61f86e28-1f2b-4bef-939e-f7d45a71a374︡︡{"html":"Thanks to the operator <x,y,z> in the chart declaration, each coordinate can be accessed directly via its name:
︡a7d1f2c3-acf7-42d3-a818-3fdb2c90d27a︡︡{"done":true,"html":"Thanks to the operator <x,y,z> in the chart declaration, each coordinate can be accessed directly via its name:
"} ︠af49aee3-58c5-41cc-a7ac-048ed515f59cs︠ z is X[3] ︡6b82b2a4-1430-463d-bdb2-af482ab0174f︡︡{"html":"Coordinates are Sage symbolic expressions:
︡30c963f2-c5ac-4677-9d20-aaf44c180254︡︡{"done":true,"html":"Coordinates are Sage symbolic expressions:
"} ︠85cf5f0b-0cd8-412c-907e-4622fa258316s︠ type(z) ︡59630093-bf12-49f1-9375-3c5e843a0894︡︡{"stdout":"Real-valued functions of the chart coordinates (mathematically speaking, functions defined on the chart codomain) are formed via the method function() acting on the chart:
︡c75dd954-3268-4133-b16e-b7b99838606f︡︡{"done":true,"html":"Real-valued functions of the chart coordinates (mathematically speaking, functions defined on the chart codomain) are formed via the method function() acting on the chart:
"} ︠afeef508-383e-4473-b64b-fa09e188f2e1s︠ f = X.function(x+y^2+z^3) ; f ︡7c14e570-e3f5-450a-942e-4dafcb79b4ab︡︡{"html":"They belong to SageManifolds class FunctionChart:
︡587ad0eb-5e20-4c61-a2eb-804902e10ada︡︡{"done":true,"html":"They belong to SageManifolds class FunctionChart:
"} ︠bee06e9d-d061-4194-93c0-0f2bd1afefb8s︠ type(f) ︡a447a757-e206-4dc8-a92e-62a406a04471︡︡{"stdout":"and differ from Sage standard symbolic functions by automatic simplifications in all operations. For instance, adding the two symbolic functions
︡c6ccbd4d-9701-4799-9912-a8a8603df377︡︡{"done":true,"html":"and differ from Sage standard symbolic functions by automatic simplifications in all operations. For instance, adding the two symbolic functions
"} ︠ef1362ec-9962-4a3c-a9a1-e749dc828b42s︠ f0(x,y,z) = cos(x)^2 ; g0(x,y,z) = sin(x)^2 ︡27d1d8ca-ce65-4328-ac3a-bca6ee18b819︡︡{"done":true} ︠0a625372-ca0c-4cd7-85ba-f35b14d499b0i︠ %htmlresults in
︡94339f1b-6d74-48d3-842b-fd915201877f︡︡{"done":true,"html":"results in
"} ︠0b3badba-d5ce-4657-bb21-5347cbbfcb86s︠ f0 + g0 ︡2d97d2c0-d149-4645-9310-cd022521ba86︡︡{"html":"while the sum of the corresponding FunctionChart functions is automatically simplified:
︡e86af1af-c377-4b88-a8f3-267188b8a149︡︡{"done":true,"html":"while the sum of the corresponding FunctionChart functions is automatically simplified:
"} ︠d341d206-8d8d-454b-a633-c91b7109b157s︠ f1 = X.function(cos(x)^2) ; g1 = X.function(sin(x)^2) f1 + g1 ︡33a74b89-0f33-4d1b-a4e0-3ff373b9dfe6︡︡{"html":"To get the same output with symbolic functions, one has to call the method simplify_trig():
︡715030fd-c5dc-4d8c-99b3-09282e4cbaa2︡︡{"done":true,"html":"To get the same output with symbolic functions, one has to call the method simplify_trig():
"} ︠c8864dc4-678c-4d37-8d1e-04def4f6a347s︠ (f0 + g0).simplify_trig() ︡1fa1cc1a-9ec5-4ee2-9f3f-bb1305b699a1︡︡{"html":"Another difference regards the display; if we ask for the symbolic function f0, we get:
︡db126ec2-51d9-4564-8267-a0bcdc52c525︡︡{"done":true,"html":"Another difference regards the display; if we ask for the symbolic function f0, we get:
"} ︠6a15149d-5fc3-4e52-923a-403be15e8e2as︠ f0 ︡31d4bd94-2b3f-4227-9179-e3826db0f9c2︡︡{"html":"while if we ask for the chart function f1, we get only the coordinate expression:
︡dc9b038a-c3da-4708-b77e-18530adccb9f︡︡{"done":true,"html":"while if we ask for the chart function f1, we get only the coordinate expression:
"} ︠b0dc8b96-4dd6-48af-a613-3d1ae759d9ces︠ f1 ︡f8439266-fb4f-45a6-9563-58ebe540ff03︡︡{"html":"To get an output similar to that of f0, one should call the method display():
︡cbc8071d-d080-4116-a895-a8b30a802481︡︡{"done":true,"html":"To get an output similar to that of f0, one should call the method display():
"} ︠38c5a959-3350-4d42-a5da-2dab040a5b78s︠ f1.display() ︡f77d5183-f875-4315-af5a-568d4521259e︡︡{"html":"Note that the method expr() returns the corresponding Sage symbolic expression:
︡476a46df-ccee-40d2-9ff5-059de8d6000f︡︡{"done":true,"html":"Note that the method expr() returns the corresponding Sage symbolic expression:
"} ︠a85190b3-f0ab-4857-b973-b49e69c4bfd4s︠ f1.expr() ︡93b9f8ae-fd7f-4f74-8741-5a8e0856b702︡︡{"html":"Let us first consider an open subset of $\mathcal{M}$, for instance the complement $U$ of the region defined by $\{y=0, x\geq 0\}$ (note that (y!=0, x<0) stands for $y\not=0$ OR $x<0$; the condition $y\not=0$ AND $x<0$ would have been written [y!=0, x<0] instead):
︡09863b0b-8c08-40de-a796-239d528229e0︡︡{"done":true,"html":"Let us first consider an open subset of $\\mathcal{M}$, for instance the complement $U$ of the region defined by $\\{y=0, x\\geq 0\\}$ (note that (y!=0, x<0) stands for $y\\not=0$ OR $x<0$; the condition $y\\not=0$ AND $x<0$ would have been written [y!=0, x<0] instead):
"} ︠559f1141-41a2-46ec-b64e-2d11d8511cfas︠ U = M.open_subset('U', coord_def={X: (y!=0, x<0)}) ︡cf9b5199-0b73-4de0-b129-05a61c374eb7︡︡{"done":true} ︠b00a0a5c-ac5b-4fa8-8a1a-f4610b6d92ddi︠ %htmlLet us call X_U the restriction of the chart X to the open subset $U$:
︡d995a041-57db-4b0c-95bc-d2b375067d9a︡︡{"done":true,"html":"Let us call X_U the restriction of the chart X to the open subset $U$:
"} ︠1ea60cfd-3a77-4676-a6b7-59932f80641bs︠ X_U = X.restrict(U) ; X_U ︡c9a80ba1-2a6e-4894-a580-2c1f641cf527︡︡{"html":"We introduce another chart on $U$, with spherical-type coordinates $(r,\theta,\phi)$:
︡2ba3e346-e253-4c87-b8b6-039b96e744af︡︡{"done":true,"html":"We introduce another chart on $U$, with spherical-type coordinates $(r,\\theta,\\phi)$:
"} ︠f98412c9-09bd-4fd1-9362-392101e3b322s︠ Y.The function chart() has now some argument; it is a string, which contains specific LaTeX symbols, hence the prefix 'r' to it (for raw string). It also contains the coordinate ranges, since they are different from the default value, which is $(-\infty, +\infty)$. For a given coordinate, the various fields are separated by the character ':' and a space character separates the coordinates. Note that for $r$, there is only two fields, since the LaTeX symbol has not to be specified. The LaTeX symbols are used for the outputs:
︡efe51328-3f45-4213-9ecc-ea1d7bdb8f28︡︡{"done":true,"html":"The function chart() has now some argument; it is a string, which contains specific LaTeX symbols, hence the prefix 'r' to it (for raw string). It also contains the coordinate ranges, since they are different from the default value, which is $(-\\infty, +\\infty)$. For a given coordinate, the various fields are separated by the character ':' and a space character separates the coordinates. Note that for $r$, there is only two fields, since the LaTeX symbol has not to be specified. The LaTeX symbols are used for the outputs:
"} ︠cfeb27cc-49a3-477d-854b-58262454fd9as︠ th, ph ︡9bd8f811-210e-4bf0-b9f1-fb55d3ea0b16︡︡{"html":"The declared coordinate ranges are now known to Sage, as we may check by means of the command assumptions():
︡860caac2-aa10-4f64-8591-8a7dde49185d︡︡{"done":true,"html":"The declared coordinate ranges are now known to Sage, as we may check by means of the command assumptions():
"} ︠d5136124-65e7-4de4-936e-b3b7af7f6797s︠ assumptions() ︡7b81bba2-0b1d-4dbb-b537-05b84f84fc1a︡︡{"html":"They are used in simplifications:
︡97dc1283-4224-4edd-9aae-bc878c4385da︡︡{"done":true,"html":"They are used in simplifications:
"} ︠49e86a98-3a4c-4524-ad07-75f33deb1ec1s︠ simplify(abs(r)) ︡502433d5-c038-4286-bdea-70674b614fe6︡︡{"html":"After having been declared, the chart Y can be fully specified by its relation to the chart X_U, via a transition map:
︡d803bda2-002b-402b-a4aa-2e72858539bf︡︡{"done":true,"html":"After having been declared, the chart Y can be fully specified by its relation to the chart X_U, via a transition map:
"} ︠5d5c73bc-69e9-4579-9e76-62111bdf26a0s︠ transit_Y_to_X = Y.transition_map(X_U, [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)]) ︡fb08ef9b-b746-411c-876a-d509a2f3d8a6︡︡{"done":true} ︠ec5d7e5e-d23e-4bb5-8a75-a747e9c234d5s︠ transit_Y_to_X ︡1daaa2e4-04e6-41d4-ae21-44b7d1312c50︡︡{"html":"The inverse of the transition map can be specified by means of the method set_inverse():
︡7ef4388e-6195-47c4-bda7-a07055b24b35︡︡{"done":true,"html":"The inverse of the transition map can be specified by means of the method set_inverse():
"} ︠30a47d3a-032a-4740-860d-3c094f566b7as︠ transit_Y_to_X.set_inverse(sqrt(x^2+y^2+z^2), atan2(sqrt(x^2+y^2),z), atan2(y, x)) transit_Y_to_X.inverse().display() ︡544de7d0-4173-4604-9e88-84e4fa6a3633︡︡{"stdout":"Check of the inverse coordinate transformation:","done":false}︡{"stdout":"\n r == ","done":false}︡{"stdout":"r\n th == ","done":false}︡{"stdout":"arctan2(r*sin(th), r*cos(th))\n ph == ","done":false}︡{"stdout":"arctan2(r*sin(ph)*sin(th), r*cos(ph)*sin(th))\n x == ","done":false}︡{"stdout":"x\n y == ","done":false}︡{"stdout":"y\n z == ","done":false}︡{"stdout":"z\n","done":false}︡{"html":"The check is passed, although some simplifications related to the arctan2 function are not performed.
At this stage, the manifold's atlas (the "user atlas", not the maximal atlas!) contains three charts:
︡fde748b8-efdb-4d92-b76b-43df9624ddbe︡︡{"done":true,"html":"The check is passed, although some simplifications related to the arctan2 function are not performed.
\nAt this stage, the manifold's atlas (the \"user atlas\", not the maximal atlas!) contains three charts:
"} ︠b93444d2-f094-44e5-8295-3bef9bfb072ds︠ M.atlas() ︡e00125ba-ac1c-4aff-a35e-5b4b8e782572︡︡{"html":"The first chart defined on the manifold is considered as the manifold's default chart (it can be changed by the method set_default_chart()):
︡54c2c384-3503-41f3-a556-805fd6b8dccd︡︡{"done":true,"html":"The first chart defined on the manifold is considered as the manifold's default chart (it can be changed by the method set_default_chart()):
"} ︠6e9e04e6-79c9-4d33-8f62-320e65b797abs︠ M.default_chart() ︡eaaef19d-4739-4afb-b0f2-6657656af3df︡︡{"html":"Each open subset has its own atlas:
︡6a2bcb32-1dd2-45fc-8f18-452e0b84cb7b︡︡{"done":true,"html":"Each open subset has its own atlas:
"} ︠dc39bd79-0cf9-4f9a-a3f0-41e4a1a6e3dbs︠ U.atlas() ︡f1c3d06d-e747-42ae-8aed-07c9e580dbed︡︡{"html":"We can draw the chart $Y$ in terms of the chart $X$: the plot shows lines of constant coordinates from the $Y$ chart in a "Cartesian frame" based on the $X$ coordinates:
︡40b2e269-bba9-4e8e-a08b-030ed6cc1cce︡︡{"done":true,"html":"We can draw the chart $Y$ in terms of the chart $X$: the plot shows lines of constant coordinates from the $Y$ chart in a \"Cartesian frame\" based on the $X$ coordinates:
"} ︠81f7b0a7-aa6c-4dbf-9e73-c82c4844541fs︠ Y.plot(X) ︡2d002718-48c6-40aa-ac9a-6490fc10761f︡︡{"done":false,"file":{"uuid":"a37018b7-355d-4856-92bf-2e1d8367e2bf","filename":"a37018b7-355d-4856-92bf-2e1d8367e2bf.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠bda40007-62c4-4943-baab-0d417295b60ci︠ %htmlThe command plot() allows for many options, to control the number of coordinate lines to be drawn, their style and color, as well as the coordinate ranges:
︡db241a50-03ba-49cb-960d-fb2e477e90f3︡︡{"done":true,"html":"The command plot() allows for many options, to control the number of coordinate lines to be drawn, their style and color, as well as the coordinate ranges:
"} ︠5cc4c9e3-43a8-4990-9d89-a206fdf7d7d7s︠ graph = Y.plot(X, ranges={r:(1,2), th:(0,pi/2)}, nb_values=4, color={r:'blue', th:'green', ph:'red'}) show(graph, aspect_ratio=1) ︡0a27df7e-3be6-4d7b-a858-24aab294f611︡︡{"done":false,"file":{"uuid":"ec83fe16-a149-484f-898e-3cf8162fcb7b","filename":"ec83fe16-a149-484f-898e-3cf8162fcb7b.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠c702309b-7bf6-409c-b811-761429629921i︠ %htmlConversly, the chart $X|_{U}$ can be plotted in terms of the chart $Y$ (this is not possible for the whole chart $X$ since its domain is larger than that of chart $Y$):
︡562c6839-3252-4902-9278-903baefbc01a︡︡{"done":true,"html":"Conversly, the chart $X|_{U}$ can be plotted in terms of the chart $Y$ (this is not possible for the whole chart $X$ since its domain is larger than that of chart $Y$):
"} ︠38737723-3021-446b-89a2-ee200673268es︠ X_U.plot(Y) ︡47a6f637-919c-4b74-9510-c1fcf7b96352︡︡{"done":false,"file":{"uuid":"29e55d4f-87da-4608-86dd-aafb8d5b0991","filename":"29e55d4f-87da-4608-86dd-aafb8d5b0991.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠952c9e65-4fb3-4cd9-b657-b347fb52e9f4i︠ %htmlA point on $\mathcal{M}$ is defined by its coordinates in a given chart:
︡ba7aea90-0a78-4fb9-a938-f1e9e4cc3bb9︡︡{"done":true,"html":"A point on $\\mathcal{M}$ is defined by its coordinates in a given chart:
"} ︠498833b4-b6d8-481c-93ef-e8517434ac06s︠ p = M.point((1,2,-1), X, name='p') ; print p ; p ︡7d64b2ff-99ff-4009-bf82-c616ae9384df︡︡{"stdout":"point 'p' on 3-dimensional manifold 'M'\n","done":false}︡{"html":"Since $X=(\mathcal{M}, (x,y,z))$ is the manifold's default chart, its name can be omitted:
︡22f8611f-24fc-45d9-ba70-48719438fc2f︡︡{"done":true,"html":"Since $X=(\\mathcal{M}, (x,y,z))$ is the manifold's default chart, its name can be omitted:
"} ︠a39a3bd6-3013-4a9f-a579-ea004d91718ds︠ p = M.point((1,2,-1), name='p') ; print p ; p ︡5bae03ae-e824-42e8-a11a-b41a99f709c1︡︡{"stdout":"point 'p' on 3-dimensional manifold 'M'\n","done":false}︡{"html":"Of course, $p$ belongs to $\mathcal{M}$:
︡37a938b5-7ba2-40c3-bf08-d68feb17a727︡︡{"done":true,"html":"Of course, $p$ belongs to $\\mathcal{M}$:
"} ︠ccd77682-847e-4451-b98f-a3d933950876s︠ p in M ︡8c4adf86-985f-47c1-86ff-66ba4a9d0a30︡︡{"html":"It is also in $U$:
︡e9848e49-c680-40a6-8358-610940386ae0︡︡{"done":true,"html":"It is also in $U$:
"} ︠9a69d7f1-fb79-451d-8af3-ed81ac814abfs︠ p in U ︡89592849-d29d-41d0-b81d-cc80463b985d︡︡{"html":"Indeed the coordinates of $p$ have $y\not=0$:
︡6db75923-bb62-4d37-9544-5cf3799b3224︡︡{"done":true,"html":"Indeed the coordinates of $p$ have $y\\not=0$:
"} ︠cc1957ee-7878-46fa-b2a1-395bb805cc29s︠ p.coord(X) ︡b1c9abec-6e8c-460b-8e73-ca2282ccffb0︡︡{"html":"Note in passing that since $X$ is the default chart on $\mathcal{M}$, its name can be omitted in the arguments of coord():
︡3ccec839-9567-40d0-9c33-38bba7217cba︡︡{"done":true,"html":"Note in passing that since $X$ is the default chart on $\\mathcal{M}$, its name can be omitted in the arguments of coord():
"} ︠2769fbfb-a20f-4074-b09e-890761797120s︠ p.coord() ︡08b4be7c-4591-4f80-bbdc-62b4c84dccf2︡︡{"html":"The coordinates of $p$ can also be obtained by letting the chart acting of the point (from the very definition of a chart!):
︡bbbaefa4-a7f2-4bde-b4b1-593d8f8069d0︡︡{"done":true,"html":"The coordinates of $p$ can also be obtained by letting the chart acting of the point (from the very definition of a chart!):
"} ︠0c15dbdc-a887-4942-b0f3-9e328180a541s︠ X(p) ︡aab693e5-67cd-47f8-8802-765bee5515f4︡︡{"html":"Let $q$ be a point with $y = 0$ and $x \geq 0$:
︡7c0b41d2-582a-4c63-b57b-a4d3f724b562︡︡{"done":true,"html":"Let $q$ be a point with $y = 0$ and $x \\geq 0$:
"} ︠047dffdf-dfcd-4f5a-9de4-1090ddae2dc2s︠ q = M.point((1,0,2), name='q') ︡9e1068ba-c3a9-434b-a117-aebe28837f20︡︡{"done":true} ︠cbf21a00-b03d-4852-8f0c-69cea726fa98i︠ %htmlThis time, the point does not belong to $U$:
︡05d16cf3-9cf0-4acc-af7c-83e9db646b9f︡︡{"done":true,"html":"This time, the point does not belong to $U$:
"} ︠d14f8cd1-81fe-487a-9d68-9d5d931844dbs︠ q in U ︡cb6ef562-2a25-4160-abf4-55894c780d36︡︡{"html":"Accordingly, we cannot ask for the coordinates of $q$ in the chart $Y=(U, (r,\theta,\phi))$:
︡4701bf91-ce1e-41a1-8e29-d6d7daf505da︡︡{"done":true,"html":"Accordingly, we cannot ask for the coordinates of $q$ in the chart $Y=(U, (r,\\theta,\\phi))$:
"} ︠7fa4b373-29e5-4e73-8024-8830e8bc0fb4s︠ q.coord(Y) ︡262b7764-2de8-4307-acb9-5a60442a4401︡︡{"done":false,"stderr":"Error in lines 1-1\nTraceback (most recent call last):\n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py\", line 905, in execute\n exec compile(block+'\\n', '', 'single') in namespace, locals\n File \"\", line 1, inbut we can for point $p$:
︡cbc3a34a-ce64-43f5-9f16-f80f53dc51c6︡︡{"done":true,"html":"but we can for point $p$:
"} ︠690034b2-272b-4b1b-a136-34c6dfa214b6s︠ p.coord(Y) ︡fa2fde27-9fba-4a39-a09f-52e7b331edbf︡︡{"html":"Points can be compared:
︡6d997649-b7a9-4132-a315-7da28bf85521︡︡{"done":true,"html":"Points can be compared:
"} ︠14060e3d-89ae-4003-b929-aca7783303f4s︠ q == p ︡457c90e6-4d7d-4eb6-8cc9-1b427292ba22︡︡{"html":"In Sage's terminology, points are elements, whose parents are the manifold on which they have been defined:
︡167d237d-5881-4f3c-aa32-443b061e7eb8︡︡{"done":true,"html":"In Sage's terminology, points are elements, whose parents are the manifold on which they have been defined:
"} ︠57503e04-49f0-4578-9edf-297141d64fd3s︠ p.parent() ︡397756eb-fce4-4f81-bec5-4f985ca9805a︡︡{"html":"A scalar field is a differentiable mapping $U\subset \mathcal{M} \longrightarrow \mathbb{R}$, where $U$ is an open subset of $\mathcal{M}$.
The scalar field is defined by its expressions in terms of charts covering its domain (in general more than one chart is necessary to cover all the domain):
︡9222f493-9487-499b-b704-ba9084efc37a︡︡{"done":true,"html":"A scalar field is a differentiable mapping $U\\subset \\mathcal{M} \\longrightarrow \\mathbb{R}$, where $U$ is an open subset of $\\mathcal{M}$.
\nThe scalar field is defined by its expressions in terms of charts covering its domain (in general more than one chart is necessary to cover all the domain):
"} ︠69035756-9941-4186-8204-744961e2bf7es︠ f = U.scalar_field({X_U: x+y^2+z^3}, name='f') ; print f ︡7b6333ab-3fa1-4691-bd2a-4a4e81c94b51︡︡{"stdout":"scalar field 'f' on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠8dfb8a56-f3e7-48e4-ba3a-17d257d4aaaai︠ %htmlThe coordinate expressions of the scalar field are passed as a Python dictionary, with the charts as keys, hence the writing {X_U: x+y^2+z^3}.
Since in the present case, there is only one chart in the dictionary, an alternative writing is
︡d135ecf8-2a0f-4699-a597-0330523878b3︡︡{"done":true,"html":"The coordinate expressions of the scalar field are passed as a Python dictionary, with the charts as keys, hence the writing {X_U: x+y^2+z^3}.
\nSince in the present case, there is only one chart in the dictionary, an alternative writing is
"} ︠502482c4-a4ea-4567-a806-95a261de3fcds︠ f = U.scalar_field(x+y^2+z^3, chart=X_U, name='f') ; print f ︡534065d0-ad4d-4720-8462-89a3066f3ae4︡︡{"stdout":"scalar field 'f' on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠045bd2f6-9fa8-406e-bb52-de568dd22182i︠ %htmlSince X_U is the domain's default chart, it can be omitted in the above declaration:
︡d802591a-99c0-4886-9ae2-05035dbbb7c1︡︡{"done":true,"html":"Since X_U is the domain's default chart, it can be omitted in the above declaration:
"} ︠63f4cf0a-db13-4745-95b0-0bebe903c8bes︠ f = U.scalar_field(x+y^2+z^3, name='f') ; print f ︡a90c35e1-9d06-468c-9b35-e78f18486832︡︡{"stdout":"scalar field 'f' on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠942ee39c-93da-4bf5-b145-e288f267c9a1i︠ %htmlAs a mapping $U\subset\mathcal{M}\longrightarrow\mathbb{R} $, a scalar field acts on points, not on coordinates:
︡677dcf33-06e0-440a-b71b-dbf1112e135e︡︡{"done":true,"html":"As a mapping $U\\subset\\mathcal{M}\\longrightarrow\\mathbb{R} $, a scalar field acts on points, not on coordinates:
"} ︠991a6d36-57c4-47ca-ba6e-61be3f9c0e0ds︠ f(p) ︡b2c2dcc3-264c-41d8-be5e-456fd62c0255︡︡{"html":"The expression of the scalar field in terms of the coordinates $(x,y,z)$:
︡89f75986-d279-40f9-93ff-8ea7a090dfad︡︡{"done":true,"html":"The expression of the scalar field in terms of the coordinates $(x,y,z)$:
"} ︠684b9bbd-16fd-4890-9d8b-6c41521a1f78s︠ f.display(X_U) ︡04e78a81-daee-4805-ae60-46fbc1aaa2ed︡︡{"html":"If the method display() is used without any argument, it displays the coordinate expression of the scalar field in all the charts defined on the domain (except for subcharts, i.e. the restrictions of some chart to a subdomain):
︡9b91bbcf-ad0d-44f3-b176-f91f41412a5a︡︡{"done":true,"html":"If the method display() is used without any argument, it displays the coordinate expression of the scalar field in all the charts defined on the domain (except for subcharts, i.e. the restrictions of some chart to a subdomain):
"} ︠7c49be27-cf92-45a5-99b4-6f97401cd0e6s︠ f.display() ︡796e627d-7245-40d4-ac95-7c90e8b1d4cd︡︡{"html":"Note that the expression of the scalar field in terms of the coordinates $(r,\theta,\phi)$ has not been provided by the user: it has been automatically computed via the change-of-coordinate formula declared above in the transition map.
︡29fabc5b-0f93-4807-9241-6b6122cd61a7︡︡{"done":true,"html":"Note that the expression of the scalar field in terms of the coordinates $(r,\\theta,\\phi)$ has not been provided by the user: it has been automatically computed via the change-of-coordinate formula declared above in the transition map.
"} ︠7365e7bf-d8dc-491f-8f42-42e95a64cc0cs︠ f.display(Y) ︡c65ff8a9-c2d6-49f0-a526-6ae9281ecc25︡︡{"html":"In each chart, the scalar field is represented by a function of the chart coordinates (an object of the type FunctionChart described above), which is accessible via the method function_chart():
︡65725614-563e-4840-9ce1-3cb03d9de69f︡︡{"done":true,"html":"In each chart, the scalar field is represented by a function of the chart coordinates (an object of the type FunctionChart described above), which is accessible via the method function_chart():
"} ︠e6256b98-c685-42fa-ad50-662578867819s︠ f.function_chart(X_U) ︡3fe9ab2e-591f-4d07-ad3d-bbef8017ca78︡︡{"html":"The "raw" symbolic expression is returned by the method expr():
︡e363ccbb-6332-4a46-9b2f-f83669933f92︡︡{"done":true,"html":"The \"raw\" symbolic expression is returned by the method expr():
"} ︠ce986779-77f2-4dda-9ab6-5f6c57d13236s︠ f.expr(X_U) ︡28f2362c-18ee-4d85-bedd-8e956c34a7c3︡︡{"html":"A scalar field can also be defined by some unspecified function of the coordinates:
︡c7946a6a-13f2-4f4a-8cdb-85bc777984fc︡︡{"done":true,"html":"A scalar field can also be defined by some unspecified function of the coordinates:
"} ︠f466129b-1972-4612-8898-a7058afdb9afs︠ h = U.scalar_field(function('H', x, y, z), name='h') ; print h ︡252e4e8b-7385-4c19-a74f-907b995c4477︡︡{"stdout":"scalar field 'h' on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠73c9bad5-8dbb-4f2f-ad42-53f6ffd62b40s︠ h.display() ︡96dcf13a-f3de-47e6-8961-f34818151638︡︡{"html":"The parent of $f$ is the set $C^\infty(U)$ of all smooth scalar fields on $U$, which is a commutative algebra over $\mathbb{R}$:
︡6378e5b9-795d-4b9b-a1e3-818650a426a2︡︡{"done":true,"html":"The parent of $f$ is the set $C^\\infty(U)$ of all smooth scalar fields on $U$, which is a commutative algebra over $\\mathbb{R}$:
"} ︠35d1ef9f-7f4a-4ee1-b5e3-001a15fa9d67s︠ CU = f.parent() ; CU ︡7560d918-b126-4916-83af-435384d12194︡︡{"html":"The base ring of the algebra is the field $\mathbb{R}$, which is represented by Sage's Symbolic Ring (SR):
The base ring of the algebra is the field $\\mathbb{R}$, which is represented by Sage's Symbolic Ring (SR):
\nArithmetic operations on scalar fields are defined through the algebra structure:
︡c61742fa-c198-4ca1-ae24-7b9aef2bc5fc︡︡{"done":true,"html":"Arithmetic operations on scalar fields are defined through the algebra structure:
"} ︠a94be8c6-1c05-435d-8771-0dbd73b7b114s︠ s = f + 2*h ; print s ︡50b3751d-50a7-4776-b54a-e62013686aac︡︡{"stdout":"scalar field on the open subset 'U' of the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠44a130ac-b203-4bb9-92de-c98f780b5e52s︠ s.display() ︡8c0b9a52-90d4-4887-8848-e76b6d134c07︡︡{"html":"The tangent vector space to the manifold at point $p$ is obtained as follows:
︡96b0f07e-b7b4-4937-ba13-066970aaab90︡︡{"done":true,"html":"The tangent vector space to the manifold at point $p$ is obtained as follows:
"} ︠e6e117f8-fb8c-4f85-938d-7f54800d131bs︠ Tp = p.tangent_space() ; Tp ︡588f4c69-16cf-46ad-b53d-80d14e3b3f0c︡︡{"html":"$T_p\, \mathcal{M}$ is a 2-dimensional vector space over $\mathbb{R}$ (represented here by Sage Symbolic Ring (SR)) :
︡51525781-9de6-48ee-98fc-b5ad8c97e911︡︡{"done":true,"html":"$T_p\\, \\mathcal{M}$ is a 2-dimensional vector space over $\\mathbb{R}$ (represented here by Sage Symbolic Ring (SR)) :
"} ︠bdd11192-8232-4fcc-b818-240033c91b05s︠ Tp.category() ︡3a09b67f-d955-4412-ae82-44cb7b89512f︡︡{"html":"$T_p\, \mathcal{M}$ is automatically endowed with vector bases deduced from the vector frames defined around the point:
︡66ce2299-7f92-4c5f-a8b4-0359691d687b︡︡{"done":true,"html":"$T_p\\, \\mathcal{M}$ is automatically endowed with vector bases deduced from the vector frames defined around the point:
"} ︠2f94d774-6c18-4edc-a2f4-4cfd6b87ac17s︠ Tp.bases() ︡039c3cef-1d1f-4fde-a729-789343eb89f1︡︡{"html":"For the tangent space at the point $q$, on the contrary, there is only one pre-defined basis, since $q$ is not in the domain $U$ of the frame associated with coordinates $(r,\theta,\phi)$:
︡cced461d-9c77-40b1-91c6-b04e531f635c︡︡{"done":true,"html":"For the tangent space at the point $q$, on the contrary, there is only one pre-defined basis, since $q$ is not in the domain $U$ of the frame associated with coordinates $(r,\\theta,\\phi)$:
"} ︠74226716-bb76-47d5-bdfa-34d48bec0c44s︠ Tq = q.tangent_space() Tq.bases() ︡9f2b3063-1be5-4476-9ea8-50f5f846191f︡︡{"html":"A random element:
︡bd6e8b16-9769-4f57-912f-6c411ba4ef25︡︡{"done":true,"html":"A random element:
"} ︠40521dc4-d9a1-4ede-aada-a4a6c3f7ba35s︠ v = Tp.an_element() ; print v ︡06460d49-c652-4138-89c5-79774c75e4e7︡︡{"stdout":"tangent vector at point 'p' on 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠ef5202fe-7cb3-4545-920a-9959e7c6474fs︠ v.display() ︡b6176b63-4738-44ac-89b1-2b26cac65166︡︡{"html":"Note that, despite what the above simplified writing may suggest (the mention of the point $p$ or $q$ is omitted in the basis vectors), $u$ and $v$ are different vectors, for they belong to different vector spaces:
︡0b7e9c89-94e4-41e9-b9d8-e2d0f59cb0c0︡︡{"done":true,"html":"Note that, despite what the above simplified writing may suggest (the mention of the point $p$ or $q$ is omitted in the basis vectors), $u$ and $v$ are different vectors, for they belong to different vector spaces:
"} ︠9c3db090-a91e-49fe-a06d-f18b80b7652ds︠ v.parent() ︡47a401aa-ca07-43b9-aa21-297d01c489fd︡︡{"html":"In particular, it is not possible to add $u$ and $v$:
︡732a978e-157c-4b33-aa25-d8b47709373a︡︡{"done":true,"html":"In particular, it is not possible to add $u$ and $v$:
"} ︠f350ee40-04e8-41fd-bd34-c6fba642a948s︠ s = u + v ︡bc0b96ae-aaba-48ae-af48-bca4a7b170aa︡︡{"done":false,"stderr":"Error in lines 1-1\nTraceback (most recent call last):\n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py\", line 905, in execute\n exec compile(block+'\\n', '', 'single') in namespace, locals\n File \"\", line 1, inEach chart defines a vector frame on the chart domain: the so-called coordinate basis:
︡78d40f80-348b-415f-b2e7-fc90a8f7bf5a︡︡{"done":true,"html":"Each chart defines a vector frame on the chart domain: the so-called coordinate basis:
"} ︠1f18223f-3a9a-4db4-8607-d5945dd020des︠ X.frame() ︡c2973171-8821-47f1-9873-e30947390931︡︡{"html":"The list of frames defined on a given open subset is returned by the method frames():
︡33968ee2-1d7e-46c8-8086-44c452950a58︡︡{"done":true,"html":"The list of frames defined on a given open subset is returned by the method frames():
"} ︠1059d39b-9aa9-4c67-8d34-73bfe3ac1e79s︠ M.frames() ︡43d12752-a96b-425a-9005-ec37e0875b64︡︡{"html":"Unless otherwise specified (via the command set_default_frame()), the default frame is that associated with the default chart:
︡f7fd3a82-f24f-4c29-8cde-d082b077383b︡︡{"done":true,"html":"Unless otherwise specified (via the command set_default_frame()), the default frame is that associated with the default chart:
"} ︠3829147c-c075-4ffe-a4cd-f62c01aabe84s︠ M.default_frame() is M.default_chart().frame() ︡b5a54154-31d1-49fd-b505-e362beef8d01︡︡{"html":"Individual elements of a frame can be accessed by means of their indices:
︡06441bdf-cabe-481a-ab7e-ede791c18291︡︡{"done":true,"html":"Individual elements of a frame can be accessed by means of their indices:
"} ︠e4a28517-138a-4794-8db7-c2e62bebaa4bs︠ e = U.default_frame() ; e2 = e[2] ; e2 ︡f2df27ec-777f-4314-ad52-2265029dbd07︡︡{"html":"We may define a new vector field as follows:
︡6a86a239-a9d3-43df-804e-98712a5065dd︡︡{"done":true,"html":"We may define a new vector field as follows:
"} ︠4f999c8d-998b-4506-b193-4d13b3bcbd08s︠ v = e[2] + 2*x*e[3] ; print v ︡ba61b0fc-b055-4f1e-a9b9-767050484814︡︡{"stdout":"vector field on the open subset 'U' of the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠29d26594-ef19-41ee-8a57-243036526eads︠ v.display() ︡66e55e10-6983-4c12-af8b-51d0f1e2dfeb︡︡{"html":"A vector field can be defined by its components with respect to a given vector frame. When the latter is not specified, the open set's default frame is of course assumed:
︡f79548a9-8b47-4cf1-b657-3ee5bbdcad92︡︡{"done":true,"html":"A vector field can be defined by its components with respect to a given vector frame. When the latter is not specified, the open set's default frame is of course assumed:
"} ︠49f874cd-a6a5-4832-80a8-1ca1f9b71542s︠ v = U.vector_field(name='v') # vector field defined on the open set U v[1] = 1+y v[2] = -x v[3] = x*y*z v.display() ︡a9b8892a-a2f8-4522-b55f-8f025f17fbd0︡︡{"html":"Vector fields on $U$ are Sage element objects, whose parent is the set $\mathcal{X}(U)$ of vector fields defined on $U$:
︡6d743779-6cfa-44e1-a5fa-2c9df191ecf9︡︡{"done":true,"html":"Vector fields on $U$ are Sage element objects, whose parent is the set $\\mathcal{X}(U)$ of vector fields defined on $U$:
"} ︠76044b89-7499-400f-8e04-71705eb86c70s︠ v.parent() ︡7f31be47-5f80-4bc4-a333-c9ea41967482︡︡{"html":"The set $\mathcal{X}(U)$ is a module over the commutative algebra $C^\infty(U)$ of scalar fields on $U$:
︡2bb4455d-aa1b-4844-b80d-af07f028a093︡︡{"done":true,"html":"The set $\\mathcal{X}(U)$ is a module over the commutative algebra $C^\\infty(U)$ of scalar fields on $U$:
"} ︠7296404e-eab1-4604-9459-a32f207bd60ds︠ print v.parent() ︡72cfd650-3b69-4c9e-9627-84ac2d9c1d0b︡︡{"stdout":"free module X(U) of vector fields on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠60b32e80-130d-4e6c-b412-a582305a9965s︠ print v.parent().category() ︡87e147e3-da7d-43e9-ae4a-856e99449ff3︡︡{"stdout":"Category of modules over algebra of scalar fields on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠38fa55c6-fd1a-4d0c-877d-0f7cdc8e5f68s︠ v.parent().base_ring() ︡8575ac63-dfcb-4b62-a68d-e50c8cac29aa︡︡{"html":"A vector field acts on scalar fields:
︡e73d49da-a371-4098-abe8-2a4768291e69︡︡{"done":true,"html":"A vector field acts on scalar fields:
"} ︠7db8af20-e34e-4a70-93f3-3e0f4e4eb379s︠ f.display() ︡80be631f-820f-4c7b-814e-7991b3afa529︡︡{"html":"Unset components are assumed to be zero:
︡6a23a9e1-744b-4ab1-bf38-481ae18753f2︡︡{"done":true,"html":"Unset components are assumed to be zero:
"} ︠c7fb6ac5-eed1-499d-8b0c-afbe6eefe339s︠ w = U.vector_field(name='w') w[2] = 3 w.display() ︡fad5e45b-b4ef-43e2-be22-c9cbd2279233︡︡{"html":"A vector field on $U$ can be expanded in the vector frame associated with the chart $(r,\theta,\phi)$:
︡04f60ab8-ecf2-4734-8685-f6ba0dd9ec16︡︡{"done":true,"html":"A vector field on $U$ can be expanded in the vector frame associated with the chart $(r,\\theta,\\phi)$:
"} ︠603e8358-f481-4449-b9be-69261dcb6489s︠ v.display(Y.frame()) ︡b3ea5aa3-5a29-4478-be11-4ab1bf3bd7d3︡︡{"html":"By default, the components are expressed in terms of the default coordinates $(x,y,z)$. To express them in terms of the coordinates $(r,\theta,\phi)$, one should add the corresponding chart as the second argument of the method display():
︡1a49e0b5-14e9-4b62-ab88-905f0ed19916︡︡{"done":true,"html":"By default, the components are expressed in terms of the default coordinates $(x,y,z)$. To express them in terms of the coordinates $(r,\\theta,\\phi)$, one should add the corresponding chart as the second argument of the method display():
"} ︠1cebadaa-e7cb-4b9e-8534-94be5bcf55a6s︠ v.display(Y.frame(), Y) ︡cd733111-bca3-4273-a94f-e33cec64c9b9︡︡{"html":"The components of a tensor field w.r.t. the default frame can also be obtained as a list, via the command [:]:
︡b0ec1a23-51ad-4194-952c-c64ff5c72e9d︡︡{"done":true,"html":"The components of a tensor field w.r.t. the default frame can also be obtained as a list, via the command [:]:
"} ︠e8f59a30-fd36-493f-aadd-81e0dbebb484s︠ v[:] ︡4448efe1-24ad-44bb-b6a5-fc440bb7c8b3︡︡{"html":"An alternative is to use the method display_comp():
︡24d093a2-5cf4-41d8-9d27-7211b055d837︡︡{"done":true,"html":"An alternative is to use the method display_comp():
"} ︠dd1d6940-3510-4ae0-abc9-1164d122389ds︠ v.display_comp() ︡cc6fcdd8-8452-4a4d-b4cf-3d9f04711a86︡︡{"html":"To obtain the components w.r.t. to another frame, one may go through the method comp() and specify the frame:
︡a8b2dd4a-b437-46a6-923b-7611e5f6394a︡︡{"done":true,"html":"To obtain the components w.r.t. to another frame, one may go through the method comp() and specify the frame:
"} ︠0dd59daf-ce78-4946-a631-df5053bad785s︠ v.comp(Y.frame())[:] ︡5ecd8365-a172-481d-9722-59dd816714f3︡︡{"html":"However a shortcut is to provide the frame as the first argument of the square brackets:
︡24cf2a82-a082-4c04-af8b-b387e70e9372︡︡{"done":true,"html":"However a shortcut is to provide the frame as the first argument of the square brackets:
"} ︠d2a317fe-ee6d-43ef-916f-34c7276a9973s︠ v[Y.frame(), :] ︡ab4e9d56-0edf-49b7-b9ba-2f203d863f86︡︡{"html":"Components are shown expressed in terms of the default's coordinates; to get them in terms of the coordinates $(r,\theta,\phi)$ instead, add the chart name as the last argument in the square brackets:
︡fd277014-709c-4bec-a135-0944590e586d︡︡{"done":true,"html":"Components are shown expressed in terms of the default's coordinates; to get them in terms of the coordinates $(r,\\theta,\\phi)$ instead, add the chart name as the last argument in the square brackets:
"} ︠ce774e5d-04c6-41e1-95c1-5c6e0470f647s︠ v[Y.frame(), :, Y] ︡aadaf102-874b-446a-af86-741e5dd59f77︡︡{"html":"or specify the chart in display_comp():
or specify the chart in display_comp():
To get some components of a vector as a scalar field, instead of a coordinate expression, use double square brackets:
︡dff5a2f8-3805-4b40-b1b4-6f886de263a5︡︡{"done":true,"html":"To get some components of a vector as a scalar field, instead of a coordinate expression, use double square brackets:
"} ︠cffc2620-80e7-4fed-9269-6218fb93baf4s︠ print v[[1]] ︡5f286bcf-ef1b-4be9-97af-3a323112726b︡︡{"stdout":"scalar field on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠c41da077-3f28-4c42-80a6-ab3b2c124accs︠ v[[1]].display() ︡1c65a74f-9cbf-4d4f-b3ed-f4b29858f590︡︡{"html":"A vector field can be defined with components being unspecified functions of the coordinates:
︡c5e71eed-5da0-4b71-86c6-96c22d742bb6︡︡{"done":true,"html":"A vector field can be defined with components being unspecified functions of the coordinates:
"} ︠ff2c2093-f5f3-4a4b-9403-9a2fbc07aff8s︠ u = U.vector_field(name='u') u[:] = [function('u_x', x,y,z), function('u_y', x,y,z), function('u_z', x,y,z)] u.display() ︡fcee5829-85e4-4b28-a02a-3bb0076d90ac︡︡{"html":"The value of a vector field at some point of the manifold is obtained via the method at():
︡7dc3ad3c-e659-429f-b3cc-96902f3cfc43︡︡{"done":true,"html":"The value of a vector field at some point of the manifold is obtained via the method at():
"} ︠cf00880a-18c5-4a7c-b3df-b62a67110d36s︠ vp = v.at(p) ; print vp ︡db8af25f-118b-4a64-b2e3-19727c988194︡︡{"stdout":"tangent vector v at point 'p' on 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠41a0c19b-4e37-4e64-b9b6-8a431fffe22ds︠ vp.display() ︡579817b8-6736-45fe-8a43-008be6f84dd4︡︡{"html":"Indeed, recall that, w.r.t. chart X_U=$(x,y,z)$, the coordinates of the point $p$ and the components of the vector field $v$ are
︡70c1dacd-50ef-4e26-9505-ff34c70bbc2b︡︡{"done":true,"html":"Indeed, recall that, w.r.t. chart X_U=$(x,y,z)$, the coordinates of the point $p$ and the components of the vector field $v$ are
"} ︠4199a7ee-bb74-40d6-865b-be86b63e35e8s︠ p.coord(X_U) ︡673df1cd-c8e4-47f9-aa07-66270faf01b5︡︡{"html":"Note that to simplify the writing, the symbol used to denote the value of the vector field at point $p$ is the same as that of the vector field itself (namely $v$); this can be changed by the method set_name():
︡d7da94ac-333e-4dc5-b305-25ad242f0b7f︡︡{"done":true,"html":"Note that to simplify the writing, the symbol used to denote the value of the vector field at point $p$ is the same as that of the vector field itself (namely $v$); this can be changed by the method set_name():
"} ︠6b67d147-0a22-4dba-974f-dc95d48c1d83s︠ vp.set_name(latex_name='v|_p') vp.display() ︡8299155c-e2d0-4464-8363-86a304949748︡︡{"html":"Of course, $v|_p$ belongs to the tangent space at $p$:
︡59c1d9d1-ced3-46a2-8051-cfa1456280d1︡︡{"done":true,"html":"Of course, $v|_p$ belongs to the tangent space at $p$:
"} ︠d3c5188a-2502-4068-a493-62bfce21e19fs︠ vp.parent() ︡feae02b0-0182-4cf0-80c1-85e809512d2f︡︡{"html":"A 1-form on $\mathcal{M}$ is a field of linear forms. For instance, it can be the differential of a scalar field:
︡dc0409a9-5432-45a4-ace9-0c1dff968fbb︡︡{"done":true,"html":"A 1-form on $\\mathcal{M}$ is a field of linear forms. For instance, it can be the differential of a scalar field:
"} ︠e65dc99c-a63e-4300-be78-f169d6c51c35s︠ df = f.differential() ; print df ︡57e21bda-714f-4805-8270-f35e81b2778e︡︡{"stdout":"1-form 'df' on the open subset 'U' of the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠6ec7bc4d-e745-4861-9fc2-cdf5eec3ce6fs︠ df.display() ︡7b3439be-5add-4647-be40-75c561db2a12︡︡{"html":"In the above writing, the 1-form is expanded over the basis $(\mathrm{d}x, \mathrm{d}y, \mathrm{d}z)$ associated with the chart $(x,y,z)$. This basis can be accessed via the member coframe:
︡9ccb804d-8273-4b10-95e2-c141eeec731f︡︡{"done":true,"html":"In the above writing, the 1-form is expanded over the basis $(\\mathrm{d}x, \\mathrm{d}y, \\mathrm{d}z)$ associated with the chart $(x,y,z)$. This basis can be accessed via the member coframe:
"} ︠071e7b4c-8b16-4f28-bbe1-cbf691876bc1s︠ dX = X.coframe() ; dX ︡d8993cef-b5a0-41bc-a38d-4b3ddd7e55a3︡︡{"html":"The list of all coframes defined on a given manifold open subset is returned by the method coframes():
︡8498b451-410a-4848-8d84-b3a6a9ec7610︡︡{"done":true,"html":"The list of all coframes defined on a given manifold open subset is returned by the method coframes():
"} ︠ff0954e7-8771-4b5b-b675-27219e3e7d38s︠ M.coframes() ︡0f473e60-2c74-4ac7-b932-df7f72eeb478︡︡{"html":"As for a vector field, the value of the differential form at some point on the manifold is obtained by the method at():
︡97e7ef04-8d4b-4bab-9627-8d1651907f07︡︡{"done":true,"html":"As for a vector field, the value of the differential form at some point on the manifold is obtained by the method at():
"} ︠1f678271-c6d1-4fa8-8054-a74c61e3dafbs︠ dfp = df.at(p) ; print dfp ︡359a8785-9207-4d5f-8808-aeb661c82209︡︡{"stdout":"Linear form df on the tangent space at point 'p' on 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠136a8bdc-c34e-448d-8464-2875f9b26e5fs︠ dfp.display() ︡dcf5f0e4-cd48-4487-ba24-bea88b39c2e3︡︡{"html":"Recall that
︡70750078-1fee-4543-899b-3f796c18e392︡︡{"done":true,"html":"Recall that
"} ︠0f45b2fc-91c3-47b1-ace1-d1b4cc42abf3s︠ p.coord() ︡f753b109-a67d-408e-9fb2-13e169c91953︡︡{"html":"The linear form $\mathrm{d}f|_p$ belongs to the dual of the tangent vector space at $p$:
︡92679c1e-b0ef-4a4f-b340-d62a4a5301c7︡︡{"done":true,"html":"The linear form $\\mathrm{d}f|_p$ belongs to the dual of the tangent vector space at $p$:
"} ︠bf610105-f0a1-4a30-96bb-2dc999ea9e9es︠ dfp.parent() ︡1804cd08-6eec-4e20-a03b-3ca2c5324b41︡︡{"html":"As such, it is acting on vectors at $p$, yielding a real number:
︡a2f0e4e7-2231-43de-acb0-89b89c4cf558︡︡{"done":true,"html":"As such, it is acting on vectors at $p$, yielding a real number:
"} ︠c617a2d0-5f0b-42a5-8b4f-5354a069c02bs︠ print vp ; vp.display() ︡a6451967-903a-41a0-a12d-7a3855693820︡︡{"stdout":"tangent vector v at point 'p' on 3-dimensional manifold 'M'\n","done":false}︡{"html":"The differential 1-form of the unspecified scalar field $h$:
︡8d29f777-6b68-489e-973a-2abb3b4577fb︡︡{"done":true,"html":"The differential 1-form of the unspecified scalar field $h$:
"} ︠a9804882-586b-4ba0-8049-db1d9fb4f155s︠ h.display() ; dh = h.differential() ; dh.display() ︡b197b119-100c-4dea-9623-b33915fa8157︡︡{"html":"A 1-form can also be defined from scratch:
︡3d65565f-3b9c-49dc-b08a-7addd67e1cea︡︡{"done":true,"html":"A 1-form can also be defined from scratch:
"} ︠837a6ee9-7756-45ff-860c-a623714c66ebs︠ om = U.one_form('omega', r'\omega') ; print om ︡d5df3dd0-fbba-41c2-893b-3c002b273e90︡︡{"stdout":"1-form 'omega' on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠eb50457b-43b7-4db6-9974-56c562896101i︠ %htmlIt can be specified by providing its components in a given coframe:
︡af2ba825-100d-4c86-9bf2-f34b03fdb431︡︡{"done":true,"html":"It can be specified by providing its components in a given coframe:
"} ︠5b2e4110-e2f3-44a4-89cd-071c9b5610a8s︠ om[:] = [x^2+y^2, z, x-z] # components in the default coframe (dx,dy,dz) om.display() ︡be6021aa-9342-4a51-ab4b-3a6ec0e99ffa︡︡{"html":"Of course, one may set the components in a frame different from the default one:
︡dfe75692-0ca7-4ce8-b9a0-0c85e8b59bd5︡︡{"done":true,"html":"Of course, one may set the components in a frame different from the default one:
"} ︠cc8162dd-9e2c-492d-8196-a0a008030bfes︠ om[Y.frame(), :, Y] = [r*sin(th)*cos(ph), 0, r*sin(th)*sin(ph)] om.display(Y.frame(), Y) ︡91eeab69-18c2-41d2-9288-fa54c2b08963︡︡{"html":"The components in the coframe $(\mathrm{d}x,\mathrm{d}y,\mathrm{d}z)$ are updated automatically:
︡f328aeec-5390-4bea-9be4-3d0222f18234︡︡{"done":true,"html":"The components in the coframe $(\\mathrm{d}x,\\mathrm{d}y,\\mathrm{d}z)$ are updated automatically:
"} ︠c95134ec-35a4-44b2-a2bb-a226e09907afs︠ om.display() ︡afc54b3c-70a0-466f-b07e-b51e34b3dd84︡︡{"html":"Let us revert to the values set previously:
︡f59e9614-968c-4f11-b02b-9f2b2b61861f︡︡{"done":true,"html":"Let us revert to the values set previously:
"} ︠772e57b8-a0fc-48f9-8bb2-57949287c818s︠ om[:] = [x^2+y^2, z, x-z] om.display() ︡df8c7bda-37aa-42e9-b761-2c8e6596f4d0︡︡{"html":"This time, the components in the coframe $(\mathrm{d}r, \mathrm{d}\theta,\mathrm{d}\phi)$ are those that are updated:
︡97da5448-df65-41a2-8741-739ee2388018︡︡{"done":true,"html":"This time, the components in the coframe $(\\mathrm{d}r, \\mathrm{d}\\theta,\\mathrm{d}\\phi)$ are those that are updated:
"} ︠d7201736-3ab6-4b2a-8142-d500f169473bs︠ om.display(Y.frame(), Y) ︡f12a9c4e-2188-4b4a-9095-c1dff1ce5017︡︡{"html":"A 1-form acts on vector fields, resulting in a scalar field:
︡983e7e9b-5141-46a7-9e81-805e91bd9f17︡︡{"done":true,"html":"A 1-form acts on vector fields, resulting in a scalar field:
"} ︠e124d205-b130-4c96-b72f-9df2f1349679s︠ v.display() ; om.display() ; print om(v) ; om(v).display() ︡e763bf86-f7fe-405b-a539-783c61c5692a︡︡{"html":"In the case of a differential 1-form, the following identity holds:
︡265289bd-a632-44af-9b10-b1bb2c176c56︡︡{"done":true,"html":"In the case of a differential 1-form, the following identity holds:
"} ︠3648caee-fb1d-4240-a00d-c76280d4e606s︠ df(v) == v(f) ︡c288cd7a-4ece-4d71-bc10-9d12d8616210︡︡{"html":"1-forms are Sage element objects, whose parent is the set $\mathcal{T}^{(0,1)}(U)$ of all tensor fields of type (0,1) defined on $U$:
︡2261e9ab-676f-4948-b4dc-1452c0642381︡︡{"done":true,"html":"1-forms are Sage element objects, whose parent is the set $\\mathcal{T}^{(0,1)}(U)$ of all tensor fields of type (0,1) defined on $U$:
"} ︠685526f7-8f46-430b-b760-f14dc9832abbs︠ df.parent() ︡fca3088a-13b7-47fe-9d35-658257c82efc︡︡{"html":"$\mathcal{T}^{(0,1)}(U)$ is actually the dual of the free module $\mathcal{X}(U)$:
︡2bbf3bb7-7657-4920-ae52-b9d42012e203︡︡{"done":true,"html":"$\\mathcal{T}^{(0,1)}(U)$ is actually the dual of the free module $\\mathcal{X}(U)$:
"} ︠4572f86b-c600-4f2d-ab61-b7263ddb52ces︠ df.parent() is v.parent().dual() ︡cb2317f6-260c-470f-b6df-8f981d02aa8a︡︡{"html":"The exterior product of two 1-forms is taken via the method wedge() and results in a 2-form:
︡6cf7ac81-2d65-4a43-985a-59bbba304577︡︡{"done":true,"html":"The exterior product of two 1-forms is taken via the method wedge() and results in a 2-form:
"} ︠cdbcd466-322f-43b7-b910-7269ae86e20cs︠ a = om.wedge(df) ; print a ; a.display() ︡d3787221-4a47-463e-86b2-0c4598a7e5ed︡︡{"stdout":"2-form 'omega/\\df' on the open subset 'U' of the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"html":"A matrix view of the components:
︡7559bfc1-6ce6-4a2d-935c-da9719ac7b68︡︡{"done":true,"html":"A matrix view of the components:
"} ︠ee024ba9-ad34-4b45-b06b-9fde68c852abs︠ a[:] ︡d584a816-676b-4b77-9467-36fc73ef6e88︡︡{"html":"Displaying only the non-vanishing components, skipping the redundant ones (i.e. those that can be deduced by antisymmetry):
︡c1a9fc87-dcc3-4df5-b4e8-4399aad87fea︡︡{"done":true,"html":"Displaying only the non-vanishing components, skipping the redundant ones (i.e. those that can be deduced by antisymmetry):
"} ︠fe5e0d33-25cb-4a79-9463-d150ad5f1020s︠ a.display_comp(only_nonredundant=True) ︡4a80426d-2b25-4615-a25f-40d39979942b︡︡{"html":"The 2-form $\omega\wedge\mathrm{d}f$ can be expanded on the $(\mathrm{d}r,\mathrm{d}\theta,\mathrm{d}\phi)$ coframe:
︡4e033815-13b4-4728-8aba-325154033d81︡︡{"done":true,"html":"The 2-form $\\omega\\wedge\\mathrm{d}f$ can be expanded on the $(\\mathrm{d}r,\\mathrm{d}\\theta,\\mathrm{d}\\phi)$ coframe:
"} ︠90892827-1e73-4dda-b97b-0b99b3e5f144s︠ a.display(Y.frame(), Y) ︡7d8d44b8-4024-4517-a7c4-d2dd1c6e81c4︡︡{"html":"As a 2-form, $A:=\omega\wedge\mathrm{d}f$ can be applied to a pair of vectors and is antisymmetric:
︡90d92980-6c9c-4ac9-b082-ec939c23be00︡︡{"done":true,"html":"As a 2-form, $A:=\\omega\\wedge\\mathrm{d}f$ can be applied to a pair of vectors and is antisymmetric:
"} ︠f8b800d3-fcc6-4620-930f-49fb63db9ae0s︠ a.set_name('A') print a(u,v) ; a(u,v).display() ︡ee857686-3798-41c2-b96b-1ad124d55a1a︡︡{"stdout":"scalar field 'A(u,v)' on the open subset 'U' of the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"html":"The exterior derivative of a differential form is taken by invoking the function xder():
︡4a52d8f3-3cd8-4f00-a6c0-d4561152343f︡︡{"done":true,"html":"The exterior derivative of a differential form is taken by invoking the function xder():
"} ︠a074dfe9-5720-42ad-9ddb-a063f2e4665fs︠ dom = xder(om) ; print dom ; dom.display() ︡7da96cf4-94e9-4a14-959e-f76d7b06136f︡︡{"stdout":"2-form 'domega' on the open subset 'U' of the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"html":"The exterior derivative is nilpotent:
︡66516c2d-e1c7-44e7-845c-fd7cc748675f︡︡{"done":true,"html":"The exterior derivative is nilpotent:
"} ︠68e0b720-6502-41bb-8d23-4506ba147be5s︠ ddf = xder(df) ; ddf.display() ︡121e1eb1-5b14-4e07-8524-7649a7f2e23a︡︡{"html":"The Lie derivative of any tensor field with respect to a vector field is computed by the method lie_der(), with the vector field as argument:
︡73167f5f-85ae-41f8-9580-3fa0468e2033︡︡{"done":true,"html":"The Lie derivative of any tensor field with respect to a vector field is computed by the method lie_der(), with the vector field as argument:
"} ︠c595d077-2e76-4bcc-8ad9-814927510997s︠ lv_om = om.lie_der(v) ; print lv_om ; lv_om.display() ︡9d8c8cdb-8719-4da3-ac51-d7833cbe4e67︡︡{"stdout":"1-form on the open subset 'U' of the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"html":"Let us check Cartan identity on the 1-form $\omega$:
$\mathcal{L}_v \omega = v\cdot \mathrm{d}\omega + \mathrm{d}\langle \omega, v\rangle$
and on the 2-form $A$:
$\mathcal{L}_v A = v\cdot \mathrm{d}A + \mathrm{d}(v\cdot A)$
︡33261be4-6d21-410d-a54e-a3c4b9c1f50b︡︡{"done":true,"html":"Let us check Cartan identity on the 1-form $\\omega$:
\n$\\mathcal{L}_v \\omega = v\\cdot \\mathrm{d}\\omega + \\mathrm{d}\\langle \\omega, v\\rangle$
\nand on the 2-form $A$:
\n$\\mathcal{L}_v A = v\\cdot \\mathrm{d}A + \\mathrm{d}(v\\cdot A)$
"} ︠e9205808-6997-4ded-9afc-4aa275703dd0s︠ om.lie_der(v) == v.contract(xder(om)) + xder(om(v)) ︡b667f588-0b1f-46f5-b30a-698240e7f288︡︡{"html":"The Lie derivative of a vector field along another one is the commutator of the two vectors fields:
︡3fa8aeae-b2c6-4ab2-89c3-bdc8bdc8baf8︡︡{"done":true,"html":"The Lie derivative of a vector field along another one is the commutator of the two vectors fields:
"} ︠4ec55a35-169e-44ac-ae61-fe813fa985cfs︠ v.lie_der(u)(f) == u(v(f)) - v(u(f)) ︡d9c2577c-cad5-4fe7-b7b2-bf94949529ae︡︡{"html":"Up to now, we have encountered tensor fields
More generally, tensor fields of any type $(p,q)$ can be introduced in SageManifolds. For instance a tensor field of type (1,2) on the open subset $U$ is declared as follows:
︡6b42f3ad-a8d9-4de3-9600-37a62b3d98d0︡︡{"done":true,"html":"Up to now, we have encountered tensor fields
\nMore generally, tensor fields of any type $(p,q)$ can be introduced in SageManifolds. For instance a tensor field of type (1,2) on the open subset $U$ is declared as follows:
"} ︠3a20fdd8-5d93-44ec-a8dc-e4f0f0790b7as︠ t = U.tensor_field(1, 2, name='T') ; print t ︡f208003c-51bd-4cf8-8eff-7413df8f52cb︡︡{"stdout":"tensor field 'T' of type (1,2) on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠20f1653c-ca54-46d9-84ff-aae74d33a75bi︠ %htmlAs for vectors or 1-forms, the tensor's components with respect to the domain's default frame are set by means of square brackets:
︡fdd7fb6c-c019-4eee-9c87-e04595d8a5e1︡︡{"done":true,"html":"As for vectors or 1-forms, the tensor's components with respect to the domain's default frame are set by means of square brackets:
"} ︠cb5af8eb-abb3-48ce-9ff9-466e4ac28d36s︠ t[1,2,1] = 1 + x^2 t[3,2,1] = x*y*z ︡72414bc7-f1b0-49da-a651-761f9452fb77︡︡{"done":true} ︠249dbdea-e836-443a-ac2d-ef39079bddd6i︠ %htmlUnset components are zero:
︡67596379-1da2-4e02-b020-1d671ddd3ad7︡︡{"done":true,"html":"Unset components are zero:
"} ︠f9b11e12-0fdb-4426-a2ea-b6ea2b788da6s︠ t.display() ︡25d99353-4bff-4da9-8d20-6b4cf6b3d1bb︡︡{"html":"Display of the nonzero components:
︡710c5a54-f4b6-4f52-ace6-800de1c520b2︡︡{"done":true,"html":"Display of the nonzero components:
"} ︠3e83f60d-03a4-45b0-bacc-7da3de698f24s︠ t.display_comp() ︡50a1422a-2ff6-40c5-bcca-8d80a47971f7︡︡{"html":"Double square brackets return the component (still w.r.t. the default frame) as a scalar field, while single square brackets return the expression of this scalar field in terms of the domain's default coordinates:
︡a8c9c0f9-b28e-437f-8e35-a5424df3da17︡︡{"done":true,"html":"Double square brackets return the component (still w.r.t. the default frame) as a scalar field, while single square brackets return the expression of this scalar field in terms of the domain's default coordinates:
"} ︠959974bf-9d5c-4da2-b3ef-3e0bddd1be1bs︠ print t[[1,2,1]] ; t[[1,2,1]].display() ︡c0671986-dbcb-4844-be7e-7a43bce8a2cd︡︡{"stdout":"scalar field on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"html":"A tensor field of type (1,2) maps a 3-tuple (1-form, vector field, vector field) to a scalar field:
︡bd345d02-b702-4458-b5af-103a8025f3fe︡︡{"done":true,"html":"A tensor field of type (1,2) maps a 3-tuple (1-form, vector field, vector field) to a scalar field:
"} ︠f8185a4f-1fc4-4753-995a-1e6d185cb26es︠ print t(om, u, v) ; t(om, u, v).display() ︡7a1aef37-07b1-4836-bc8e-b035c14e782d︡︡{"stdout":"scalar field 'T(omega,u,v)' on the open subset 'U' of the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"html":"As for vectors and differential forms, the tensor components can be taken in any frame defined on the manifold:
︡cf8c653c-cad0-499f-849e-ac80e3848f16︡︡{"done":true,"html":"As for vectors and differential forms, the tensor components can be taken in any frame defined on the manifold:
"} ︠5148bf65-c257-43fd-abf9-a4b878f8a43fs︠ t[Y.frame(), 1,1,1, Y] ︡7976ba93-9da0-4b26-9c0a-67e19a93cd9c︡︡{"html":"The tensor product is denoted by *:
︡72ff4cd9-d3cb-4473-990a-502f41dfa493︡︡{"done":true,"html":"The tensor product is denoted by *:
"} ︠86b76f4e-1b85-48fe-9a5a-f44deea02aa8s︠ v.tensor_type() ; a.tensor_type() ︡f2291d3f-9a13-40e9-b0b7-bc01dbe3e6cf︡︡{"html":"The tensor product preserves the (anti)symmetries: since $A$ is a 2-form, it is antisymmetric with respect to its two arguments (positions 0 and 1); as a result, b is antisymmetric with respect to its last two arguments (positions 1 and 2):
︡68f7052a-2330-4cd4-a610-92fecd2f52a9︡︡{"done":true,"html":"The tensor product preserves the (anti)symmetries: since $A$ is a 2-form, it is antisymmetric with respect to its two arguments (positions 0 and 1); as a result, b is antisymmetric with respect to its last two arguments (positions 1 and 2):
"} ︠97fced17-1f49-449e-91c3-a13e624cdcafs︠ a.symmetries() ︡78143a78-00e6-479d-9088-6cbadcd6f05f︡︡{"stdout":"no symmetry; antisymmetry: (0, 1)\n","done":false}︡{"done":true} ︠02772967-2ce3-4cb0-834d-a8c7e618200es︠ b.symmetries() ︡7b51744e-ab27-4fb6-8b08-115d6692ea66︡︡{"stdout":"no symmetry; antisymmetry: (1, 2)\n","done":false}︡{"done":true} ︠f2e69fa9-110c-4b14-8ef7-aefbc80e03d8i︠ %htmlStandard tensor arithmetics is implemented:
︡6589c29f-213f-4503-838d-28b81df17911︡︡{"done":true,"html":"Standard tensor arithmetics is implemented:
"} ︠2bed6170-9e01-4dda-82d4-c2f300a3b2a5s︠ s = - t + 2*f* b ; print s ︡0e29b1db-549b-4a62-8f61-91a02742b961︡︡{"stdout":"tensor field of type (1,2) on the open subset 'U' of the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠b3c788a3-2301-4a2d-8f69-c49172c530afi︠ %htmlTensor contractions are dealt with by the methods trace() and contract(): for instance, let us contract the tensor $T$ w.r.t. its first two arguments (positions 0 and 1), i.e. let us form the tensor $c$ of components $c_i = T^k_{\ \, k i}$:
︡9b4b538b-24a2-4c9e-b87c-9779eceda162︡︡{"done":true,"html":"Tensor contractions are dealt with by the methods trace() and contract(): for instance, let us contract the tensor $T$ w.r.t. its first two arguments (positions 0 and 1), i.e. let us form the tensor $c$ of components $c_i = T^k_{\\ \\, k i}$:
"} ︠76df2640-66b0-4d2a-88b5-cb7ae2a68bafs︠ c = t.trace(0,1) print c ︡77c7f3df-7f4e-4ba8-9d7f-f62d95e160b1︡︡{"stdout":"1-form on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠b9393234-cd3a-4f40-bec5-3e526ad8dcadi︠ %htmlAn alternative to the writing trace(0,1) is to use the index notation to denote the contraction: the indices are given in a string inside the [] operator, with '^' in front of the contravariant indices and '_' in front of the covariant ones:
︡298519db-e601-4900-8771-5cb7523ceacc︡︡{"done":true,"html":"An alternative to the writing trace(0,1) is to use the index notation to denote the contraction: the indices are given in a string inside the [] operator, with '^' in front of the contravariant indices and '_' in front of the covariant ones:
"} ︠85545833-2773-45da-8a51-c8d2831d1dfbs︠ c1 = t['^k_ki'] print c1 c1 == c ︡efc4df97-c64c-420e-a20d-c73f1b412678︡︡{"stdout":"1-form on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"html":"The contraction is performed on the repeated index (here k); the letter denoting the remaining index (here i) is arbitrary:
︡dc56652e-d362-499a-9317-5ece116c73a4︡︡{"done":true,"html":"The contraction is performed on the repeated index (here k); the letter denoting the remaining index (here i) is arbitrary:
"} ︠a75e6ef6-5401-4dc5-ba33-50d452664cd7s︠ t['^k_kj'] == c ︡0ae17370-673c-4537-a25c-932cba2cc1fe︡︡{"html":"It can even be replaced by a dot:
︡f55d76e5-3e36-48a7-8dc1-a40d89890734︡︡{"done":true,"html":"It can even be replaced by a dot:
"} ︠5fd3de9a-ab18-43ac-b984-058027c0ea21s︠ t['^k_k.'] == c ︡ffa34b39-228f-44b7-ac60-cdc0410e33df︡︡{"html":"LaTeX notations are allowed:
︡a9029b9e-96d9-4f50-9b8d-12dd941a0e5d︡︡{"done":true,"html":"LaTeX notations are allowed:
"} ︠80866545-96bb-4afd-98a1-867bdfaae525s︠ t['^{k}_{ki}'] == c ︡9dfebe80-59de-4afe-85b0-4a7b82a7b757︡︡{"html":"The contraction $T^i_{\ j k} v^k$ of the tensor fields $T$ and $v$ is taken as follows (2 refers to the last index position of $T$ and 0 to the only index position of v):
︡53ef268c-7983-42d1-950a-27773344054b︡︡{"done":true,"html":"The contraction $T^i_{\\ j k} v^k$ of the tensor fields $T$ and $v$ is taken as follows (2 refers to the last index position of $T$ and 0 to the only index position of v):
"} ︠b64fb6dc-0a17-46a2-ac5b-00bf157dce82s︠ tv = t.contract(2, v, 0) print tv ︡bf0124ea-4ff9-46cf-bd43-14377ee7b550︡︡{"stdout":"tensor field of type (1,1) on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠bf5cb16a-1ad0-4549-bddc-5550010a8d1ci︠ %htmlSince 2 corresponds to the last index position of $T$ and 0 to the first index position of $v$, a shortcut for the above is
︡da080a42-3922-44d1-8301-f788fcd77710︡︡{"done":true,"html":"Since 2 corresponds to the last index position of $T$ and 0 to the first index position of $v$, a shortcut for the above is
"} ︠b0a73b1f-38e9-4888-9054-6eb7f2fc1e4as︠ tv1 = t.contract(v) print tv1 ︡c890f9de-3713-45fd-a67f-4a97eaeb8ef3︡︡{"stdout":"tensor field of type (1,1) on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠ca3f7783-689e-4c5e-9d81-1ce4c698809cs︠ tv1 == tv ︡33c1c20f-c593-4eaa-9912-1e2fd49596ad︡︡{"html":"Instead of contract(), the index notation, combined with the * operator, can be used to denote the contraction:
︡c1476adb-6b29-4416-b6e6-9cea064a11be︡︡{"done":true,"html":"Instead of contract(), the index notation, combined with the * operator, can be used to denote the contraction:
"} ︠1b282387-5d3b-471b-919e-97256358bba3s︠ t['^i_jk']*v['^k'] == tv ︡9fb5ea94-1192-4cc8-835c-630728be90d4︡︡{"html":"The non-repeated indices can be replaced by dots:
︡42cbbd8f-68df-4857-ae05-9042afaed9bd︡︡{"done":true,"html":"The non-repeated indices can be replaced by dots:
"} ︠8f6625b8-4da5-4888-8084-edf0510632dbs︠ t['^._.k']*v['^k'] == tv ︡c96583ad-4082-468d-91fb-3f7c597e5762︡︡{"html":"A Riemannian metric on the manifold $\mathcal{M}$ is declared as follows:
︡9cffd833-31a0-475b-99e6-6dcb893f28fb︡︡{"done":true,"html":"A Riemannian metric on the manifold $\\mathcal{M}$ is declared as follows:
"} ︠9d5bf35b-20a9-44ec-83d5-0f1ef584a6f4s︠ g = M.riemann_metric('g') print g ︡975c4f60-126d-4dbb-a03f-29cac8af8ffa︡︡{"stdout":"Riemannian metric 'g' on the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠44b554ec-1646-43d3-8216-dc3157f4054ei︠ %htmlIt is a symmetric tensor field of type (0,2):
︡ada03e88-f457-49fb-9883-a799e34b8c54︡︡{"done":true,"html":"It is a symmetric tensor field of type (0,2):
"} ︠04210564-e32b-4268-9f19-be7681d0b2efs︠ g.parent() ︡025c51f5-7c9f-45a5-a228-5c4768a820b1︡︡{"html":"The metric is initialized by its components with respect to some vector frame. For instance, using the default frame of $\mathcal{M}$:
︡36956fdc-f0ac-4d9a-b104-d6c3f1758f91︡︡{"done":true,"html":"The metric is initialized by its components with respect to some vector frame. For instance, using the default frame of $\\mathcal{M}$:
"} ︠b1c02716-9417-4bfd-843c-80973251d05bs︠ g[1,1], g[2,2], g[3,3] = 1, 1, 1 g.display() ︡9c08052d-5071-4b44-96d4-79daecd8fc0e︡︡{"html":"The components w.r.t. another vector frame are obtained as for any tensor field:
︡9ae05b0c-ac16-41a0-bbbe-3627bc4d9442︡︡{"done":true,"html":"The components w.r.t. another vector frame are obtained as for any tensor field:
"} ︠baec1f49-d947-4c69-af41-42ee1f0d2113s︠ g.display(Y.frame(), Y) ︡32d6c770-0fa4-49ff-a006-3c80cef12db3︡︡{"html":"Of course, the metric acts on vector pairs:
︡deb520ac-17e2-4641-a79f-269aaa4a06c2︡︡{"done":true,"html":"Of course, the metric acts on vector pairs:
"} ︠b0d53b67-83d4-4a06-b4d1-b5a7fa1f7ac9s︠ u.display() ; v.display(); print g(u,v) ; g(u,v).display() ︡ae2bc0bd-99d8-4e73-919d-9e97a8326d7f︡︡{"html":"The Levi-Civita connection associated to the metric $g$:
︡e2a9da74-83b9-465c-ae61-23a4ead61a3a︡︡{"done":true,"html":"The Levi-Civita connection associated to the metric $g$:
"} ︠1991f56b-abe9-437e-a0e5-c6091c62cdfas︠ nabla = g.connection() print nabla ; nabla ︡41f5591c-891d-4011-97b3-7b8f282f8def︡︡{"stdout":"Levi-Civita connection 'nabla_g' associated with the Riemannian metric 'g' on the 3-dimensional manifold 'M'\n","done":false}︡{"html":"The Christoffel symbols with respect to the manifold's default coordinates:
︡5f924d68-21a3-429c-a3c7-d5732a2a8267︡︡{"done":true,"html":"The Christoffel symbols with respect to the manifold's default coordinates:
"} ︠917faa6d-4906-49ec-991e-a67a72a4fd5ds︠ nabla.coef()[:] ︡8f9f7857-5788-4e03-a135-5a0c48ec65fd︡︡{"html":"The Christoffel symbols with respect to the coordinates $(r,\theta,\phi)$:
︡e173ae67-b835-438b-853b-5043fe9dbbed︡︡{"done":true,"html":"The Christoffel symbols with respect to the coordinates $(r,\\theta,\\phi)$:
"} ︠98141ea7-6af1-4da0-9062-fdf4261c02b9s︠ nabla.coef(Y.frame())[:, Y] ︡0e457a90-14c1-42ae-b1f9-c5fe2e5e7772︡︡{"html":"A nice view is obtained via the method display() (by default, only the nonzero connection coefficients are shown):
A nice view is obtained via the method display() (by default, only the nonzero connection coefficients are shown):
The connection acting as a covariant derivative:
︡bea1d9b8-16b8-4292-b28a-2f51c6ed5085︡︡{"done":true,"html":"The connection acting as a covariant derivative:
"} ︠c5be84fe-5707-41ab-b94e-eb5b9c1acf24s︠ nab_v = nabla(v) print nab_v ; nab_v.display() ︡de7a5f17-cbcf-4714-98f6-998288884d46︡︡{"stdout":"tensor field 'nabla_g v' of type (1,1) on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"html":"Being a Levi-Civita connection, $\nabla_g$ is torsion.free:
︡c6c178f1-7d25-4e02-8fac-a58bbd972c34︡︡{"done":true,"html":"Being a Levi-Civita connection, $\\nabla_g$ is torsion.free:
"} ︠23778a94-f0d1-4d5f-9001-0fdd00680989s︠ print nabla.torsion() ; nabla.torsion().display() ︡fb1efd90-cdae-4e05-b5a6-35b4b64129e9︡︡{"stdout":"tensor field of type (1,2) on the 3-dimensional manifold 'M'\n","done":false}︡{"html":"In the present case, it is also flat:
︡5aad82fe-a656-4b98-aa4e-adf6c66e80ab︡︡{"done":true,"html":"In the present case, it is also flat:
"} ︠dfdfca5b-64ec-408c-809e-692780b7c919s︠ print nabla.riemann() ; nabla.riemann().display() ︡633ae123-0a8c-433a-8bb4-dd3c3b511409︡︡{"stdout":"tensor field 'Riem(g)' of type (1,3) on the 3-dimensional manifold 'M'","done":false}︡{"stdout":"\n","done":false}︡{"html":"Let us consider a non-flat metric, by changing $g_{rr}$ to $1/(1+r^2)$:
︡9b75096d-e58f-4dc7-a04a-c036d37e4081︡︡{"done":true,"html":"Let us consider a non-flat metric, by changing $g_{rr}$ to $1/(1+r^2)$:
"} ︠c3ba7164-a626-469b-bdfc-52996cc5a9b7s︠ g[Y.frame(), 1,1, Y] = 1/(1+r^2) g.display(Y.frame(), Y) ︡49d5f4ca-286e-43f2-8ed7-9823fe541f06︡︡{"html":"For convenience, we change the default chart on the domain $U$ to Y=$(U,(r,\theta,\phi))$:
︡2fd3f73e-4258-4e93-ac75-3df3631ee995︡︡{"done":true,"html":"For convenience, we change the default chart on the domain $U$ to Y=$(U,(r,\\theta,\\phi))$:
"} ︠cf8ccc4a-a496-4bcc-a6cf-fb56310b1b8as︠ U.set_default_chart(Y) ︡9d952d68-173d-447d-8929-b5f702afc188︡︡{"done":true} ︠5ce42986-d5ac-4470-9b60-0778dede0b71i︠ %htmlIn this way, we do not have to specify Y when asking for coordinate expressions in terms of $(r,\theta,\phi)$:
︡b9fe5291-663c-4928-b2e7-7fdaae8809a0︡︡{"done":true,"html":"In this way, we do not have to specify Y when asking for coordinate expressions in terms of $(r,\\theta,\\phi)$:
"} ︠9e73c46d-be39-4356-8e69-33334c7bf5f2s︠ g.display(Y.frame()) ︡781b70f7-8dcc-4e2b-9fc9-824eff2432b9︡︡{"html":"We recognize the metric of the hyperbolic space $\mathbb{H}^3$. Its expression in terms of the chart $(U,(x,y,z))$ is
︡5d974623-1b44-41bf-84e5-c023d4aca48f︡︡{"done":true,"html":"We recognize the metric of the hyperbolic space $\\mathbb{H}^3$. Its expression in terms of the chart $(U,(x,y,z))$ is
"} ︠b0c761bd-9315-4307-8fb8-4c3bb06e2dbas︠ g.display(X_U.frame(), X_U) ︡63724edc-ea65-414a-b7c9-22ccdbfbfffe︡︡{"html":"A matrix view of the components may be more appropriate:
︡650a277c-2ebf-49e8-9087-03185d7b8c3d︡︡{"done":true,"html":"A matrix view of the components may be more appropriate:
"} ︠4fa75af8-0d7d-4ca3-823f-222dbba69b41s︠ g[X_U.frame(), :, X_U] ︡ec83ceba-5fcb-45fe-9d08-537ecadfe17a︡︡{"html":"We extend these components, a priori defined only on $U$, to the whole manifold $\mathcal{M}$, by demanding the same coordinate expressions in the frame associated to the chart X=$(\mathcal{M},(x,y,z))$:
︡9dec611c-38a1-4a75-b582-789987883b89︡︡{"done":true,"html":"We extend these components, a priori defined only on $U$, to the whole manifold $\\mathcal{M}$, by demanding the same coordinate expressions in the frame associated to the chart X=$(\\mathcal{M},(x,y,z))$:
"} ︠acfb7abd-12bb-4ba2-8004-b9a1fddb44fes︠ g.add_comp_by_continuation(X.frame(), U, X) g.display() ︡b75a897d-c293-461e-b34e-4bca57552609︡︡{"html":"The Levi-Civita connection is automatically recomputed, after the change in $g$:
︡e2916ed8-7390-4390-9a78-aa63e79332cb︡︡{"done":true,"html":"The Levi-Civita connection is automatically recomputed, after the change in $g$:
"} ︠0a3920a5-6533-45d8-aceb-96f40d96fc68s︠ nabla = g.connection() ︡902c130a-141c-42b7-b8eb-e8638c869d7d︡︡{"done":true} ︠135cc092-d5d6-4c3a-aed2-4342fe995aeei︠ %htmlIn particular, the Christoffel symbols are different:
︡16d455a5-a71b-4dee-b551-91a947d69138︡︡{"done":true,"html":"In particular, the Christoffel symbols are different:
"} ︠2c81a460-a8d3-4add-b57b-1e3c98e7fcc2s︠ nabla.display(only_nonredundant=True) ︡4a4722b5-7f10-4225-b51c-f425b845394d︡︡{"html":"The Riemann tensor is now
︡cbc1d96b-c730-47e4-b1b8-2d51361562c4︡︡{"done":true,"html":"The Riemann tensor is now
"} ︠11bf71e2-8ab6-4b1a-a0d3-e7024b918082s︠ Riem = nabla.riemann() print Riem ; Riem.display(Y.frame()) ︡802f808f-b5af-4325-ad1e-b192fc01c807︡︡{"stdout":"tensor field 'Riem(g)' of type (1,3) on the 3-dimensional manifold 'M'\n","done":false}︡{"html":"Note that it can be accessed directely via the metric, without any explicit mention of the connection:
︡3f9766d2-f9c7-4a94-bcfe-6816374d545f︡︡{"done":true,"html":"Note that it can be accessed directely via the metric, without any explicit mention of the connection:
"} ︠a9b92f83-b2d3-4155-a29d-a382de8e3924s︠ g.riemann() is nabla.riemann() ︡7a447441-71dd-4ba9-b629-3b6cefb1c0ac︡︡{"html":"The Ricci tensor is
︡51943f88-752f-4ad9-b22a-d28b23d79a15︡︡{"done":true,"html":"The Ricci tensor is
"} ︠cb6b9141-93cc-4a5f-8001-54ebc190ff71s︠ Ric = g.ricci() print Ric ; Ric.display(Y.frame()) ︡b7d09729-d9f8-49cd-8ac7-edd841ef7eb6︡︡{"stdout":"field of symmetric bilinear forms 'Ric(g)' on the 3-dimensional manifold 'M'\n","done":false}︡{"html":"The Weyl tensor is:
︡4add3c18-ad14-4962-ac9b-35fe1d764db9︡︡{"done":true,"html":"The Weyl tensor is:
"} ︠8634ff12-8217-40ab-a758-900d263a9c18s︠ C = g.weyl() print C ; C.display() ︡7d02d6da-37d5-4293-ab38-732e4c7de97a︡︡{"stdout":"tensor field 'C(g)' of type (1,3) on the 3-dimensional manifold 'M'\n","done":false}︡{"html":"The Weyl tensor vanishes identically because the dimension of $\mathcal{M}$ is 3.
Finally, the Ricci scalar is
︡ef7886b9-2897-490c-94b9-469a7b767100︡︡{"done":true,"html":"The Weyl tensor vanishes identically because the dimension of $\\mathcal{M}$ is 3.
\nFinally, the Ricci scalar is
"} ︠43d4a442-1800-47b8-8a9a-9e4d20dd4ce7s︠ R = g.ricci_scalar() print R ; R.display() ︡e895f3fb-7a60-47be-82ee-de9be680d4ed︡︡{"stdout":"scalar field 'r(g)' on the 3-dimensional manifold 'M'\n","done":false}︡{"html":"We recover the fact that $\mathbb{H}^3$ is a Riemannian manifold of constant negative curvature.
The most important tensor transformation induced by the metric $g$ is the so-called musical isomorphism, or index raising and index lowering:
︡49cc508c-4f1f-4156-b53f-bc6019a2f043︡︡{"done":true,"html":"We recover the fact that $\\mathbb{H}^3$ is a Riemannian manifold of constant negative curvature.
\n\nThe most important tensor transformation induced by the metric $g$ is the so-called musical isomorphism, or index raising and index lowering:
"} ︠732ebf74-388c-4874-ab55-7ef1461f36f5s︠ print t ︡f6476f99-db13-42b6-8484-1ad3552f4dc4︡︡{"stdout":"tensor field 'T' of type (1,2) on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠ff9b8aca-e9da-423c-be72-86464cecc882s︠ t.display() ︡d9cb9a23-e938-4d2f-957d-eb91d4e78a4d︡︡{"html":"Raising the last index of $T$ with $g$:
︡27bcb89b-0e5a-4a45-be30-cc2cd4282f55︡︡{"done":true,"html":"Raising the last index of $T$ with $g$:
"} ︠89e83860-f3f1-4580-991b-f113d14ae1a2s︠ s = t.up(g, 2) print s ︡8675c2a0-9224-4d3c-ab6c-73a55f5a8524︡︡{"stdout":"tensor field of type (2,1) on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠a50d23fa-d070-45fc-ac34-8c4888533117i︠ %htmlRaising all the covariant indices of $T$ (i.e. those at the positions 1 and 2):
︡0b39397f-0e5e-4b2a-b716-88501d541a56︡︡{"done":true,"html":"Raising all the covariant indices of $T$ (i.e. those at the positions 1 and 2):
"} ︠b880edd0-6207-4f6c-9d1d-e2a685ae2df6s︠ s = t.up(g) print s ︡81303aaa-5884-4d65-9d94-d3e4d3509fce︡︡{"stdout":"tensor field of type (3,0) on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠5d0e20e5-d3bd-443a-9146-bcc9d6a14f66s︠ s = t.down(g) print s ︡5fe9dada-527b-4d5a-9d18-5652fbd239dc︡︡{"stdout":"tensor field of type (0,3) on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠67519935-7144-4496-b89d-bbadc444f927i︠ %htmlThe volume 3-form (Levi-Civita tensor) associated with the metric $g$ is
︡c7ede082-7585-498e-a084-cae27c80b79d︡︡{"done":true,"html":"The volume 3-form (Levi-Civita tensor) associated with the metric $g$ is
"} ︠65887ac0-f787-4dd1-b1ca-ff2016d9f08ds︠ epsilon = g.volume_form() print epsilon ; epsilon.display() ︡d7e4c523-8e30-437c-8f68-d8382945f430︡︡{"stdout":"3-form 'eps_g' on the 3-dimensional manifold 'M'\n","done":false}︡{"html":"We check the classical formula $\star f = f\, \epsilon_g$, or, more precisely, $\star f = f\, \epsilon_g|_U$ (for $f$ is defined on $U$ only):
︡809fcd3e-e60e-4771-b75d-2db4fe94d506︡︡{"done":true,"html":"We check the classical formula $\\star f = f\\, \\epsilon_g$, or, more precisely, $\\star f = f\\, \\epsilon_g|_U$ (for $f$ is defined on $U$ only):
"} ︠b7246e2d-f9c8-4473-99b2-55d239cb3babs︠ sf == f * epsilon.restrict(U) ︡9c676497-4212-40e7-aa63-19d14d33a089︡︡{"html":"The Hodge dual of a 1-form is a 2-form:
︡b90a203c-9789-4b26-85d2-f8645168f2d8︡︡{"done":true,"html":"The Hodge dual of a 1-form is a 2-form:
"} ︠fbebd271-9f81-4b65-8ede-78ffdb787d3es︠ print om ; om.display() ︡e37ee568-c698-45ee-afef-fd7276a8302a︡︡{"stdout":"1-form 'omega' on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"html":"The Hodge dual of a 2-form is a 1-form:
︡981a9711-2d11-4b43-b173-28952364309b︡︡{"done":true,"html":"The Hodge dual of a 2-form is a 1-form:
"} ︠547381c1-023e-4e41-aa47-37ee551c5ac0s︠ print a ︡ee1e982e-0511-4d2a-b1a1-ef74603b990d︡︡{"stdout":"2-form 'A' on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"done":true} ︠d2ae98b1-b60a-4f8a-8326-1d5a7a1775f5s︠ sa = a.hodge_star(g) print sa ; sa.display() ︡eb736761-57f9-4d13-b273-72d9c02024db︡︡{"stdout":"1-form '*A' on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"html":"Finally, the Hodge dual of a 3-form is a 0-form:
︡b4cce5f3-f290-4fab-ac44-fcb82d5fc087︡︡{"done":true,"html":"Finally, the Hodge dual of a 3-form is a 0-form:
"} ︠cfb88882-2011-409c-89a3-d3fc3c03c0fes︠ print da ; da.display() ︡976b50d1-6ffb-468e-8a69-ff10ab450f85︡︡{"stdout":"3-form 'dA' on the open subset 'U' of the 3-dimensional manifold 'M'\n","done":false}︡{"html":"In dimension 3 and for a Riemannian metric, the Hodge star is idempotent:
︡995bd5ea-92b0-4eaa-b8b4-84f5c22e5f66︡︡{"done":true,"html":"In dimension 3 and for a Riemannian metric, the Hodge star is idempotent:
"} ︠33f99e54-62d4-49dc-bc82-38a687632a9es︠ sf.hodge_star(g) == f ︡2ff916af-90b5-4d44-9b1b-d563cee9357c︡︡{"html":"To get the list of functions (methods) that can be called on a object, type the name of the object, followed by a dot and the TAB key, e.g.
sa.<tab>
To get information on an object or a method, use the question mark:
︡8215884f-8894-45c9-b50c-d6278f0fad21︡︡{"done":true,"html":"To get the list of functions (methods) that can be called on a object, type the name of the object, followed by a dot and the TAB key, e.g.
\nsa.<tab>
To get information on an object or a method, use the question mark:
"} ︠132c4234-6c89-4208-8ecf-7db1f57641ebs︠ nabla? ︡5f9d7214-83fa-4dc9-8435-203b999a16cc︡︡{"code":{"source":"File: /projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/geometry/manifolds/connection.py\nSignature : nabla()\nDocstring :\nLevi-Civita connection on a pseudo-Riemannian manifold.\n\nGiven a differentiable manifold M endowed with a pseudo-Riemannian\nmetric g and denoting by mathcal{X}(M) the C^infty(M)-module of\nvector fields on M (cf. \"VectorFieldModule\"), the *Levi-Civita\nconnection associated with* g is the unique operator\n\n begin{array}{cccc} nabla: & mathcal{X}(M) x\n mathcal{X}(M) & longrightarrow & mathcal{X}(M) \n & (u,v) & longmapsto & nabla_u v end{array}\n\nthat\n\n* is bilinear when considering mathcal{X}(M) as a vector space\n over RR\n\n* is C^infty(M)-linear w.r.t. the first argument: forall fin\n C^infty(M), nabla_{fu} v = fnabla_u v\n\n* obeys Leibniz rule w.r.t. the second argument: forall fin\n C^infty(M), nabla_u (f v) = df(u), v + f nabla_u v\n\n* is torsion-free\n\n* is compatible with g: forall (u,v,w)in mathcal{X}(M)^3,\n u(g(v,w)) = g(nabla_u v, w) + g(v, nabla_u w)\n\nThe Levi-Civita connection nabla gives birth to the *covariant\nderivative operator* acting on tensor fields, denoted by the same\nsymbol:\n\n begin{array}{cccc} nabla: & T^{(k,l)}(M) & longrightarrow &\n T^{(k,l+1)}(M) & t & longmapsto & nabla t\n end{array}\n\nwhere T^{(k,l)}(M) stands for the C^infty(M)-module of tensor\nfields of type (k,l) on M (cf. \"TensorFieldModule\"), with the\nconvention T^{(0,0)}(M):=C^infty(M). For a vector field v, the\ncovariant derivative nabla v is a type-(1,1) tensor field such\nthat\n\n forall u inmathcal{X}(M), nabla_u v = nabla v(., u)\n\nMore generally for any tensor field tin T^{(k,l)}(M), we have\n\n forall u inmathcal{X}(M), nabla_u t = nabla t(...,\n u)\n\nNote: The above convention means that, in terms of index\n notation, the \"derivation index\" in nabla t is the *last* one:\n\n nabla_c t^{a_1... a_k}_{quadquad b_1... b_l} =\n (nabla t)^{a_1... a_k}_{quadquad b_1... b_l c}\n\nINPUT:\n\n* \"metric\" -- the metric g defining the Levi-Civita connection,\n as an instance of class \"Metric\"\n\n* \"name\" -- name given to the connection\n\n* \"latex_name\" -- (default: \"None\") LaTeX symbol to denote the\n connection\n\n* \"init_coef\" -- (default: True) determines whether the\n Chrsitoffel symbols are initialized (in the top charts on the\n domain, i.e. disregarding the subcharts)\n\nEXAMPLES:\n\nLevi-Civita connection associated with the Euclidean metric on\nRR^3 expressed in spherical coordinates:\n\n sage: Manifold._clear_cache_() # for doctests only\n sage: forget() # for doctests only\n sage: M = Manifold(3, 'R^3', start_index=1)\n sage: c_spher.Using a double question mark leads directly to the Python source code (Sage and SageManifolds are open source, aren't they ?)
Using a double question mark leads directly to the Python source code (Sage and SageManifolds are open source, aren't they ?)
Have a look at the examples on SageManifolds page, especially the 2-dimensional sphere example for usage on a non-parallelizable manifold (each scalar field has to be defined in at least two coordinate charts, the module $\mathcal{X}(\mathcal{M})$ is no longer free and each tensor field has to be defined in at least two vector frames).
︡1d1096db-ee4b-4748-a241-7dbbc6e6e1cb︡︡{"done":true,"html":"Have a look at the examples on SageManifolds page, especially the 2-dimensional sphere example for usage on a non-parallelizable manifold (each scalar field has to be defined in at least two coordinate charts, the module $\\mathcal{X}(\\mathcal{M})$ is no longer free and each tensor field has to be defined in at least two vector frames).
"} ︠0f555c16-ec94-4067-bb79-9ae21d4a7238︠