This Sage worksheet provides some introduction to tensors on free modules of finite rank. This is a pure algebraic subpart of SageManifolds (version 0.8), which does not depend on other parts of SageManifolds and which has been integrated in Sage 6.6.
This worksheet is released under the GNU General Public License version 2.
(c) Eric Gourgoulhon, Michal Bejger (2015)
The whole worksheet file can be downloaded from here.
Let $R$ be a commutative ring and $M$ a free module of finite rank over $R$, i.e. a module over $R$ that admits a finite basis (finite family of linearly independent generators). Since $R$ is commutative, it has the invariant basis number property: all bases of $M$ have the same cardinality, which is called the rank of $M$. In this tutorial, we consider a free module of rank 3 over the integer ring $\mathbb{Z}$:
︡efcffe9e-f0e8-410b-9651-88fae861cb94︡︡{"done":true,"html":"This Sage worksheet provides some introduction to tensors on free modules of finite rank. This is a pure algebraic subpart of SageManifolds (version 0.8), which does not depend on other parts of SageManifolds and which has been integrated in Sage 6.6.
\nThis worksheet is released under the GNU General Public License version 2.
\n(c) Eric Gourgoulhon, Michal Bejger (2015)
\nThe whole worksheet file can be downloaded from here.
\nLet $R$ be a commutative ring and $M$ a free module of finite rank over $R$, i.e. a module over $R$ that admits a finite basis (finite family of linearly independent generators). Since $R$ is commutative, it has the invariant basis number property: all bases of $M$ have the same cardinality, which is called the rank of $M$. In this tutorial, we consider a free module of rank 3 over the integer ring $\\mathbb{Z}$:
"} ︠6eaf06af-35be-4b59-b8e7-a699694c6a8cs︠ M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1) ︡10184741-bf76-4e90-8f5a-f88c79a18670︡︡{"done":true} ︠0425fd41-8eb8-47a6-b548-ef6b389a5affi︠ %htmlThe first two arguments are the ring and the rank; the third one is a string to denote the module and the last one defines the range of indices to be used for tensor components on the module: setting it to 1 means that indices will range in $\{1,2,3\}$. The default value is start_index=0.
The command print returns a short description of the just constructed module:
︡fd926209-82e7-46d6-8133-b196c7931dcd︡︡{"done":true,"html":"The first two arguments are the ring and the rank; the third one is a string to denote the module and the last one defines the range of indices to be used for tensor components on the module: setting it to 1 means that indices will range in $\\{1,2,3\\}$. The default value is start_index=0.
\nThe command print returns a short description of the just constructed module:
"} ︠fa32e62c-7368-4ebc-b8b0-0eb74dbe8568s︠ print M ︡c95a8060-303d-479d-b6af-0c7df2d9f20a︡︡{"stdout":"Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠82666b42-6b88-4e69-b01b-840ec6f8017di︠ %htmlIf we ask just for M, the module's LaTeX symbol is returned (provided that the worksheet's Typeset box has been selected); by default, this is the same as the argument name in the constructor (this can be changed by providing the optional argument latex_name):
︡b2f1fb12-a50b-40b2-b000-d4f2fa7ba3a3︡︡{"done":true,"html":"If we ask just for M, the module's LaTeX symbol is returned (provided that the worksheet's Typeset box has been selected); by default, this is the same as the argument name in the constructor (this can be changed by providing the optional argument latex_name):
"} ︠ef813805-685f-4efa-b2d5-b7bfb4b6b3e8s︠ M ︡bf38df4c-6ad8-4324-8a90-575b00537ba2︡︡{"html":"The indices of basis elements or tensor components on the module are generated by the method irange(), to be used in loops:
︡40ade4e8-3573-4f23-9cc0-a4653a91a60b︡︡{"done":true,"html":"The indices of basis elements or tensor components on the module are generated by the method irange(), to be used in loops:
"} ︠4ca62212-208c-4a13-adfa-ec444ef0c9fds︠ for i in M.irange(): print i, ︡b6675d37-92ca-4b37-87fc-a0f234afa14b︡︡{"stdout":"1 2 3","done":false}︡{"done":true} ︠f75deed3-07b7-4d43-8356-4dca1e612265i︠ %htmlIf the parameter start_index had not been specified, the default range of the indices would have been $\{0,1,2\}$ instead:
︡d9be8a76-2fa0-41c8-9c0f-28f9dcdb3184︡︡{"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:
"} ︠c756043e-9c87-4a08-8e2f-5fb0fb680e80s︠ M0 = FiniteRankFreeModule(ZZ, 3, name='M') for i in M0.irange(): print i, ︡93eac87b-def4-4e48-9810-9c48f043e36f︡︡{"stdout":"0 1 2","done":false}︡{"done":true} ︠58e1cc63-9208-4bd4-86b2-8dafc01d0f2ci︠ %html$M$ is the category of modules over $\mathbb{Z}$:
︡01fcae53-6f96-4a9f-b577-26d7d75b5314︡︡{"done":true,"html":"$M$ is the category of modules over $\\mathbb{Z}$:
"} ︠5d32bb51-77fb-47b2-b37d-e123eb10ef20s︠ M.category() ︡eaf30477-d400-48c2-8822-d348b00b4b5e︡︡{"html":"Self-inquiry commands are
︡96f24c26-b55c-4837-88e0-77c43daa7a14︡︡{"done":true,"html":"Self-inquiry commands are
"} ︠ed98e3ae-5f41-4783-91fc-7d5645578df4s︠ M.base_ring() ︡17cd8abc-d3fa-474f-b059-560f3d21b987︡︡{"html":"At construction, the free module $M$ has no pre-defined basis:
︡bbd766bb-1613-49e7-b668-82acb7a78434︡︡{"done":true,"html":"At construction, the free module $M$ has no pre-defined basis:
"} ︠b7d6d172-525b-42df-9a8d-3db523dc06b2s︠ M.print_bases() ︡456897b0-5e7e-4483-ba6a-a568d88ed61c︡︡{"stdout":"Bases defined on the Rank-3 free module M over the Integer Ring:\n - (e_1,e_2,e_3) (default basis)\n - (f_1,f_2,f_3)\n","done":false}︡{"done":true} ︠6335970f-7270-4dec-ba8d-9b89994f803es︠ M.bases() ︡c6c8acdb-70ae-4fcc-83c8-c835e384dd9b︡︡{"html":"For this reason, the class FiniteRankFreeModule does not inherit from Sage class CombinatorialFreeModule:
︡2465492d-a3d0-40a9-ba35-776d90deb6cb︡︡{"done":true,"html":"For this reason, the class FiniteRankFreeModule does not inherit from Sage class CombinatorialFreeModule:
"} ︠f61715ca-9ebf-47da-ae9f-be61ed39cd66s︠ isinstance(M, CombinatorialFreeModule) ︡5d31a0b8-2395-45c8-a529-a8871fdb6c39︡︡{"html":"and $M$ does not belong to the category of modules with a distinguished basis:
︡43dd4e55-bc67-46e8-971d-1fcae3a63cda︡︡{"done":true,"html":"and $M$ does not belong to the category of modules with a distinguished basis:
"} ︠e3ef50b8-4619-4dd7-9230-aafdb6851418s︠ M in ModulesWithBasis(ZZ) ︡7c90024f-81c3-4c49-98c2-dfe14cdd7e02︡︡{"html":"It simply belongs to the category of modules over $\mathbb{Z}$:
︡d9192115-4e0b-4440-bb52-ef1355350dbc︡︡{"done":true,"html":"It simply belongs to the category of modules over $\\mathbb{Z}$:
"} ︠0c864572-21b3-4e0c-a6ac-c4bf395144fbs︠ M in Modules(ZZ) ︡32c4c9b9-7182-4c5f-a325-00fed73619c5︡︡{"html":"We define a first basis on $M$ as follows:
︡de1696e7-00c3-4228-9caf-c0e8f785300f︡︡{"done":true,"html":"We define a first basis on $M$ as follows:
"} ︠8fc7380f-064f-4409-a19e-dd6dd621e55ds︠ e = M.basis('e') ; e ︡2e38d732-f40e-4378-bcfe-3012b2df5737︡︡{"html":"The elements of the basis are accessed via their indices:
︡277b52af-6fe2-4771-b0d2-7274b9794798︡︡{"done":true,"html":"The elements of the basis are accessed via their indices:
"} ︠9fa6a641-bcbc-41e7-98ea-7d3255df9a24s︠ e[1] ︡2f317643-24e3-4606-9a1b-dca5ed0f497c︡︡{"html":"Let us introduce a second basis on the free module $M$ from a family of 3 linearly independent module elements:
︡42044d50-9472-4d34-a43e-79de2c94c93d︡︡{"done":true,"html":"Let us introduce a second basis on the free module $M$ from a family of 3 linearly independent module elements:
"} ︠84ff46b8-0abb-46ed-be76-0de493a72f0as︠ f = M.basis('f', from_family=(-e[1]+2*e[2]-4*e[3], e[2]+2*e[3], e[2]+3*e[3])) print f ; f ︡242c1d52-7642-45ed-be1d-2b0151c31493︡︡{"stdout":"Basis (f_1,f_2,f_3) on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"html":"We may ask to view each element of basis $f$ in terms of its expansion onto basis $e$, via the method display(), abridged as disp():
︡4b8a9e0e-2c58-42a1-8215-cd61abcc1b2d︡︡{"done":true,"html":"We may ask to view each element of basis $f$ in terms of its expansion onto basis $e$, via the method display(), abridged as disp():
"} ︠42368610-b27f-44c8-9676-49d29f5bbc48s︠ f[1].disp(e) ︡d845bd97-5b46-4193-a6df-d06ffb76df76︡︡{"html":"Conversely, the expression of basis $e$ is terms of basis $f$ is
︡f1ed24c0-8346-4114-b83a-1d8cdf110a95︡︡{"done":true,"html":"Conversely, the expression of basis $e$ is terms of basis $f$ is
"} ︠c7a8ac4e-2f9a-4460-b5e8-ed549b0651afs︠ e[1].disp(f) ︡780fe26c-c180-46e1-9783-d90297c99380︡︡{"html":"The module automorphism $a$ relating the two bases is obtained as
︡b04683c7-32e4-416e-9174-a161c066fdc7︡︡{"done":true,"html":"The module automorphism $a$ relating the two bases is obtained as
"} ︠0db68551-bc10-4090-95e8-0386aef8ae68s︠ a = M.change_of_basis(e,f) ; a ︡b21669cd-f940-456a-b2f5-a8d22b7bc1d2︡︡{"html":"It belongs to the general linear group of the free module $M$:
︡34139a52-c11d-4137-9462-efd2d22b4aca︡︡{"done":true,"html":"It belongs to the general linear group of the free module $M$:
"} ︠606f2da0-7136-4a5e-8ceb-50c75c8c14c7s︠ a.parent() ︡0f45f421-f19e-414a-b858-66a96e347220︡︡{"html":"and its matrix w.r.t. basis $e$ is
︡42f24bf0-48ce-4051-9332-10bfb059c52e︡︡{"done":true,"html":"and its matrix w.r.t. basis $e$ is
"} ︠a3d0d2a2-d365-4f2c-83bf-75c81566225as︠ a.matrix(e) ︡13a6d0f4-786c-4b95-9372-25f9a30a8b0a︡︡{"html":"Let us check that the elements of basis $f$ are images of the elements of basis $e$ via $a$:
︡8db9439b-fad3-4edf-87bd-8680c6fec429︡︡{"done":true,"html":"Let us check that the elements of basis $f$ are images of the elements of basis $e$ via $a$:
"} ︠ce643d3f-600d-43c1-847a-e43b3cd88c61s︠ f[1] == a(e[1]), f[2] == a(e[2]), f[3] == a(e[3]) ︡a36cf81a-0862-4eb2-9a82-b6327285fe1a︡︡{"html":"The reverse change of basis is of course the inverse automorphism:
︡99e980e3-c2c8-4fcb-80d8-03adafb2e546︡︡{"done":true,"html":"The reverse change of basis is of course the inverse automorphism:
"} ︠5984817d-24d2-4cfe-9b97-6e6295205339s︠ M.change_of_basis(f,e) == a^(-1) ︡fcd3d54f-b447-4961-ac23-6eec928bb0a4︡︡{"html":"At this stage, two bases have been defined on $M$:
︡4e1a0c81-40fd-4611-8be7-279a18e57105︡︡{"done":true,"html":"At this stage, two bases have been defined on $M$:
"} ︠06a759ef-df41-4ae7-9a40-007ba7694649s︠ M.print_bases() ︡313f4f54-3f0a-4ea7-856b-9ff343e3be2f︡︡{"stdout":"Bases defined on the Rank-3 free module M over the Integer Ring:\n - (e_1,e_2,e_3) (default basis)\n - (f_1,f_2,f_3)\n","done":false}︡{"done":true} ︠b5a29d43-e0a6-4bc2-85c5-6f3e3d7c8e94i︠ %htmlThe first defined basis, $e$, is considered as the default basis, which means that it can be skipped in any method argument requirying a basis. For instance, let us consider the method display():
︡9ebfa4b3-5401-4371-b28a-edc11d7e9d0a︡︡{"done":true,"html":"The first defined basis, $e$, is considered as the default basis, which means that it can be skipped in any method argument requirying a basis. For instance, let us consider the method display():
"} ︠ca340f0c-8d3d-4448-9fc5-da679f62ec39s︠ f[1].display(e) ︡696355be-fc4b-42e5-9ed3-b164742f3be5︡︡{"html":"Since $e$ is the default basis, the above command is fully equivalent to
︡e99025f9-7263-4e77-aee5-41e30710c778︡︡{"done":true,"html":"Since $e$ is the default basis, the above command is fully equivalent to
"} ︠13f29363-4441-4c4a-b4a6-c8e0f4793607s︠ f[1].display() ︡db919abf-b1a7-4ceb-b8b1-1b469035e8ab︡︡{"html":"Of course, the names of non-default bases have to be specified:
︡2fc8009b-d710-4ef1-a68a-e497ea990f76︡︡{"done":true,"html":"Of course, the names of non-default bases have to be specified:
"} ︠df4c7944-4a35-4b5f-ae22-6cc6229d6cbfs︠ f[1].display(f) ︡192ce50c-485e-49d2-a9ef-63f8f56ff70e︡︡{"html":"Note that the concept of default basis is different from that of distinguished basis which is implemented in other free module constructions in Sage (e.g. CombinatorialFreeModule): the default basis is intended only for shorthand notations in user commands, avoiding to repeat the basis name many times; it is by no means a privileged basis on the module. For user convenience, the default basis can be changed at any moment by means of the method set_default_basis():
︡17f61e55-1c73-4c6e-8613-70ba2b476718︡︡{"done":true,"html":"Note that the concept of default basis is different from that of distinguished basis which is implemented in other free module constructions in Sage (e.g. CombinatorialFreeModule): the default basis is intended only for shorthand notations in user commands, avoiding to repeat the basis name many times; it is by no means a privileged basis on the module. For user convenience, the default basis can be changed at any moment by means of the method set_default_basis():
"} ︠65cb526e-d449-469e-b083-d3d0870ca1ees︠ M.set_default_basis(f) e[1].display() ︡8c80168a-94ca-4c49-b540-175c3f5883dc︡︡{"html":"Let us revert to $e$ as the default basis:
︡58a45c60-0dc9-4b03-946e-3372eaecc1fb︡︡{"done":true,"html":"Let us revert to $e$ as the default basis:
"} ︠4ae0176c-9b5c-4728-8faa-68d7ab6a2652s︠ M.set_default_basis(e) ︡8106c197-9e3b-46ef-a768-59a7211c5898︡︡{"done":true} ︠f4d19a77-c25d-4af3-8eeb-a12996403c8ci︠ %htmlElements of the free module $M$ are constructed by providing their components with respect to a given basis to the operator () acting on the module:
︡3e0ba5d9-78c2-4280-852f-22b82415bfeb︡︡{"done":true,"html":"Elements of the free module $M$ are constructed by providing their components with respect to a given basis to the operator () acting on the module:
"} ︠a4d1b775-f0bd-4ff1-b0b9-b0f852f9d4e2s︠ v = M([3,-4,1], basis=e, name='v') print v ︡bd39fd10-f5f5-488d-81b9-38abb2dc3a81︡︡{"stdout":"Element v of the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠d81cb282-5e6a-42d2-bc83-3d15cd16898ci︠ %htmlSince $e$ is the default basis, its mention can be skipped:
︡5833d3e0-01f2-4c0d-b8f5-8057e430b41f︡︡{"done":true,"html":"Since $e$ is the default basis, its mention can be skipped:
"} ︠1658b54b-bd60-45ef-b4f1-dd8aba072177s︠ v = M([3,-4,1], name='v') print v ︡2e4bc7fc-13a0-4833-a148-d7020739ad5d︡︡{"stdout":"Element v of the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠14df5062-fdae-42d3-bb0b-ca39c9348e83s︠ v.disp() ︡40f77b54-2098-4f95-b60a-4461ec61f13a︡︡{"html":"While $v$ has been defined from the basis $e$, its expression in terms of the basis $f$ can be evaluated, thanks to the known relation between the two bases:
︡52c1645c-bb6c-40a3-86fc-551708051547︡︡{"done":true,"html":"While $v$ has been defined from the basis $e$, its expression in terms of the basis $f$ can be evaluated, thanks to the known relation between the two bases:
"} ︠93d6b047-80ee-4b00-98fa-3b41cca29e2as︠ v.disp(f) ︡f6385e91-69e8-40d0-b2f5-822498124e2b︡︡{"html":"According to Sage terminology, the parent of $v$ is of course $M$:
︡280c4731-e5bc-4684-8bd4-72a824786e3c︡︡{"done":true,"html":"According to Sage terminology, the parent of $v$ is of course $M$:
"} ︠ba8e44af-24f6-413c-898f-79acc81e3c9ds︠ v.parent() ︡a9b84d7c-000b-4709-8f73-332c106cd4ea︡︡{"html":"We have also
︡1245a6c0-fbec-4c87-96d4-700191c6c9b9︡︡{"done":true,"html":"We have also
"} ︠601eb802-fb95-4a8a-86fa-c5c4f398ca1cs︠ v in M ︡92cde1ab-adb2-4a8d-80cb-60842a5d319e︡︡{"html":"Let us define a second module element, from the basis $f$ this time:
︡9f19efa5-2b7a-4542-9cf9-a8bd315c289e︡︡{"done":true,"html":"Let us define a second module element, from the basis $f$ this time:
"} ︠031c0ba5-6953-4d9c-9f54-956988495f93s︠ u = M([-1,3,5], basis=f, name='u') u.disp(f) ︡684f4332-1994-4427-ae45-1a96c895a9c1︡︡{"html":"Another way to define module elements is of course via linear combinations:
︡f20a0174-7c66-49a2-b362-0c28fb241056︡︡{"done":true,"html":"Another way to define module elements is of course via linear combinations:
"} ︠53ee1426-7ac4-45d7-beab-7458ef164501s︠ w = 2*e[1] - e[2] - 3*e[3] print w ︡54632f4c-2afd-4604-8d98-ce30ac34ca9e︡︡{"stdout":"Element of the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠518ad8cb-eac7-459a-8cb2-149e5ce3ed1as︠ w.disp() ︡f3ab4f65-9963-45f4-a6f8-58e2cc0cfe05︡︡{"html":"As the result of a linear combination, $w$ has no name; it can be given one by the method set_name() and the LaTeX symbol can be specified if different from the name:
︡865f878b-166a-473e-8ada-8d638d690788︡︡{"done":true,"html":"As the result of a linear combination, $w$ has no name; it can be given one by the method set_name() and the LaTeX symbol can be specified if different from the name:
"} ︠a6588302-cd7a-44cf-b61a-110e087c4399s︠ w.set_name('w', latex_name=r'\omega') w.disp() ︡8f3b9968-4581-45a1-83e7-52f41dce4277︡︡{"html":"Module operations are implemented, independently of the bases:
︡df01b6be-dc94-453f-b588-4d5e6354df12︡︡{"done":true,"html":"Module operations are implemented, independently of the bases:
"} ︠1c56e91d-e817-436d-b115-f6350bd3101fs︠ s = u + 3*v print s ︡acda6edd-3ae5-4f0f-8bb6-b78ba1fb365e︡︡{"stdout":"Element of the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠fa1bead7-d5e6-48e3-8c96-6ca3daf1e572s︠ s.disp() ︡89277a48-3f68-4e01-9359-a740695cfb47︡︡{"html":"The components of a module element with respect to a given basis are given by the method components():
︡b940a9f7-ca49-4f5a-afa9-479f5b6fb879︡︡{"done":true,"html":"The components of a module element with respect to a given basis are given by the method components():
"} ︠d44e3375-1c22-409e-a8b9-1b57c558cdd4s︠ v.components(f) ︡046c4aaf-e4a4-4b53-9f64-aac5ca9e2422︡︡{"html":"A shortcut is comp():
︡1191dd04-ae72-4da7-9fb6-0a686fa6acef︡︡{"done":true,"html":"A shortcut is comp():
"} ︠db332186-1bcd-440f-bf36-bd0cb6d8b103s︠ v.comp(f) is v.components(f) ︡cd23ff7d-cb46-430e-b9a2-5c356ac34528︡︡{"html":"The function display_comp() provides a list of components w.r.t. to a given basis:
︡583ca62a-eed9-4ff3-af43-d1b5335994ad︡︡{"done":true,"html":"The function display_comp() provides a list of components w.r.t. to a given basis:
"} ︠3f5e9ed6-3b36-4b0b-968c-b814f3443be4s︠ v.display_comp(f) ︡4d08bd5b-53c0-4ac0-b3db-a89262fb19fa︡︡{"html":"As a shortcut, instead of calling the method comp(), the basis can be provided as the first argument of the square bracket operator:
︡7e4c7310-4a06-4a98-a339-4bfb3b8ae368︡︡{"done":true,"html":"As a shortcut, instead of calling the method comp(), the basis can be provided as the first argument of the square bracket operator:
"} ︠93c511eb-43c9-47b6-aca4-fa044d71dd5cs︠ v[f,2] ︡95335019-195e-45e9-b440-40fd4bf20110︡︡{"html":"For the default basis, the basis can be omitted:
︡c2aca0f9-ee3d-416a-b72b-36ad1b226c3c︡︡{"done":true,"html":"For the default basis, the basis can be omitted:
"} ︠a23a905a-efcb-4584-ae10-029b61de3973s︠ v[:] ︡e19a5848-9b65-4b2e-9e15-12c7262db624︡︡{"html":"A specific module element is the zero one:
︡918caff9-73c4-4679-8e1f-17a35bc20beb︡︡{"done":true,"html":"A specific module element is the zero one:
"} ︠605ca30d-05f9-47e4-a71b-7637b12e8bf4s︠ print M.zero() ︡7c122e93-a7b4-4140-a7ac-b73945a98fda︡︡{"stdout":"Element zero of the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠f2f523b5-eabe-41ea-b89a-7e63a110418fs︠ M.zero()[:] ︡8b07e417-1002-4a3c-bf6a-f8292cfad26e︡︡{"html":"Let us introduce some linear form on the free module $M$:
︡b61031e6-cbf1-4ca5-b285-fdf19d6e793d︡︡{"done":true,"html":"Let us introduce some linear form on the free module $M$:
"} ︠4f771208-c833-4771-91d6-bcd623fe5ac6s︠ a = M.linear_form('a') print a ︡ea2d78e6-d722-4b2f-b6aa-557a9808d916︡︡{"stdout":"Linear form a on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠d67b541e-5ab7-4a1f-92d2-22449d763bbdi︠ %html$a$ is specified by its components with respect to the basis dual to $e$:
︡c782e5d5-4f03-43fa-978a-9e2f43433045︡︡{"done":true,"html":"$a$ is specified by its components with respect to the basis dual to $e$:
"} ︠bb6a86ff-7599-4fee-9fed-1abc3c95a1cds︠ a[:] = [2,-1,3] a.disp() ︡f3a23214-111d-4fbd-9e5c-253ae2d20e2f︡︡{"html":"The notation $e^i$ stands for the elements of the basis dual to $e$, i.e. the basis of the dual module $M^*$ such that \[ e^i(e_j) = \delta^i_{\ \, j} \]
Indeed
︡a2cb3705-f212-43a1-b29e-097dea5b4458︡︡{"done":true,"html":"The notation $e^i$ stands for the elements of the basis dual to $e$, i.e. the basis of the dual module $M^*$ such that \\[ e^i(e_j) = \\delta^i_{\\ \\, j} \\]
\nIndeed
"} ︠b669e171-fbeb-4f8e-96b2-5a2c46db5ee1s︠ ed = e.dual_basis() ed ︡74f779ac-ec36-4ada-878d-15f34f8022c1︡︡{"html":"The linear form $a$ can also be defined by its components with respect to the basis dual to $f$:
︡8979c619-24a7-4fb0-8189-06fba895b4a7︡︡{"done":true,"html":"The linear form $a$ can also be defined by its components with respect to the basis dual to $f$:
"} ︠5e49df00-26ed-4706-9c11-4863a418cdc9s︠ a[f,:] = [2,-1,3] a.disp(f) ︡a524d75f-30e1-498e-8b4e-e98cdaba6308︡︡{"html":"For consistency, the previously defined components with respect to the basis dual to $e$ are automatically deleted and new ones are computed from the change-of-basis formula:
︡b16db677-d6a7-4494-bccf-a099408bd15b︡︡{"done":true,"html":"For consistency, the previously defined components with respect to the basis dual to $e$ are automatically deleted and new ones are computed from the change-of-basis formula:
"} ︠c50e3cb1-9c4d-49e9-9446-438c43b3ea0bs︠ a.disp() ︡eff7b195-bc18-47a3-8e24-183f5e029967︡︡{"html":"By definition, linear forms belong to the dual module:
︡83edfda0-0540-4e85-9d26-1e05ea7dfce1︡︡{"done":true,"html":"By definition, linear forms belong to the dual module:
"} ︠bce8e6a3-f675-4947-a44a-dc17666805c4s︠ a.parent() ︡61698307-b375-4f3d-b908-54ec228abaca︡︡{"html":"The dual module is itself a free module of the same rank as $M$:
︡13e71ca8-b426-423d-859b-cf0ae8c08850︡︡{"done":true,"html":"The dual module is itself a free module of the same rank as $M$:
"} ︠403c0a80-ec8d-4059-b203-21d75de06fees︠ isinstance(M.dual(), FiniteRankFreeModule) ︡d4fc15b3-a577-486c-acf1-ade5efe61b06︡︡{"html":"Linear forms map module elements to ring elements:
︡9207f7b8-2962-4a06-a466-47db5d1a0051︡︡{"done":true,"html":"Linear forms map module elements to ring elements:
"} ︠72ec0cd3-84b5-4b51-92d7-64b44fba2b6cs︠ a(v) ︡dc04448d-40d4-4291-a57d-e4a719aaf305︡︡{"html":"in a linear way:
︡80b71b2f-8b38-4a15-a670-b07289a91fb9︡︡{"done":true,"html":"in a linear way:
"} ︠9588b865-9923-4c8b-9c74-21c1cd869196s︠ a(u+2*v) == a(u) + 2*a(v) ︡d3b2670c-82b7-4a71-98fb-4fb45fa8bbbe︡︡{"html":"Let us introduce a second linear form, $b$, on the free module $M$:
︡3bfe4f8a-23ca-47a7-bb76-5fe1655902d7︡︡{"done":true,"html":"Let us introduce a second linear form, $b$, on the free module $M$:
"} ︠477c4ac0-fc2e-4a5f-a2cc-3bfa6a6ab2a9s︠ b = M.linear_form('b') b[:] = [-4,2,5] ︡c7660022-0447-4bc0-9def-f033a1a3714c︡︡{"done":true} ︠094efd2c-a98a-4ff0-b86d-b65322fb90f7i︠ %htmland take its exterior product with the linear form $a$:
︡27d14af9-ae57-4a2e-8ebd-0c691f7ff5fb︡︡{"done":true,"html":"and take its exterior product with the linear form $a$:
"} ︠213e5823-c1f8-42f7-b140-cea86ab90bfcs︠ c = a.wedge(b) print c c ︡776bd79b-d7f8-4988-bf87-9b6c641c7915︡︡{"stdout":"Alternating form a/\\b of degree 2 on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"html":"$c$ is antisymmetric:
︡162dc140-bcd8-41ee-bc4d-bf6515b9730c︡︡{"done":true,"html":"$c$ is antisymmetric:
"} ︠16da0823-2f6d-4d80-a0bc-816faad8ef1as︠ c(v,u) ︡61d96e2c-cd30-41ca-b2ea-f98b6eb67a4a︡︡{"html":"and is multilinear:
︡532ede59-8732-4956-8a45-8861654ade3c︡︡{"done":true,"html":"and is multilinear:
"} ︠ae2f325c-1b30-44cf-9c38-f32bffd4996bs︠ c(u+4*w,v) == c(u,v) + 4*c(w,v) ︡59e65e20-83fd-4e19-8bb9-b85b0bda09a9︡︡{"html":"We may check the standard formula for the exterior product of two linear forms:
︡f9a75bc8-9889-491d-b38e-f02b13177dd5︡︡{"done":true,"html":"We may check the standard formula for the exterior product of two linear forms:
"} ︠09bbd09b-0c5d-4198-afc5-0d79993527bbs︠ c(u,v) == a(u)*b(v) - b(u)*a(v) ︡28b39b06-f583-4f87-92ba-644444402010︡︡{"html":"In terms of tensor product (denoted here by *), it reads
︡05140ce2-b81e-4e14-8d44-d006e90fc85c︡︡{"done":true,"html":"In terms of tensor product (denoted here by *), it reads
"} ︠e84d19f8-09c9-482b-9321-b73297d78d1bs︠ c == a*b - b*a ︡16d3fdca-a706-4d02-bc3d-5688d54bd534︡︡{"html":"The parent of the alternating form $c$ is the second external power of the dual module $M^*$, which is denoted by $\Lambda^2(M^*)$:
︡7b1c7818-6512-403b-a60f-75254d688c2b︡︡{"done":true,"html":"The parent of the alternating form $c$ is the second external power of the dual module $M^*$, which is denoted by $\\Lambda^2(M^*)$:
"} ︠f39ccc87-e83e-45e2-97b0-3ff66fa0149es︠ c.parent() ︡0c0e6cae-c257-4744-9184-6bb7b74f6914︡︡{"html":"$c$ is a tensor field of type $(0,2)$:
︡fd59729c-a86e-4e80-8370-a20a45b9ad46︡︡{"done":true,"html":"$c$ is a tensor field of type $(0,2)$:
"} ︠c23507d5-c65d-40a9-adcc-9707aa3b845fs︠ c.tensor_type() ︡c0513ed3-f524-4e3f-807d-d895f007d7ea︡︡{"html":"whose components with respect to any basis are antisymmetric:
︡2d783004-30f9-4b53-86d1-2a3c2261a9cc︡︡{"done":true,"html":"whose components with respect to any basis are antisymmetric:
"} ︠02d66e6a-4fcd-4385-b65b-b5baf5a33c36s︠ c[:] # components with respect to the default basis (e) ︡7f00cb2c-8d67-440e-9988-619b8d92de0b︡︡{"html":"An alternating form can be constructed from scratch:
︡38f00d1d-c2b6-47e3-8602-e68204438522︡︡{"done":true,"html":"An alternating form can be constructed from scratch:
"} ︠0356a753-cc39-4349-8f7f-79abdb56f0e7s︠ c1 = M.alternating_form(2) # 2 stands for the degree ︡9e9da605-131a-4876-85c3-c096091a5724︡︡{"done":true} ︠69b2c9ba-9672-45b2-9e3d-d8db844f104ai︠ %htmlOnly the non-zero and non-redundant components are to be defined (the others are deduced by antisymmetry); for the components with respect to the default basis, we write:
︡f9810e5b-e7b5-4471-a121-e3059c73d2b4︡︡{"done":true,"html":"Only the non-zero and non-redundant components are to be defined (the others are deduced by antisymmetry); for the components with respect to the default basis, we write:
"} ︠470caefc-e0cf-461c-a3c1-62bcb05026b6s︠ c1[1,2] = -108 c1[1,3] = -164 c1[2,3] = -53 ︡8d6be67c-2bab-40e9-bb74-04065078d93e︡︡{"done":true} ︠1c3c167e-e03e-499b-a92a-b92c42d8358ai︠ %htmlThen
︡1a120890-d874-496c-8664-645ec61abdad︡︡{"done":true,"html":"Then
"} ︠55acf566-fe8f-4a3f-8dbf-7926a5ac4575s︠ c1[:] ︡2d5f7095-79ba-47f1-a49c-bde5b1e84223︡︡{"html":"Internally, only non-redundant components are stored, in a dictionary whose keys are the indices:
︡fff29494-65ba-4413-b65a-900d50b07295︡︡{"done":true,"html":"Internally, only non-redundant components are stored, in a dictionary whose keys are the indices:
"} ︠4bc34a18-9300-4412-babf-6834a5e1a007s︠ c.comp(e)._comp ︡af113c28-8642-4c61-b788-29e37fe57416︡︡{"html":"The other components are deduced by antisymmetry.
The exterior product of a linear form with an alternating form of degree 2 leads to an alternating form of degree 3:
︡ce8ec0db-9198-401c-b852-ecaa63ad3707︡︡{"done":true,"html":"The other components are deduced by antisymmetry.
\n\nThe exterior product of a linear form with an alternating form of degree 2 leads to an alternating form of degree 3:
"} ︠3255b225-4172-4eaa-8d22-adf31d8a2ea5s︠ d = M.linear_form('d') d[:] = [-1,-2,4] s = d.wedge(c) print s ︡fdfc64e2-08c3-4af5-8571-1f39c5310023︡︡{"stdout":"Alternating form d/\\a/\\b of degree 3 on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠b68d73e7-5c94-46f9-8b47-7b0f090b5d00s︠ s.disp() ︡7064f349-d7e0-444b-9233-d8cac6ec8aa6︡︡{"html":"$s$ is antisymmetric:
︡24bbb8e4-6107-47fa-9b9e-19cc3176712e︡︡{"done":true,"html":"$s$ is antisymmetric:
"} ︠d9215e2a-d702-48c8-b85b-72c8d023de55s︠ s(u,v,w), s(u,w,v), s(v,w,u), s(v,u,w), s(w,u,v), s(w,v,u) ︡9805948f-eb5f-40f5-ab0b-328be1547898︡︡{"html":"$k$ and $l$ being non negative integers, a tensor of type $(k,l)$ on the free module $M$ is a multilinear map
\[ t: \underbrace{M^*\times\cdots\times M^*}_{k\ \; \mbox{times}}
\times \underbrace{M\times\cdots\times M}_{l\ \; \mbox{times}}
\longrightarrow R \]
In the present case the ring $R$ is $\mathbb{Z}$.
For free modules of finite rank, we have the canonical isomorphism $M^{**} \simeq M$, so that the set of all tensors of type $(k,l)$ can be identified with the tensor product
\[ T^{(k,l)}(M) = \underbrace{M\otimes\cdots\otimes M}_{k\ \; \mbox{times}}
\otimes \underbrace{M^*\otimes\cdots\otimes M^*}_{l\ \; \mbox{times}} \]
In particular, tensors of type $(1,0)$ are identified with elements of $M$:
︡e6f208e1-b257-4a70-83be-2b3ebdc330fb︡︡{"done":true,"html":"$k$ and $l$ being non negative integers, a tensor of type $(k,l)$ on the free module $M$ is a multilinear map
\n\\[ t: \\underbrace{M^*\\times\\cdots\\times M^*}_{k\\ \\; \\mbox{times}}
\\times \\underbrace{M\\times\\cdots\\times M}_{l\\ \\; \\mbox{times}}
\\longrightarrow R \\]
In the present case the ring $R$ is $\\mathbb{Z}$.
\nFor free modules of finite rank, we have the canonical isomorphism $M^{**} \\simeq M$, so that the set of all tensors of type $(k,l)$ can be identified with the tensor product
\n\\[ T^{(k,l)}(M) = \\underbrace{M\\otimes\\cdots\\otimes M}_{k\\ \\; \\mbox{times}}
\\otimes \\underbrace{M^*\\otimes\\cdots\\otimes M^*}_{l\\ \\; \\mbox{times}} \\]
In particular, tensors of type $(1,0)$ are identified with elements of $M$:
"} ︠16c4ab44-f976-42a4-8f1d-89ecf2a87ec4s︠ M.tensor_module(1,0) is M ︡0ae14258-aa07-4a1b-ab48-0e625b76bdf3︡︡{"html":"According to the above definition, linear forms are tensors of type (0,1):
︡b35b6ac8-61bd-4635-aa44-191cd96bcb06︡︡{"done":true,"html":"According to the above definition, linear forms are tensors of type (0,1):
"} ︠b21c248c-927d-41cb-896b-39af657bb6cfs︠ a in M.tensor_module(0,1) ︡74f3b2f6-4dbe-4284-b216-92a91d3342c2︡︡{"html":"Note that, at the Python level, we do not have the identification of $T^{(0,1)}(M)$ with $M^*$:
︡edbe1c6b-cd5b-4b3a-bd5e-73a3de9805a7︡︡{"done":true,"html":"Note that, at the Python level, we do not have the identification of $T^{(0,1)}(M)$ with $M^*$:
"} ︠79f7387d-b097-4de2-bc24-b097bb05cb93s︠ M.tensor_module(0,1) is M.dual() ︡3bc0f4ed-c468-413e-8243-96757461a676︡︡{"html":"This is because $T^{(0,1)}(M)$ and $M^*$ are different objects:
︡e773b556-1988-4ebe-b80b-5604f14e0ffd︡︡{"done":true,"html":"This is because $T^{(0,1)}(M)$ and $M^*$ are different objects:
"} ︠e174df76-aa98-4b66-9805-f2a036a359b8s︠ type(M.tensor_module(0,1)) ︡7ebacdfc-5c30-4520-8e07-946fadd083d8︡︡{"stdout":"However, we have coercion (automatic conversion) of elements of $M^*$ into elements of $T^{(0,1)}(M)$:
︡5c650e80-8df8-4daa-b779-0429a37b654a︡︡{"done":true,"html":"However, we have coercion (automatic conversion) of elements of $M^*$ into elements of $T^{(0,1)}(M)$:
"} ︠068ea03c-9a0c-4d11-8098-d5fb675dbb6es︠ M.tensor_module(0,1).has_coerce_map_from(M.dual()) ︡ce001663-ab87-45b1-9724-d2fc5c2ed7c2︡︡{"html":"as well as coercion in the reverse direction:
︡0c18dd61-db6a-4893-992c-8812b11914d4︡︡{"done":true,"html":"as well as coercion in the reverse direction:
"} ︠bc91684e-da69-48c5-8d9e-9c600ac3059ds︠ M.dual().has_coerce_map_from(M.tensor_module(0,1)) ︡971a5356-c7e7-4dc8-9139-c2080bf7497d︡︡{"html":"Arbitrary tensors are constructed via the module method tensor(), by providing the tensor type $(k,l)$ and possibly the symbol to denote the tensor:
︡c6179048-dd5e-46ef-8268-513569bfb185︡︡{"done":true,"html":"Arbitrary tensors are constructed via the module method tensor(), by providing the tensor type $(k,l)$ and possibly the symbol to denote the tensor:
"} ︠5d743fe7-09fb-441f-9f84-b85a39343937s︠ t = M.tensor((1,1), name='t') print t ︡0682221b-c6b1-4fa3-bc8b-742b068b9466︡︡{"stdout":"Type-(1,1) tensor t on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠68e88cba-6ddb-43e3-a613-249684c90487i︠ %htmlLet us set some component of $t$ in the basis $e$, for instance the component $t^1_{\ \, 2}$:
︡10297ca4-3910-491f-ac89-b227c468f108︡︡{"done":true,"html":"Let us set some component of $t$ in the basis $e$, for instance the component $t^1_{\\ \\, 2}$:
"} ︠f97c6880-18a5-4842-9b3f-2b4394671d4es︠ t[e,1,2] = -3 ︡1b53180b-81d7-4416-b4a0-ea0e512ab3d7︡︡{"done":true} ︠003ceebc-5a77-4d36-ae5f-5f3c63349bc2i︠ %htmlSince $e$ is the default basis, a shortcut for the above is
︡d89ab4a5-67ed-4ff6-a2a5-f749f55d87a6︡︡{"done":true,"html":"Since $e$ is the default basis, a shortcut for the above is
"} ︠6837ec1e-ec92-4215-a823-455e75158e52s︠ t[1,2] = -3 ︡a64846cd-4375-45de-a625-b687aa795323︡︡{"done":true} ︠8a18386e-6091-4481-93fb-2df471de5d70i︠ %htmlThe unset components are zero:
︡0cb0bb33-8ad7-4b26-adbf-dd18de295c8a︡︡{"done":true,"html":"The unset components are zero:
"} ︠08a1d601-d7b5-4ee1-b3b9-4e90d14783eas︠ t[:] ︡cb01f60f-b6fd-4649-815d-605b58faf968︡︡{"html":"Components can be set at any time:
︡e492dbbc-25cd-43cd-99b8-5505f3cfd611︡︡{"done":true,"html":"Components can be set at any time:
"} ︠05833adf-e402-4450-9084-735c81e34fe5s︠ t[2,3] = 4 t[:] ︡2cd366eb-5c54-4d43-8ab3-3425b36ea57d︡︡{"html":"The components with respect to the basis $f$ are evaluated by the change-of-basis formula $e\rightarrow f$:
︡0913aa6c-c8c3-4818-994e-c40aca3259a1︡︡{"done":true,"html":"The components with respect to the basis $f$ are evaluated by the change-of-basis formula $e\\rightarrow f$:
"} ︠d55382ae-c8b0-4fd3-8ebf-4bc976a8cc5fs︠ t[f,:] ︡d606ebf7-c303-40ef-babd-1e395d185b59︡︡{"html":"Another view of $t$, which reflects the fact that $T^{(1,1)}(M) = M\otimes M^*$, is
︡7d734589-2bc5-46d1-90c1-2704b4fcc29c︡︡{"done":true,"html":"Another view of $t$, which reflects the fact that $T^{(1,1)}(M) = M\\otimes M^*$, is
"} ︠27dc2132-7ac8-4133-9c00-48ff45d1279ds︠ t.disp() ︡cb264584-74ad-47c7-8d9e-00b09755e1d5︡︡{"html":"Recall that $(e^i)$ is the basis of $M^*$ that is dual to the basis $(e_i)$ of $M$.
In term of the basis $(f_i)$ and its dual basis $(f^i)$, we have
︡b296c0bd-02a0-44b3-a7e6-a831f475ab6c︡︡{"done":true,"html":"Recall that $(e^i)$ is the basis of $M^*$ that is dual to the basis $(e_i)$ of $M$.
\nIn term of the basis $(f_i)$ and its dual basis $(f^i)$, we have
"} ︠e39cf6a9-ed6f-4cda-97d1-56f931958fb6s︠ t.disp(f) ︡1a4ff923-10c5-4cbe-830f-cfe8db2bc8f5︡︡{"html":"As a tensor of type (1,1), $t$ maps pairs (linear form, module element) to ring elements:
︡616e9a4b-a629-4c66-8bd9-8027894f2f11︡︡{"done":true,"html":"As a tensor of type (1,1), $t$ maps pairs (linear form, module element) to ring elements:
"} ︠4af4b4dc-e80b-4e41-82ab-9d67fafe1af9s︠ t(a,v) ︡2995236a-94b8-47f4-8756-eb1033cd3e35︡︡{"html":"Tensors of type (1,1) can be considered as endomorphisms, thanks to the isomorphism
\[ \begin{array}{cccccc} \mathrm{End}(M) & \longrightarrow & T^{(1,1)}(M) \\ \tilde t & \longmapsto & t: & M^*\times M & \longrightarrow & R \\& & & (a,v) & \longmapsto & a(\tilde t(v)) \end{array} \]
︡fec22667-bc50-4602-ab9a-ee2d463b64f9︡︡{"done":true,"html":"Tensors of type (1,1) can be considered as endomorphisms, thanks to the isomorphism
\n\\[ \\begin{array}{cccccc} \\mathrm{End}(M) & \\longrightarrow & T^{(1,1)}(M) \\\\ \\tilde t & \\longmapsto & t: & M^*\\times M & \\longrightarrow & R \\\\& & & (a,v) & \\longmapsto & a(\\tilde t(v)) \\end{array} \\]
"} ︠7de13d09-ef5e-40fd-b553-424b8bbe3279s︠ tt = End(M)(t) print tt ︡dbc83a76-0fb8-492b-bbf4-30aceccaa69e︡︡{"stdout":"Generic endomorphism of Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠c4c93275-d751-498f-a868-3e19ded71c3ds︠ tt.parent() ︡ed004e75-d7ca-49c1-b33f-a2d1aadd86a1︡︡{"html":"In a given basis, the matrix ${\tilde t}^i_{\ \, j}$ of the endomorphism $\tilde t$ is identical to the matrix of the tensor $t$:
︡18713ed6-fae9-422e-9f45-3a0240bdcf44︡︡{"done":true,"html":"In a given basis, the matrix ${\\tilde t}^i_{\\ \\, j}$ of the endomorphism $\\tilde t$ is identical to the matrix of the tensor $t$:
"} ︠17ef7ba9-a3cf-4b42-bc96-4e7a7a3a946bs︠ tt.matrix(e) ︡9b297a07-9f08-45cd-8c98-5796423d3749︡︡{"html":"As an endomorphism, $t$ maps module elements to module elements:
︡ef6b0392-61fe-470a-96af-2489a0bd1d3e︡︡{"done":true,"html":"As an endomorphism, $t$ maps module elements to module elements:
"} ︠81181037-9dcd-4d3d-b6e7-b768fd681e3es︠ tt(v) ︡30ae1f7a-890d-41fd-bb02-104ccd4856a5︡︡{"html":"$t$ belongs to the module $T^{(1,1)}(M)$:
︡deb2c21a-1783-4d9e-a224-7632d9cc0c58︡︡{"done":true,"html":"$t$ belongs to the module $T^{(1,1)}(M)$:
"} ︠f4533393-8f7e-438e-a53e-4ef8a2dc4002s︠ t in M.tensor_module(1,1) ︡fa4f17f8-657e-43c3-8a94-89c1c1f26699︡︡{"html":"or, in Sage terminology,
︡fa98acae-852e-43b5-86d3-526bddf2781a︡︡{"done":true,"html":"or, in Sage terminology,
"} ︠76796530-2af1-4805-be9c-d6df72c8b473s︠ t.parent() is M.tensor_module(1,1) ︡5627ac64-061e-44d5-8ff9-fcaee8e33d38︡︡{"html":"$T^{(1,1)}(M)$ is itself a free module of finite rank over $\mathbb{Z}$:
︡28cf52f6-e970-4132-9e09-07b7e9a01f9b︡︡{"done":true,"html":"$T^{(1,1)}(M)$ is itself a free module of finite rank over $\\mathbb{Z}$:
"} ︠7359d2c6-4e3a-48b5-be95-be11cba74bbds︠ isinstance(M.tensor_module(1,1), FiniteRankFreeModule) ︡710d5a3e-7bd7-46e6-95c0-20d506ab61f2︡︡{"html":"In addition to the arithmetic operations inherent to the module structure of $T^{(k,l)}(M)$, the following operations are implemented:
The tensor product is formed with the * operator. For instance the tensor product $t\otimes a$ is
︡fb0b5dcb-0eef-4824-bd55-54f78a6f8ecf︡︡{"done":true,"html":"In addition to the arithmetic operations inherent to the module structure of $T^{(k,l)}(M)$, the following operations are implemented:
\nThe tensor product is formed with the * operator. For instance the tensor product $t\\otimes a$ is
"} ︠cf1bb683-771c-49e4-a2b3-0c2b3c46cd8bs︠ ta = t*a print ta ︡38d5a183-7925-4ae2-be3d-461f1592422a︡︡{"stdout":"Type-(1,2) tensor t*a on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠7b9e4551-5094-48f8-97b5-1accc8446130s︠ ta ︡1a467e46-041d-4b71-bb65-75ae7db55a93︡︡{"html":"The components w.r.t. a given basis can also be displayed as an array:
︡00e6624e-c0b7-4000-989d-198eaf6e294f︡︡{"done":true,"html":"The components w.r.t. a given basis can also be displayed as an array:
"} ︠e3aee260-a111-4dbb-8394-e0165351d428s︠ ta[:] # components w.r.t. the default basis (e) ︡34bb809c-bb8c-4f16-a32a-40b3d4ebef9a︡︡{"html":"Each component ca be accessed individually:
︡9e6ea204-2fe8-4fbb-adea-3bf8fad73f18︡︡{"done":true,"html":"Each component ca be accessed individually:
"} ︠22bf2f20-715b-4147-9df0-9cae98fd1134s︠ ta[1,2,3] # access to a component w.r.t. the default basis ︡94064e0a-0a67-4cd1-bd23-8fb0bcddf3a9︡︡{"html":"The tensor product is not commutative:
︡c5915fdf-3491-4567-8ae6-cdcd400883ed︡︡{"done":true,"html":"The tensor product is not commutative:
"} ︠9a24f9e3-4686-4831-bb63-7b7ff4117cf6s︠ print a*t ︡75bed9bf-31e8-40db-9322-61d4e6e478b3︡︡{"stdout":"Type-(1,2) tensor a*t on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠11b4e6ee-1203-4674-88da-bbad543af889s︠ a*t == t*a ︡7d833fa9-dc07-413d-b078-b09dfb4b887f︡︡{"html":"Forming a tensor of rank 4:
︡f2481675-45d3-448c-80b7-26c3bda3b3c3︡︡{"done":true,"html":"Forming a tensor of rank 4:
"} ︠321c113d-87c3-4cd5-8e13-4462b83068bds︠ tav = ta*v print tav ︡190df842-d283-42b7-9455-3a2c198073d5︡︡{"stdout":"Type-(2,2) tensor t*a*v on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠5474fe50-3ab5-4120-9f22-0dbe67f72954s︠ tav.disp() ︡b689f602-2d3c-4e53-9fb9-d3ab12b7ae1e︡︡{"html":"The (anti)symmetrization of a tensor $t$ over $n$ arguments involve the division by $n!$, which does not always make sense in the base ring $R$. In the present case, $R=\mathbb{Z}$ and to (anti)symmetrize over 2 arguments, we restrict to tensors with even components:
︡72787b63-a7fa-4ce9-9ec4-2d418cf682b0︡︡{"done":true,"html":"The (anti)symmetrization of a tensor $t$ over $n$ arguments involve the division by $n!$, which does not always make sense in the base ring $R$. In the present case, $R=\\mathbb{Z}$ and to (anti)symmetrize over 2 arguments, we restrict to tensors with even components:
"} ︠a326f000-0713-4c84-a4e4-2bee67a121b4s︠ g = M.tensor((0,2), name='g') g[1,2], g[2,1], g[2,2], g[3,2], g[3,3] = 2, -4, 8, 2, -6 g[:] ︡572ef5ce-a3c5-4d8c-96d0-a832857246f6︡︡{"html":"Symmetrization can be performed on an arbitray number of arguments, by providing their positions (first position = 0). In the present case
︡efd07253-342d-4e69-a5f9-057cc5141d24︡︡{"done":true,"html":"Symmetrization can be performed on an arbitray number of arguments, by providing their positions (first position = 0). In the present case
"} ︠cc91775c-8850-4778-8e54-02235d2430e6s︠ s == g.symmetrize(0,1) ︡81fe5849-bf85-41d9-bf39-1f575e6e9d41︡︡{"html":"One may use index notation to specify the symmetry:
︡d7c727a4-7df2-4fbf-8c0c-f3a3397107ab︡︡{"done":true,"html":"One may use index notation to specify the symmetry:
"} ︠36138493-47c4-4c27-8377-97914b1ee698s︠ s == g['_(ab)'] ︡f8515f9b-f5d7-4ae9-8f4e-ed3f8727c8b5︡︡{"html":"Of course, since $s$ is already symmetric:
︡9bf29be2-78d6-4a2d-9be5-48b575d35aba︡︡{"done":true,"html":"Of course, since $s$ is already symmetric:
"} ︠4b20931f-39f7-4d53-8fa6-668dc1c842c6s︠ s.symmetrize() == s ︡608f4ef0-c581-44c1-bb66-42ba1bd0a967︡︡{"html":"The antisymmetrization proceeds accordingly:
︡429a0bb0-d2d6-45b7-853a-3c685332369d︡︡{"done":true,"html":"The antisymmetrization proceeds accordingly:
"} ︠ef0dc442-719a-470b-aa6a-efa50cf12bcds︠ s = g.antisymmetrize() ; s ︡9d544d5b-44d1-4d5d-9132-a7aa42b638cb︡︡{"html":"As for symmetries, index notation can be used, instead of antisymmetrize():
︡3597d8a3-e6cf-429f-ac47-251b49254544︡︡{"done":true,"html":"As for symmetries, index notation can be used, instead of antisymmetrize():
"} ︠b92a197f-9152-40b8-b5e9-7524acd7d74es︠ s == g['_[ab]'] ︡e8903ab6-d757-42d1-817a-869a668c827b︡︡{"html":"Of course, since $s$ is already antisymmetric:
︡1e2568ac-70e3-40aa-85ae-e28bd3f0c881︡︡{"done":true,"html":"Of course, since $s$ is already antisymmetric:
"} ︠f630dffd-64c1-4f8c-9a5d-fe0783348539s︠ s == s.antisymmetrize() ︡828bccd7-55d4-48f0-83f7-06a341e80321︡︡{"html":"Contracting the type-(1,1) tensor $t$ with the module element $v$ results in another module element:
︡9f1dbab1-d3e0-4856-b12b-357e8dd16e1b︡︡{"done":true,"html":"Contracting the type-(1,1) tensor $t$ with the module element $v$ results in another module element:
"} ︠a357b84a-7a26-4e05-8dfb-7c74b17acde6s︠ t.contract(v) ︡d305921c-8894-4d95-a9e2-33e88bb1964e︡︡{"html":"The components (w.r.t. a given basis) of the contraction are of course $t^i_{\\ j} v^j$:
︡a49a339f-4cc7-4baa-9dd4-05f6caa6069c︡︡{"done":true,"html":"The components (w.r.t. a given basis) of the contraction are of course $t^i_{\\\\ j} v^j$:
"} ︠2122494f-efb5-4f9d-b673-dec3c7b1cef1s︠ t.contract(v)[i] == sum(t[i,j]*v[j] for j in M.irange()) ︡e9c2dc6b-8f92-46b6-9b40-e8b26b81b2e6︡︡{"html":"This contraction coincides with the action of $t$ as an endomorphism:
︡a43d07d1-fa7e-423f-a41d-f041c7ccbe67︡︡{"done":true,"html":"This contraction coincides with the action of $t$ as an endomorphism:
"} ︠6483b658-75d3-4cb1-a979-43e8e51feaees︠ t.contract(v) == tt(v) ︡1bc377be-825b-4b56-adde-c68e1381cab5︡︡{"html":"Instead of contract(), index notations can be used to denote the contraction:
︡68425b71-b760-4572-9b49-4bde60c0f727︡︡{"done":true,"html":"Instead of contract(), index notations can be used to denote the contraction:
"} ︠0fc19335-9987-46a0-9b55-1bf9ed942945s︠ t['^i_j']*v['j'] == t.contract(v) ︡4f858a2b-489c-4c94-92e7-2ad81535caa9︡︡{"html":"Contracting the linear form $a$ with the module element $v$ results in a ring element:
︡f17fa339-0dc5-4fb5-bbe4-4969e32f5e50︡︡{"done":true,"html":"Contracting the linear form $a$ with the module element $v$ results in a ring element:
"} ︠687e046e-ebce-4ffb-917b-1273e52b08eds︠ a.contract(v) ︡b847d46a-75e4-4726-812c-7e215a38a657︡︡{"html":"It is of course the result of the linear form acting on the module element:
︡3bc06f27-50e3-46c8-b2dc-1b350dc985af︡︡{"done":true,"html":"It is of course the result of the linear form acting on the module element:
"} ︠03daaad0-d11c-4b97-aa7e-295bb4169c9as︠ a.contract(v) == a(v) ︡e330556a-db74-4e6c-8517-94844ded7a37︡︡{"html":"By default, the contraction is performed on the last index of the first tensor and the first index of the second one. To perform contraction on other indices, one should specify the indices positions (with the convention position=0 for the first index): for instance to get the contraction $z^i_{\ \, j} = T^i_{\ \, kj} v^k$ (with $T=t\otimes a$):
︡ded4dd14-8f1e-49ab-857e-775cb04eb24d︡︡{"done":true,"html":"By default, the contraction is performed on the last index of the first tensor and the first index of the second one. To perform contraction on other indices, one should specify the indices positions (with the convention position=0 for the first index): for instance to get the contraction $z^i_{\\ \\, j} = T^i_{\\ \\, kj} v^k$ (with $T=t\\otimes a$):
"} ︠5bbedc26-4ff0-4db6-80fa-d1ea9d092557s︠ z = ta.contract(1,v) # 1 -> second index of ta print z ︡1ce6bfa3-21a5-441f-a772-4451c6277116︡︡{"stdout":"Type-(1,1) tensor on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠2e30bdb8-15ad-4908-a652-b6258e267271i︠ %htmlTo get $z^i_{\ \, jk} = t^l_{\ \, j} T^i_{\ \, l k}$:
︡da0575cb-2209-445f-be5e-8cdc46c8828c︡︡{"done":true,"html":"To get $z^i_{\\ \\, jk} = t^l_{\\ \\, j} T^i_{\\ \\, l k}$:
"} ︠fff3bce4-c271-4465-b926-eb5b223dc7ebs︠ z = t.contract(0, ta, 1) # 0 -> first index of t, 1 -> second index of ta print z ︡949c0b98-87ad-49f4-962e-c74867e891a5︡︡{"stdout":"Type-(1,2) tensor on the Rank-3 free module M over the Integer Ring\n","done":false}︡{"done":true} ︠116d6e1f-054c-4f21-a7ce-486ff869aa51i︠ %htmlor, in terms of index notation:
︡e096599a-5bc0-4a92-a0ad-bd63ffd57287︡︡{"done":true,"html":"or, in terms of index notation:
"} ︠e1aeef8d-507d-47e6-81c8-1a750ec8f16cs︠ z1 = t['^l_j']*ta['^i_lk'] z1 == z ︡7f7042b9-90cd-4dc8-a601-9e7c43e804d5︡︡{"html":"As for any function, inline documentation is obtained via the quotation mark:
︡c3b2f0ce-4019-4713-9680-18f51ede7934︡︡{"done":true,"html":"As for any function, inline documentation is obtained via the quotation mark:
"} ︠1613314a-ad72-4b3c-8ff2-1b5e1f4b4815s︠ t.contract? ︡2af0823d-c99a-40eb-91b3-490b088179e4︡︡{"code":{"source":"File: /projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/tensor/modules/free_module_tensor.py\nSignature : t.contract(*args)\nDocstring :\nContraction on one or more indices with another tensor.\n\nINPUT:\n\n* \"pos1\" -- positions of the indices in \"self\" involved in the\n contraction; \"pos1\" must be a sequence of integers, with 0\n standing for the first index position, 1 for the second one, etc;\n if \"pos1\" is not provided, a single contraction on the last index\n position of \"self\" is assumed\n\n* \"other\" -- the tensor to contract with\n\n* \"pos2\" -- positions of the indices in \"other\" involved in the\n contraction, with the same conventions as for \"pos1\"; if \"pos2\"\n is not provided, a single contraction on the first index position\n of \"other\" is assumed\n\nOUTPUT:\n\n* tensor resulting from the contraction at the positions \"pos1\"\n and \"pos2\" of \"self\" with \"other\"\n\nEXAMPLES:\n\nContraction of a tensor of type (0,1) with a tensor of type (1,0):\n\n sage: M = FiniteRankFreeModule(ZZ, 3, name='M')\n sage: e = M.basis('e')\n sage: a = M.linear_form() # tensor of type (0,1) is a linear form\n sage: a[:] = [-3,2,1]\n sage: b = M([2,5,-2]) # tensor of type (1,0) is a module element\n sage: s = a.contract(b) ; s\n 2\n sage: s in M.base_ring()\n True\n sage: s == a[0]*b[0] + a[1]*b[1] + a[2]*b[2] # check of the computation\n True\n\nThe positions of the contraction indices can be set explicitely:\n\n sage: s == a.contract(0, b, 0)\n True\n sage: s == a.contract(0, b)\n True\n sage: s == a.contract(b, 0)\n True\n\nInstead of the explicit call to the method \"contract()\", the index\nnotation can be used to specify the contraction, via Einstein\nconventation (summation on repeated indices); it suffices to pass\nthe indices as a string inside square brackets:\n\n sage: s1 = a['_i']*b['^i'] ; s1\n 2\n sage: s1 == s\n True\n\nIn the present case, performing the contraction is identical to\napplying the linear form to the module element:\n\n sage: a.contract(b) == a(b)\n True\n\nor to applying the module element, considered as a tensor of type\n(1,0), to the linear form:\n\n sage: a.contract(b) == b(a)\n True\n\nWe have also:\n\n sage: a.contract(b) == b.contract(a)\n True\n\nContraction of a tensor of type (1,1) with a tensor of type (1,0):\n\n sage: a = M.tensor((1,1))\n sage: a[:] = [[-1,2,3],[4,-5,6],[7,8,9]]\n sage: s = a.contract(b) ; s\n Element of the Rank-3 free module M over the Integer Ring\n sage: s.display()\n 2 e_0 - 29 e_1 + 36 e_2\n\nSince the index positions have not been specified, the contraction\ntakes place on the last position of a (i.e. no. 1) and the first\nposition of \"b\" (i.e. no. 0):\n\n sage: a.contract(b) == a.contract(1, b, 0)\n True\n sage: a.contract(b) == b.contract(0, a, 1)\n True\n sage: a.contract(b) == b.contract(a, 1)\n True\n\nUsing the index notation with Einstein convention:\n\n sage: a['^i_j']*b['^j'] == a.contract(b)\n True\n\nThe index \"i\" can be replaced by a dot:\n\n sage: a['^._j']*b['^j'] == a.contract(b)\n True\n\nand the symbol \"^\" may be omitted, the distinction between\ncontravariant and covariant indices being the position with respect\nto the symbol \"_\":\n\n sage: a['._j']*b['j'] == a.contract(b)\n True\n\nContraction is possible only between a contravariant index and a\ncovariant one:\n\n sage: a.contract(0, b)\n Traceback (most recent call last):\n ...\n TypeError: contraction on two contravariant indices not permitted\n\nContraction of a tensor of type (2,1) with a tensor of type (0,2):\n\n sage: a = a*b ; a\n Type-(2,1) tensor on the Rank-3 free module M over the Integer Ring\n sage: b = M.tensor((0,2))\n sage: b[:] = [[-2,3,1], [0,-2,3], [4,-7,6]]\n sage: s = a.contract(1, b, 1) ; s\n Type-(1,2) tensor on the Rank-3 free module M over the Integer Ring\n sage: s[:]\n [[[-9, 16, 39], [18, -32, -78], [27, -48, -117]],\n [[36, -64, -156], [-45, 80, 195], [54, -96, -234]],\n [[63, -112, -273], [72, -128, -312], [81, -144, -351]]]\n\nCheck of the computation:\n\n sage: all(s[i,j,k] == a[i,0,j]*b[k,0]+a[i,1,j]*b[k,1]+a[i,2,j]*b[k,2]\n ....: for i in range(3) for j in range(3) for k in range(3))\n True\n\nUsing index notation:\n\n sage: a['il_j']*b['_kl'] == a.contract(1, b, 1)\n True\n\nLaTeX notation are allowed:\n\n sage: a['^{il}_j']*b['_{kl}'] == a.contract(1, b, 1)\n True\n\nIndices not involved in the contraction may be replaced by dots:\n\n sage: a['.l_.']*b['_.l'] == a.contract(1, b, 1)\n True\n\nThe two tensors do not have to be defined on the same basis for the\ncontraction to take place, reflecting the fact that the contraction\nis basis-independent:\n\n sage: A = M.automorphism()\n sage: A[:] = [[0,0,1], [1,0,0], [0,-1,0]]\n sage: h = e.new_basis(A, 'h')\n sage: b.comp(h)[:] # forces the computation of b's components w.r.t. basis h\n [-2 -3 0]\n [ 7 6 -4]\n [ 3 -1 -2]\n sage: b.del_other_comp(h) # deletes components w.r.t. basis e\n sage: b._components.keys() # indeed:\n [Basis (h_0,h_1,h_2) on the Rank-3 free module M over the Integer Ring]\n sage: a._components.keys() # while a is known only in basis e:\n [Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring]\n sage: s1 = a.contract(1, b, 1) ; s1 # yet the computation is possible\n Type-(1,2) tensor on the Rank-3 free module M over the Integer Ring\n sage: s1 == s # ... and yields the same result as previously:\n True\n\nThe contraction can be performed on more than a single index; for\ninstance a 2-indices contraction of a type-(2,1) tensor with a\ntype-(1,2) one is:\n\n sage: a # a is a tensor of type-(2,1)\n Type-(2,1) tensor on the Rank-3 free module M over the Integer Ring\n sage: b = M([1,-1,2])*b ; b # a tensor of type (1,2)\n Type-(1,2) tensor on the Rank-3 free module M over the Integer Ring\n sage: s = a.contract(1,2,b,1,0) ; s # the double contraction\n Type-(1,1) tensor on the Rank-3 free module M over the Integer Ring\n sage: s[:]\n [ -36 30 15]\n [-252 210 105]\n [-204 170 85]\n sage: s == a['^.k_l']*b['^l_k.'] # the same thing in index notation\n True","mode":"text/x-rst","lineno":-1,"filename":null},"done":false}︡{"done":true} ︠91fc3390-d378-4e91-82a0-5c428d6d9033︠