Introduction to the atlas 2 package
<Text-field bookmark="info" style="Heading 2" layout="Heading 2">Description: </Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font foreground="[0,0,128]">Modern differential geometry</Font></Text-field> Modern differential geometry is the basis for the atlas package. Such entities as manifolds, mappings, p-forms, tensor fields, bundles, connections are very important in the modern differential geometry. The atlas package allows to work with these entities without extra efforts, just define an entity with corresponding obvious definition and work with it just as you usually do. The following declarations are trivial and self explanatory: Domain - manifold and domain declaration, Constants - constants declaration, Functions - functions declaration, Tensors - tensors declaration, Forms - forms declaration, Vectors - vectors declaration, Mapping - declaration of a mapping between manifolds or domains, Coframe - coframe declaration, Frame - frame declaration, Metric - metric tensor declaration.
<Text-field style="Heading 3" layout="Heading 3"><Font foreground="[0,0,128]">No programming just differential geometry</Font></Text-field> When working on your problem you think in terms of manifolds, mappings, embeddings, submersions, p-forms, tensor fields etc. The atlas package allows you concentrate on the differential geometry problem not on the programming. You can use predefined declaration operators to declare various differential geometry objects, which are calculated on the fly: Projectors - automatic calculation of projectors of a mapping, Invariants - automatic calculation of invariants of a mapping, Connection - automatic calculation of connection 1-forms, Curvature - automatic calculation of curvature 2-forms, Torsion - automatic calculation of torsion 2-forms, Riemann - automatic Riemann tensor calculation, Ricci - automatic Ricci tensor calculation, RicciScalar - automatic Ricci scalar calculation.
<Text-field style="Heading 3" layout="Heading 3"><Font foreground="[0,0,128]">No ugly output just standard notation</Font></Text-field> The atlas package uses standard differential geometry notations: d - exterior derivative, \342\204\222 - Lie derivative, \316\271 - interior product, \342\213\200 - exterior product, \342\212\227 - tensor product, \342\213\206 - Hodge star, \342\210\207 - covariant differentiation, \316\264 - Kronecker's delta symbol etc. You always get output as you expected like the following: atlas package output example with Lie derivative calculation: 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 atlas package output example with exterior derivative calculation: LUkkYW5kRyUqcHJvdGVjdGVkRzYkLUYjNiQvLUkiZEc2IjYjM0kmb21lZ2FHRitJJnNpZ21hR0YrRi8sJi1GKjYjRi4iIiJGLiEiIi1GKjYjRi8= atlas package output example with tensor product calculation: L0kiZ0c2IiomLUkiKkclKnByb3RlY3RlZEc2JCIiJSwmLUkpJm90aW1lcztHRiQ2JCZJImVHRiQ2IyIiIkYvRjIqJiZGMDYjIiIjRjItRi02I0Y0RjJGMkYyKSwmRjJGMiomSSdsYW1iZGFHRiRGMiwmKiQpSSJ4R0YkRjZGMkYyKiQpSSJ5R0YkRjZGMkYyRjJGMkY2ISIiJSFH atlas package output example with covariant derivative calculation: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2I1EhRictRiM2Ky1JJW1zdWJHRiQ2JS1GLDYlUSgmbmFibGE7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiktRiw2JVEiWEYnRjdGOi8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnL0Y7USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGUS8lJmZlbmNlR0ZNLyUqc2VwYXJhdG9yR0ZNLyUpc3RyZXRjaHlHRk0vJSpzeW1tZXRyaWNHRk0vJShsYXJnZW9wR0ZNLyUubW92YWJsZWxpbWl0c0dGTS8lJ2FjY2VudEdGTS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmRvLUkobWZlbmNlZEdGJDYkLUYjNitGKy1GIzYrLUYsNiVRIllGJ0Y3RjotRlc2LVEiK0YnRlFGWkZmbkZobkZqbkZcb0Zeb0Zgby9GY29RLDAuMjIyMjIyMmVtRicvRmZvRmVwLUYsNiVRIlpGJ0Y3RjpGQkZFRkhGS0ZORlFGK0ZCRkVGSEZLRk5GUUZRRkJGRUZIRktGTkZRLUZXNi1RIj1GJ0ZRRlpGZm5GaG5Gam5GXG9GXm9GYG8vRmNvUSwwLjI3Nzc3NzhlbUYnL0Zmb0ZecS1GIzYtRistRiM2K0YxRlYtRmhvNiQtRiM2KUZecEZCRkVGSEZLRk5GUUZRRkJGRUZIRktGTkZRRmFwLUYjNitGMUZWLUZobzYkLUYjNilGZ3BGQkZFRkhGS0ZORlFGUUZCRkVGSEZLRk5GUUYrRkJGRUZIRktGTkZRRitGQkZFRkhGS0ZORlE= atlas package output example with interior product and Kronecker's delta symbol calculation: 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 atlas package output example with calculation in a manifold with symbolic dimention: 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JSFH Old atlas package notations are still supported for input purposes and for Classic Worksheet Interface of Maple 14.
<Text-field style="Heading 3" layout="Heading 3"><Font foreground="[0,0,128]">Single solving path for almost any problem</Font></Text-field> With the atlas package you always have one and the same solving path for almost any of your differential geometry problem. You start with definitions of manifolds, vector and tensor fields, p- forms and mappings between the manifolds. When you get your differential geometry entities defined, you use standard operators to get various quantities of your entities: Projectors - automatic calculation of projectors of a mapping, Invariants - automatic calculation of invariants of a mapping, Connection - automatic calculation of connection 1-forms, Curvature - automatic calculation of curvature 2-forms, Torsion - automatic calculation of torsion 2-forms, Riemann - automatic Riemann tensor calculation, Ricci - automatic Ricci tensor calculation, RicciScalar - automatic Ricci scalar calculation. This is standard procedure which can be automated completely. For instance for 2D and 3D differential geometry problems the atlas package offers atlasWizard - the powerfull Maplet tool to formulate and solve typical differential geometry problems for lower dimentions. You can fine atlasWizard in maplets subfolder of your atlas distribution folder: {atlas-path}/maplets/atlasWizard.maplet, where {atlas-path} is a path to your atlas distribution folder. In higher dimentions you can also automate the solving path by using our templates worksheets (look at {atlas-path}/examples).
<Text-field style="Heading 3" layout="Heading 3"><Font foreground="[0,0,128]"> All calculations are as coordinate free as possible </Font></Text-field> In the atlas package all calculations are performed in terms of tensors, vectors and p-forms (not their components!). For instance, conformally flat metric tensor of sphere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdXBHRiQ2JS1GLDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRkFGK0ZGRklGQQ== is presented as 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiM2KS1JJW1zdWJHRiQ2JS1GLDYlUSJlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0ZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGQy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGVC8lKXN0cmV0Y2h5R0ZULyUqc3ltbWV0cmljR0ZULyUobGFyZ2VvcEdGVC8lLm1vdmFibGVsaW1pdHNHRlQvJSdhY2NlbnRHRlQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zdby1GIzYoLUYsNiVRImRGJ0Y3RjotRk82LVEwJkFwcGx5RnVuY3Rpb247RidGQ0ZSRlVGV0ZZRmVuRmduRmluL0Zcb1EmMC4wZW1GJy9GX29GaW8tSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJ4RidGN0Y6RkVGSEZDRkNGRUZIRkNGK0ZFRkhGQy1GTzYtUSIsRidGQ0ZSL0ZWRjlGV0ZZRmVuRmduRmluRmhvL0Zfb1EsMC4zMzMzMzMzZW1GJy1GIzYpLUYyNiVGNC1GIzYmLUZANiRRIjJGJ0ZDRkVGSEZDRktGTi1GIzYoRmJvRmVvLUZccDYkLUYjNiYtRiw2JVEieUYnRjdGOkZFRkhGQ0ZDRkVGSEZDRitGRUZIRkNGK0ZFRkhGQw== are coframe 1-forms and symbol \342\212\227 - is tensor product operator (see examples below). To get more information about the main principle of the package structure and to look through some complete examples see atlas[examples]. To look through references list see atlas[references].
<Text-field style="Heading 3" layout="Heading 3"><Font foreground="[0,0,128]">Some calculations with symbolic dimension are available </Font></Text-field> The atlas package allows one to make some useful calculations even if the working dimension is symbolic. For example, if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEibkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi8vJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RicvRjZRJ25vcm1hbEYn is the dimension, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEia0YnRjJGNUY4RjIvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RicvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMkZARkM= are coframe 1-forms and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEiakYnRjJGNUY4RjIvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RicvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQEZD are frame vectors then decomposition 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 (of interior product of vector LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGJy9GM1Enbm9ybWFsRic= and 1-form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElJnhpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RidGMg==) - is one of the available calculation. Another example is Lie bracket decomposition: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2I1EhRictRiM2KS1JJW1zdWJHRiQ2JS1GLDYmUScmTHNjcjtGJy8lJXNpemVHUSMxNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYyNiUtRiw2JlEiRUYnRjdGOkY9LUYjNictRiw2JlEiaUYnRjdGOkY9RjcvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJy9GPlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y3RkxGT0ZSRlQtSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZSLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Znbi8lKXN0cmV0Y2h5R0Znbi8lKnN5bW1ldHJpY0dGZ24vJShsYXJnZW9wR0Znbi8lLm1vdmFibGVsaW1pdHNHRmduLyUnYWNjZW50R0Znbi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmZvLUkobWZlbmNlZEdGJDYlLUYjNictRjI2JUZELUYjNictRiw2JlEiakYnRjdGOkY9RjdGTEZPRlJGVEY3RkxGT0ZSRjdGUkY3RkxGT0ZSLUZYNi5RIj1GJ0Y3RlJGZW5GaG5Gam5GXG9GXm9GYG9GYm8vRmVvUSwwLjI3Nzc3NzhlbUYnL0Zob0ZpcC1GIzYrLUkrbXVuZGVyb3ZlckdGJDYnLUZYNjBRJiZTdW07RidGNy9GTVEuWzE0NCwxNDQsMTQ0XUYnRlIvSSttc2VtYW50aWNzR0YkUSZpbmVydEYnRmVuRmhuL0Zbb0Y8RlxvL0Zfb0Y8L0Zhb0Y8RmJvRmRvL0Zob1EsMC4xNjY2NjY3ZW1GJy1GIzYpLUYyNiUtRiw2JlEibEYnRjdGOkY9LUYjNictSSNtbkdGJDYlUSIxRidGN0ZSRjdGTEZPRlJGVEZlcEZmckY3RkxGT0ZSLUYsNiZRIm5GJ0Y3RjpGPUZiby8lLGFjY2VudHVuZGVyR0ZnbkYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSQ1LjBGJy8lJmRlcHRoR0Zkcy8lKmxpbmVicmVha0dRJWF1dG9GJy1GIzYpLUZqbzYlLUYjNitGKy1GIzYpLUYyNiUtRiw2JlEnJmlvdGE7RidGNy9GO0ZnbkZSRkBGVEZXLUZqbzYlLUYjNictRjI2JS1GLDYmUSgmb21lZ2E7RidGN0ZqdEZSLUYjNilGX3ItRlg2LlEiLEYnRjdGUkZlbi9GaW5GPEZqbkZcb0Zeb0Zgb0Zib0Zkby9GaG9RLDAuMzMzMzMzM2VtRidGYnBGN0ZMRk9GUkZURjdGTEZPRlJGN0ZSRjdGTEZPRlItRlg2LlEoJm1pbnVzO0YnRjdGUkZlbkZobkZqbkZcb0Zeb0Zgb0Ziby9GZW9RLDAuMjIyMjIyMmVtRicvRmhvRmB2LUYjNiktRjI2JUZndEZccEZURlctRmpvNiUtRiM2Jy1GMjYlRmF1LUYjNilGX3JGZnVGSUY3RkxGT0ZSRlRGN0ZMRk9GUkY3RlJGN0ZMRk9GUkYrRjdGTEZPRlJGN0ZSLUZYNi1RMSZJbnZpc2libGVUaW1lcztGJ0ZSRmVuRmhuRmpuRlxvRl5vRmBvRmJvRmRvRmdvLUYyNiVGRC1GIzYnRl9yRjdGTEZPRlJGVEY3RkxGT0ZSRitGN0ZMRk9GUkYrRjdGTEZPRlI= To get more information about this possibility see atlas[dim]
<Text-field style="Heading 3" layout="Heading 3"><Font foreground="[0,0,128]">Almost any differential geometry entity can be indexed</Font></Text-field> In the atlas package any object (constant, tensor, p-form, manifold etc.) can be indexed. This is very flexable feature. For or more information on atlas indexing facilities, see atlas[indexing].
<Text-field style="Heading 3" layout="Heading 3"><Font foreground="[0,0,128]">Easy customizable simplification of your results</Font></Text-field> Because computations with tensors and p-forms usually involve a great number of quantities, it is important to make simplification in each step of the computations. For this reason, the user can customise the simplification routine `atlas/simp` for a particular problem. For more information, see atlas[simp].
<Text-field style="Heading 3" layout="Heading 3"><Font foreground="[0,0,128]">Intallation instructions</Font></Text-field> Copy the files atlas.hdb, atlas.mla from the {atlas-path}/lib folder, and place them inside the lib folder of your Maple installation. To find out what your Maple lib folder is, execute in your Maple session: [> libname; You get strings separated by commas. First string is your path to your Maple lib folder. Before using routines of the atlas package, you must load the package with one of the commands with(atlas), with(atlas,[]), with(atlas,<function>) or atlas[<function>]. See with for details.
<Text-field style="Heading 2" layout="Heading 2">Declaration operators</Text-field> Use these operators to declare various differential geometry entities. Domain Manifold and domain declaration Constants Constants declaration Functions Functions declaration Tensors Tensors declaration Forms Forms declaration Vectors Vectors declaration Mapping Declaration of a mapping between manifolds or domains Coframe Coframe declaration Frame Frame declaration Metric Metric tensor declaration
<Text-field style="Heading 2" layout="Heading 2">Calculation operators</Text-field> Use these operators for automatic calculation of various differential geometry objects. Projectors Calculation of projectors of a mapping Invariants Calculation of invariants of a mapping Connection Calculation of connection 1-forms Curvature Calculation of curvature 2-forms Torsion Calculation of torsion 2-forms Riemann Riemann tensor calculation Ricci Ricci tensor calculation RicciScalar Ricci scalar calculation
<Text-field style="Heading 2" layout="Heading 2">Differential geometry operators</Text-field> Use these operators for standard differential geometry calculations. d Exterior derivative operator \342\204\222 Lie derivative \316\271 Interior product operator \342\213\200 Exterior product operator \342\212\227 Tensor product operator \342\213\206 Hodge operator \342\210\207 Covariant differentiation grad Gradient operator div Divergence operator &d Codifferential operator &L Hodge-de Rham Laplacian &/ Pullback of a [0,k] tensor field under a mapping &D Pushforward (differential of a mapping)
<Text-field style="Heading 2" layout="Heading 2">Utility operators</Text-field> Use these operators to control and manage your differential geometry entities. ToBasis "ToBasis" decomposision Who Lists of all declarations made and shows "who is who" kind Kind of a tensor &$ Generalized interior product operator &@ Natural vector operator dual Dual operator delta Kronecker's delta symbol
<Text-field bookmark="examples" style="Heading 2" layout="Heading 2">Examples:</Text-field> restart: with(atlas): Conformally flat metric on sphere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdXBHRiQ2JS1GLDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRkFGK0ZGRklGQQ==: Declare constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0Yy: Constants(lambda); PCksJF4jIiIiISIiRiRJI1BpRyUqcHJvdGVjdGVkR0kjX1pHNiJJJGRpbUdGKkkoQ2F0YWxhbkdGKEknbGFtYmRhR0Yq Declare functions: Functions(f=f(x,y),h=h(f)); PCRJImZHNiJJImhHRiQ= Declear forms: Forms(e[i]=1,xi=1,theta=p); PCVJI3hpRzYiSSZ0aGV0YUdGJCZJImVHRiQ2I0kiaUdGJA== Declare vectors: Vectors(E[j],X,Y,Z); PCZJIlhHNiJJIllHRiRJIlpHRiQmSSJFR0YkNiNJImpHRiQ= Declare coframe 1-forms: Coframe(e[1]=d(x),e[2]=d(y)); NyQvJkkiZUc2IjYjIiIiLUkiZEdGJjYjSSJ4R0YmLyZGJTYjIiIjLUYqNiNJInlHRiY= Now coframe 1-forms are 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. Declare frame vectors: Frame(E[j]); NyQvJkkiRUc2IjYjIiIiLUklRGlmZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYkSSFHRiZJInhHRiYvJkYlNiMiIiMtRio2JEYvSSJ5R0Ym Now frame vectors are 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. Metric declaration: Metric(g=4*(d(x)&.d(x)+d(y)&.d(y))/(1+lambda*(x^2+y^2))^2); L0kiZ0c2IiwkKigiIiUiIiIsJi1JIyYuR0YkNiQmSSJlR0YkNiNGKEYtRigtRis2JCZGLjYjIiIjRjJGKEYoKSwmRihGKComSSdsYW1iZGFHRiRGKCwmKiQpSSJ4R0YkRjRGKEYoKiQpSSJ5R0YkRjRGKEYoRihGKEY0ISIiRig= Calculate connection 1-forms: Connection(omega); Jkkmb21lZ2FHNiI2JEkiaUdGJEkiakdGJA== Calculate curvature 2-forms: Curvature(Omega); JkkmT21lZ2FHNiI2JEkiaUdGJEkiakdGJA== Riemann tensor: Riemann(R); L0kiUkc2IiwkKioiIzsiIiIpLChGKEYoKiZJJ2xhbWJkYUdGJEYoKUkieEdGJCIiI0YoRigqJkYsRigpSSJ5R0YkRi9GKEYoIiIlISIiRixGKC1JIyYuR0YkNiQtSSMmXkdGJDYkJkkiZUdGJDYjRigmRjw2I0YvRjhGKEYo Ricci tensor: Ricci(r); L0kickc2IiwmKioiIiUiIiJJJ2xhbWJkYUdGJEYoKSwoRihGKComRilGKClJInhHRiQiIiNGKEYoKiZGKUYoKUkieUdGJEYvRihGKEYvISIiLUkjJi5HRiQ2JCZJImVHRiQ2I0YoRjdGKEYoKipGJ0YoRilGKEYqRjMtRjU2JCZGODYjRi9GPUYoRig= Ricci scalar: RicciScalar(s); L0kic0c2IiwkKiYiIiMiIiJJJ2xhbWJkYUdGJEYoRig= Show 1-form 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: 'omega[2,1]'=factor(omega[2,1]); LyZJJm9tZWdhRzYiNiQiIiMiIiIsJCoqRidGKEknbGFtYmRhR0YlRigsJiomSSJ5R0YlRigmSSJlR0YlNiNGKEYoISIiKiZJInhHRiVGKCZGMDYjRidGKEYoRigsKEYoRigqJkYrRigpRjRGJ0YoRigqJkYrRigpRi5GJ0YoRihGMkYy Verify that there is no Killing vector field among frame vector fields: 'L[E[j]](g)'=L[E[j]](g); Ly0mSSJMRzYiNiMmSSJFR0YmNiNJImpHRiY2I0kiZ0dGJiwkKiwiIikiIiIsJi1JIyYuR0YmNiQmSSJlR0YmNiNGMUY2RjEtRjQ2JCZGNzYjIiIjRjtGMUYxSSdsYW1iZGFHRiZGMSksKEYxRjEqJkY+RjEpSSJ4R0YmRj1GMUYxKiZGPkYxKUkieUdGJkY9RjFGMSIiJCEiIiwmKihGPUYxRkNGMSZJJmRlbHRhR0YmNiRGMUYrRjFGMSooRj1GMUZGRjEmRkw2JEY9RitGMUYxRjFGSA== Verify that "rotation" vector field LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KS1GLDYlUSJaRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1GIzYqRistRiM2KC1GLDYlUSJ4RidGNEY3LUY7Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNGaW4tSShtZmVuY2VkR0YkNiQtRiM2KC1JJm1mcmFjR0YkNigtRiM2Ji1GOzYvUSsmUGFydGlhbEQ7RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnRj4vSSttc2VtYW50aWNzR0YkUSZpbmVydEYnRkBGQ0ZFRkdGSUZLRk1GaG5Gam4vJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRj4tRiM2KEYrLUYjNidGZW8tRiw2JVEieUYnRjRGN0ZecEZhcEY+RitGXnBGYXBGPi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYHEvJSliZXZlbGxlZEdGQkZlbkYrRl5wRmFwRj5GPkZecEZhcEY+LUY7Ni1RKCZtaW51cztGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGaXEtRiM2KEZocEZlbi1GXG82JC1GIzYoLUZhbzYoRmNvLUYjNihGKy1GIzYnRmVvRlhGXnBGYXBGPkYrRl5wRmFwRj5GW3FGXnFGYXFGY3FGZW5GK0ZecEZhcEY+Rj5GXnBGYXBGPkYrRl5wRmFwRj5GK0ZecEZhcEY+RitGXnBGYXBGPg== is a Killing one: Z:=x*E[2]-y*E[1]; 'L[Z]'(g)=eval(L[Z](g)); LCYqJkkieEc2IiIiIiZJIkVHRiU2IyIiI0YmRiYqJkkieUdGJUYmJkYoNiNGJkYmISIi Ly0mSSJMRzYiNiNJIlpHRiY2I0kiZ0dGJiIiIQ== Calculation of volume form using Hodge operator &** : '&**(1)'=radsimp(&**(1)); Ly1JJCYqKkc2IjYjIiIiLCQqKCIiJUYnKS1JJGFic0clKnByb3RlY3RlZEc2IywoRidGJyomSSdsYW1iZGFHRiVGJylJInhHRiUiIiNGJ0YnKiZGMkYnKUkieUdGJUY1RidGJ0Y1ISIiLUkjJl5HRiU2JCZJImVHRiVGJiZGPjYjRjVGJ0Yn Some more calculations: Using interior product operator - LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJmlvdGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJXNpemVHUSMxMkYnRjI=: 'iota[E[j]](&**(1))'=iota[E[j]](radsimp(&**(1))); Ly0mSSVpb3RhRzYiNiMmSSJFR0YmNiNJImpHRiY2Iy1JJCYqKkdGJjYjIiIiLCQqKCIiJUYwKS1JJGFic0clKnByb3RlY3RlZEc2IywoRjBGMComSSdsYW1iZGFHRiZGMClJInhHRiYiIiNGMEYwKiZGO0YwKUkieUdGJkY+RjBGMEY+ISIiLCYqJiZJJmRlbHRhR0YmNiRGMEYrRjAmSSJlR0YmNiNGPkYwRjAqJiZGRjYkRj5GK0YwJkZJRi9GMEZCRjBGMA== The same with metric tensor g: 'iota[E[k]](g)'=iota[E[k]](g); Ly0mSSVpb3RhRzYiNiMmSSJFR0YmNiNJImtHRiY2I0kiZ0dGJiwkKigiIiUiIiIpLChGMUYxKiZJJ2xhbWJkYUdGJkYxKUkieEdGJiIiI0YxRjEqJkY1RjEpSSJ5R0YmRjhGMUYxRjghIiIsJiomJkkmZGVsdGFHRiY2JEYxRitGMSZJImVHRiY2I0YxRjFGMSomJkZANiRGOEYrRjEmRkM2I0Y4RjFGMUYxRjE= Using exterior derivative operator - LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvRjNRJ25vcm1hbEYn: 'd(f)'=d(f); Ly1JImRHNiI2I0kiZkdGJSwmKiYtSSVEaWZmRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiRGJ0kieEdGJSIiIiZJImVHRiU2I0YxRjFGMSomLUYrNiRGJ0kieUdGJUYxJkYzNiMiIiNGMUYx 'd(h)'=d(h); Ly1JImRHNiI2I0kiaEdGJSomLUklRGlmZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkRidJImZHRiUiIiIsJiomLUYqNiRGL0kieEdGJUYwJkkiZUdGJTYjRjBGMEYwKiYtRio2JEYvSSJ5R0YlRjAmRjc2IyIiI0YwRjBGMA== 'd(f*xi)'=d(f*xi); Ly1JImRHNiI2IyomSSJmR0YlIiIiSSN4aUdGJUYpLCgqJi1JJURpZmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JEYoSSJ4R0YlRiktSSMmXkdGJTYkRiomSSJlR0YlNiNGKUYpISIiKiYtRi42JEYoSSJ5R0YlRiktRjU2JEYqJkY4NiMiIiNGKUY6KiZGKEYpLUYkNiNGKkYpRik= '&**(d(f))'=radsimp(&**(d(f))); Ly1JJCYqKkc2IjYjLUkiZEdGJTYjSSJmR0YlKigpLCgiIiJGLiomSSdsYW1iZGFHRiVGLilJInhHRiUiIiNGLkYuKiZGMEYuKUkieUdGJUYzRi5GLkYzRi4sJiomLUklRGlmZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkRipGMkYuJkkiZUdGJTYjRjNGLkYuKiYtRjo2JEYqRjZGLiZGQDYjRi5GLiEiIkYuKS1JJGFic0dGPDYjRi1GM0ZH As LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImtGJ0YyRjUvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQEZD are coframe 1-forms and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRImlGJ0YyRjUvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj1GQEZD are frame vectors then: 'iota[E[i]](e[k])'=iota[E[i]](e[k]); Ly0mSSVpb3RhRzYiNiMmSSJFR0YmNiNJImlHRiY2IyZJImVHRiY2I0kia0dGJiZJJmRlbHRhR0YmNiRGK0Yw "ToBasis" decomposition: X=ToBasis(X); xi=ToBasis(xi); L0kiWEc2IiwmKiYtJkklaW90YUdGJDYjRiM2IyZJImVHRiQ2IyIiIkYvJkkiRUdGJEYuRi9GLyomLUYoNiMmRi02IyIiI0YvJkYxRjZGL0Yv L0kjeGlHNiIsJiomLSZJJWlvdGFHRiQ2IyZJIkVHRiQ2IyIiIjYjRiNGLiZJImVHRiRGLUYuRi4qJi0mRik2IyZGLDYjIiIjRi9GLiZGMUY3Ri5GLg==
<Text-field bookmark="seealso" style="Heading 2" layout="Heading 2">See Also: </Text-field> atlas[types], atlas[simp], atlas[dim] , atlas[examples], atlas[references].