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def _SR_to_Sympy(expression):
"\n Convert an expression from ``SR`` to ``sympy``.\n\n In the case where the expression is already ``sympy``, it is simply\n returned.\n\n INPUT:\n\n - ``expression`` -- ``SR`` or ``sympy`` symbolic expression\n\n OUTPUT:\n\n - ``expression`` -- ``sympy`` symbol... |
def _Sympy_to_SR(expression):
"\n Convert an expression from ``sympy`` to ``SR``.\n\n In the case where the expression is already ``SR``, it is simply returned.\n\n INPUT:\n\n - ``expression`` -- ``sympy`` symbolic expression\n\n OUTPUT:\n\n - ``expression`` -- ``SR`` or ``sympy`` symbolic expre... |
class CalculusMethod(SageObject):
"\n Control of calculus backends used on coordinate charts of manifolds.\n\n This class stores the possible calculus methods and permits to switch\n between them, as well as to change the simplifying functions associated\n with them.\n For the moment, only two calc... |
def Minkowski(positive_spacelike=True, names=None):
'\n Generate a Minkowski space of dimension 4.\n\n By default the signature is set to `(- + + +)`, but can be changed to\n `(+ - - -)` by setting the optional argument ``positive_spacelike`` to\n ``False``. The shortcut operator ``.<,>`` can be used ... |
def Kerr(m=1, a=0, coordinates='BL', names=None):
'\n Generate a Kerr spacetime.\n\n A Kerr spacetime is a 4 dimensional manifold describing a rotating black\n hole. Two coordinate systems are implemented: Boyer-Lindquist and Kerr\n (3+1 version).\n\n The shortcut operator ``.<,>`` can be used to s... |
def Torus(R=2, r=1, names=None):
'\n Generate a 2-dimensional torus embedded in Euclidean space.\n\n The shortcut operator ``.<,>`` can be used to specify the coordinates.\n\n INPUT:\n\n - ``R`` -- (default: ``2``) distance form the center to the\n center of the tube\n - ``r`` -- (default: ``1... |
def RealProjectiveSpace(dim=2):
'\n Generate projective space of dimension ``dim`` over the reals.\n\n This is the topological space of lines through the origin in\n `\\RR^{d+1}`. The standard atlas consists of `d+2` charts, which sends\n the set `U_i = \\{[x_1, x_2, \\ldots, x_{d+1}] : x_i \\neq 0 \\... |
class Chart(UniqueRepresentation, SageObject):
'\n Chart on a topological manifold.\n\n Given a topological manifold `M` of dimension `n` over a topological\n field `K`, a *chart* on `M` is a pair `(U, \\varphi)`, where `U` is an\n open subset of `M` and `\\varphi : U \\rightarrow V \\subset K^n` is a... |
class RealChart(Chart):
'\n Chart on a topological manifold over `\\RR`.\n\n Given a topological manifold `M` of dimension `n` over `\\RR`, a *chart*\n on `M` is a pair `(U,\\varphi)`, where `U` is an open subset of `M` and\n `\\varphi : U \\to V \\subset \\RR^n` is a homeomorphism from `U` to\n an... |
class CoordChange(SageObject):
"\n Transition map between two charts of a topological manifold.\n\n Giving two coordinate charts `(U, \\varphi)` and `(V, \\psi)` on a\n topological manifold `M` of dimension `n` over a topological field `K`,\n the *transition map from* `(U, \\varphi)` *to* `(V, \\psi)`... |
class ChartFunction(AlgebraElement, ModuleElementWithMutability):
'\n Function of coordinates of a given chart.\n\n If `(U, \\varphi)` is a chart on a topological manifold `M` of\n dimension `n` over a topological field `K`, a *chart function*\n associated to `(U, \\varphi)` is a map\n\n .. MATH::... |
class ChartFunctionRing(Parent, UniqueRepresentation):
"\n Ring of all chart functions on a chart.\n\n INPUT:\n\n - ``chart`` -- a coordinate chart, as an instance of class\n :class:`~sage.manifolds.chart.Chart`\n\n EXAMPLES:\n\n The ring of all chart functions w.r.t. to a chart::\n\n s... |
class MultiCoordFunction(SageObject, Mutability):
"\n Coordinate function to some Cartesian power of the base field.\n\n If `n` and `m` are two positive integers and `(U, \\varphi)` is a chart on\n a topological manifold `M` of dimension `n` over a topological field `K`,\n a *multi-coordinate function... |
class ContinuousMap(Morphism):
"\n Continuous map between two topological manifolds.\n\n This class implements continuous maps of the type\n\n .. MATH::\n\n \\Phi: M \\longrightarrow N,\n\n where `M` and `N` are topological manifolds over the same\n topological field `K`.\n\n Continuous m... |
class ImageManifoldSubset(ManifoldSubset):
'\n Subset of a topological manifold that is a continuous image of a manifold subset.\n\n INPUT:\n\n - ``map`` -- continuous map `\\Phi`\n - ``inverse`` -- (default: ``None``) continuous map from\n ``map.codomain()`` to ``map.domain()``, which once restr... |
class AffineConnection(SageObject):
'\n Affine connection on a smooth manifold.\n\n Let `M` be a differentiable manifold of class `C^\\infty` (smooth manifold)\n over a non-discrete topological field `K` (in most applications `K=\\RR`\n or `K=\\CC`), let `C^\\infty(M)` be the algebra of smooth functio... |
class AutomorphismField(TensorField):
"\n Field of automorphisms of tangent spaces to a generic (a priori\n not parallelizable) differentiable manifold.\n\n Given a differentiable manifold `U` and a differentiable map\n `\\Phi: U \\rightarrow M` to a differentiable manifold `M`,\n a *field of tange... |
class AutomorphismFieldParal(FreeModuleAutomorphism, TensorFieldParal):
"\n Field of tangent-space automorphisms with values on a parallelizable\n manifold.\n\n Given a differentiable manifold `U` and a differentiable map\n `\\Phi: U \\rightarrow M` to a parallelizable manifold `M`,\n a *field of t... |
class AutomorphismFieldGroup(UniqueRepresentation, Parent):
"\n General linear group of the module of vector fields along a differentiable\n manifold `U` with values on a differentiable manifold `M`.\n\n Given a differentiable manifold `U` and a differentiable map\n `\\Phi: U \\rightarrow M` to a diff... |
class AutomorphismFieldParalGroup(FreeModuleLinearGroup):
"\n General linear group of the module of vector fields along a differentiable\n manifold `U` with values on a parallelizable manifold `M`.\n\n Given a differentiable manifold `U` and a differentiable map\n `\\Phi: U \\rightarrow M` to a parall... |
class BundleConnection(SageObject, Mutability):
"\n An instance of this class represents a bundle connection `\\nabla` on a\n smooth vector bundle `E \\to M`.\n\n INPUT:\n\n - ``vbundle`` -- the vector bundle on which the connection is defined\n (must be an instance of class\n :class:`~sage.... |
class CharacteristicCohomologyClassRingElement(IndexedFreeModuleElement):
"\n Characteristic cohomology class.\n\n Let `E \\to M` be a real/complex vector bundle of rank `k`. A characteristic\n cohomology class of `E` is generated by either\n\n - Chern classes if `E` is complex,\n - Pontryagin clas... |
class CharacteristicCohomologyClassRing(FiniteGCAlgebra):
"\n Characteristic cohomology class ring.\n\n Let `E \\to M` be a real or complex vector bundle of rank `k` and `R` be a\n torsion-free subring of `\\CC`.\n\n Let `BG` be the classifying space of the group `G`. As for vector bundles,\n we co... |
def multiplicative_sequence(q, n=None):
'\n Turn the polynomial ``q`` into its multiplicative sequence.\n\n Let `q` be a polynomial and `x_1, \\ldots x_n` indeterminates. The\n *multiplicative sequence of* `q` is then given by the polynomials `K_j`\n\n .. MATH::\n\n \\sum_{j=0}^n K_j(\\sigma_1,... |
def additive_sequence(q, k, n=None):
'\n Turn the polynomial ``q`` into its additive sequence.\n\n Let `q` be a polynomial and `x_1, \\ldots x_n` indeterminates. The\n *additive sequence of* `q` is then given by the polynomials `Q_j`\n\n .. MATH::\n\n \\sum_{j=0}^n Q_j(\\sigma_1, \\ldots, \\sig... |
def fast_wedge_power(form, n):
"\n Return the wedge product power of `form` using a square-and-wedge\n algorithm.\n\n INPUT:\n\n - ``form`` -- a differential form\n - ``n`` -- a non-negative integer\n\n EXAMPLES::\n\n sage: M = Manifold(4, 'M')\n sage: X.<x,y,z,t> = M.chart()\n ... |
class Algorithm_generic(SageObject):
'\n Abstract algorithm class to compute the characteristic forms of the\n generators.\n\n EXAMPLES::\n\n sage: from sage.manifolds.differentiable.characteristic_cohomology_class import Algorithm_generic\n sage: algorithm = Algorithm_generic()\n sa... |
class ChernAlgorithm(Singleton, Algorithm_generic):
"\n Algorithm class to generate Chern forms.\n\n EXAMPLES:\n\n Define a complex line bundle over a 2-dimensional manifold::\n\n sage: M = Manifold(2, 'M', structure='Lorentzian')\n sage: X.<t,x> = M.chart()\n sage: E = M.vector_bund... |
class PontryaginAlgorithm(Singleton, Algorithm_generic):
'\n Algorithm class to generate Pontryagin forms.\n\n EXAMPLES:\n\n 5-dimensional Euclidean space::\n\n sage: M = manifolds.EuclideanSpace(5)\n sage: g = M.metric()\n sage: nab = g.connection()\n sage: nab.set_immutable(... |
class EulerAlgorithm(Singleton, Algorithm_generic):
'\n Algorithm class to generate Euler forms.\n\n EXAMPLES:\n\n Consider the 2-dimensional Euclidean space::\n\n sage: M = manifolds.EuclideanSpace(2)\n sage: g = M.metric()\n sage: nab = g.connection()\n sage: nab.set_immutab... |
class PontryaginEulerAlgorithm(Singleton, Algorithm_generic):
'\n Algorithm class to generate Euler and Pontryagin forms.\n\n EXAMPLES:\n\n 6-dimensional Euclidean space::\n\n sage: M = manifolds.EuclideanSpace(6)\n sage: g = M.metric()\n sage: nab = g.connection()\n sage: nab... |
class DeRhamCohomologyClass(AlgebraElement):
"\n Define a cohomology class in the de Rham cohomology ring.\n\n INPUT:\n\n - ``parent`` -- de Rham cohomology ring represented by an instance of\n :class:`DeRhamCohomologyRing`\n - ``representative`` -- a closed (mixed) differential form representing... |
class DeRhamCohomologyRing(Parent, UniqueRepresentation):
"\n The de Rham cohomology ring of a de Rham complex.\n\n This ring is naturally endowed with a multiplication induced by the wedge\n product, called *cup product*, see :meth:`DeRhamCohomologyClass.cup`.\n\n .. NOTE::\n\n The current imp... |
class DiffForm(TensorField):
"\n Differential form with values on a generic (i.e. a priori not\n parallelizable) differentiable manifold.\n\n Given a differentiable manifold `U`, a differentiable map\n `\\Phi: U \\rightarrow M` to a differentiable manifold `M` and a positive\n integer `p`, a *diffe... |
class DiffFormParal(FreeModuleAltForm, TensorFieldParal, DiffForm):
"\n Differential form with values on a parallelizable manifold.\n\n Given a differentiable manifold `U`, a differentiable map\n `\\Phi: U \\rightarrow M` to a parallelizable manifold `M` and a positive\n integer `p`, a *differential ... |
class DiffFormModule(UniqueRepresentation, Parent):
"\n Module of differential forms of a given degree `p` (`p`-forms) along a\n differentiable manifold `U` with values on a differentiable manifold `M`.\n\n Given a differentiable manifold `U` and a differentiable map\n `\\Phi: U \\rightarrow M` to a d... |
class DiffFormFreeModule(ExtPowerDualFreeModule):
"\n Free module of differential forms of a given degree `p` (`p`-forms) along\n a differentiable manifold `U` with values on a parallelizable manifold `M`.\n\n Given a differentiable manifold `U` and a differentiable map\n `\\Phi:\\; U \\rightarrow M` ... |
class VectorFieldDualFreeModule(DiffFormFreeModule):
"\n Free module of differential 1-forms along a differentiable manifold `U`\n with values on a parallelizable manifold `M`.\n\n Given a differentiable manifold `U` and a differentiable map\n `\\Phi:\\; U \\rightarrow M` to a parallelizable manifold ... |
class DiffMap(ContinuousMap):
"\n Differentiable map between two differentiable manifolds.\n\n This class implements differentiable maps of the type\n\n .. MATH::\n\n \\Phi: M \\longrightarrow N\n\n where `M` and `N` are differentiable manifolds over the same topological\n field `K` (in mos... |
class DifferentiableSubmanifold(DifferentiableManifold, TopologicalSubmanifold):
'\n Submanifold of a differentiable manifold.\n\n Given two differentiable manifolds `N` and `M`, an *immersion* `\\phi` is a\n differentiable map `N\\to M` whose differential is everywhere\n injective. One then says that... |
class EuclideanSpace(PseudoRiemannianManifold):
'\n Euclidean space.\n\n An *Euclidean space of dimension* `n` is an affine space `E`, whose\n associated vector space is a `n`-dimensional vector space over `\\RR` and\n is equipped with a positive definite symmetric bilinear form, called\n the *scal... |
class EuclideanPlane(EuclideanSpace):
'\n Euclidean plane.\n\n An *Euclidean plane* is an affine space `E`, whose associated vector space\n is a 2-dimensional vector space over `\\RR` and is equipped with a\n positive definite symmetric bilinear form, called the *scalar product* or\n *dot product*.... |
class Euclidean3dimSpace(EuclideanSpace):
'\n 3-dimensional Euclidean space.\n\n A *3-dimensional Euclidean space* is an affine space `E`, whose associated\n vector space is a 3-dimensional vector space over `\\RR` and is equipped\n with a positive definite symmetric bilinear form, called the *scalar\... |
class OpenInterval(DifferentiableManifold):
'\n Open interval as a 1-dimensional differentiable manifold over `\\RR`.\n\n INPUT:\n\n - ``lower`` -- lower bound of the interval (possibly ``-Infinity``)\n - ``upper`` -- upper bound of the interval (possibly ``+Infinity``)\n - ``ambient_interval`` -- ... |
class RealLine(OpenInterval):
'\n Field of real numbers, as a differentiable manifold of dimension 1 (real\n line) with a canonical coordinate chart.\n\n INPUT:\n\n - ``name`` -- (default: ``\'R\'``) string; name (symbol) given to\n the real line\n - ``latex_name`` -- (default: ``r\'\\Bold{R}\... |
class Sphere(PseudoRiemannianSubmanifold):
"\n Sphere smoothly embedded in Euclidean Space.\n\n An `n`-sphere of radius `r`smoothly embedded in a Euclidean space `E^{n+1}`\n is a smooth `n`-dimensional manifold smoothly embedded into `E^{n+1}`,\n such that the embedding constitutes a standard `n`-sphe... |
class StandardSymplecticSpace(EuclideanSpace):
'\n The vector space `\\RR^{2n}` equipped with its standard symplectic form.\n '
_symplectic_form: SymplecticForm
def __init__(self, dimension: int, name: Optional[str]=None, latex_name: Optional[str]=None, coordinates: str='Cartesian', symbols: Option... |
class TestR2VectorSpace():
@pytest.fixture
def M(self):
return StandardSymplecticSpace(2, 'R2', symplectic_name='omega')
@pytest.fixture
def omega(self, M: StandardSymplecticSpace):
return M.symplectic_form()
def test_repr(self, M: StandardSymplecticSpace):
assert (str(M... |
class TestR4VectorSpace():
@pytest.fixture
def M(self):
return StandardSymplecticSpace(4, 'R4', symplectic_name='omega')
@pytest.fixture
def omega(self, M: StandardSymplecticSpace):
return M.symplectic_form()
def test_repr(self, M: StandardSymplecticSpace):
assert (str(M... |
class LeviCivitaConnection(AffineConnection):
'\n Levi-Civita connection on a pseudo-Riemannian manifold.\n\n Let `M` be a differentiable manifold of class `C^\\infty` (smooth manifold)\n over `\\RR` endowed with a pseudo-Riemannian metric `g`.\n Let `C^\\infty(M)` be the algebra of smooth functions\n... |
class DifferentiableManifold(TopologicalManifold):
'\n Differentiable manifold over a topological field `K`.\n\n Given a non-discrete topological field `K` (in most applications,\n `K = \\RR` or `K = \\CC`; see however [Ser1992]_ for `K = \\QQ_p` and\n [Ber2008]_ for other fields), a *differentiable m... |
class DifferentiableManifoldHomset(TopologicalManifoldHomset):
"\n Set of differentiable maps between two differentiable manifolds.\n\n Given two differentiable manifolds `M` and `N` over a topological field `K`,\n the class :class:`DifferentiableManifoldHomset` implements the set\n `\\mathrm{Hom}(M,N... |
class DifferentiableCurveSet(DifferentiableManifoldHomset):
'\n Set of differentiable curves in a differentiable manifold.\n\n Given an open interval `I` of `\\RR` (possibly `I = \\RR`) and\n a differentiable manifold `M` over `\\RR`, this is the set\n `\\mathrm{Hom}(I,M)` of morphisms (i.e. different... |
class IntegratedCurveSet(DifferentiableCurveSet):
'\n Set of integrated curves in a differentiable manifold.\n\n INPUT:\n\n - ``domain`` --\n :class:`~sage.manifolds.differentiable.examples.real_line.OpenInterval`\n open interval `I \\subset \\RR` with finite boundaries (domain of\n the mo... |
class IntegratedAutoparallelCurveSet(IntegratedCurveSet):
'\n Set of integrated autoparallel curves in a differentiable manifold.\n\n INPUT:\n\n - ``domain`` --\n :class:`~sage.manifolds.differentiable.examples.real_line.OpenInterval`\n open interval `I \\subset \\RR` with finite boundaries (do... |
class IntegratedGeodesicSet(IntegratedAutoparallelCurveSet):
'\n Set of integrated geodesic in a differentiable manifold.\n\n INPUT:\n\n - ``domain`` --\n :class:`~sage.manifolds.differentiable.examples.real_line.OpenInterval`\n open interval `I \\subset \\RR` with finite boundaries (domain of\... |
class PseudoRiemannianMetric(TensorField):
"\n Pseudo-Riemannian metric with values on an open subset of a\n differentiable manifold.\n\n An instance of this class is a field of nondegenerate symmetric bilinear\n forms (metric field) along a differentiable manifold `U` with\n values on a differenti... |
class PseudoRiemannianMetricParal(PseudoRiemannianMetric, TensorFieldParal):
"\n Pseudo-Riemannian metric with values on a parallelizable manifold.\n\n An instance of this class is a field of nondegenerate symmetric bilinear\n forms (metric field) along a differentiable manifold `U` with values in a\n ... |
class DegenerateMetric(TensorField):
"\n Degenerate (or null or lightlike) metric with values on an open subset of a\n differentiable manifold.\n\n An instance of this class is a field of degenerate symmetric bilinear\n forms (metric field) along a differentiable manifold `U` with\n values on a dif... |
class DegenerateMetricParal(DegenerateMetric, TensorFieldParal):
"\n Degenerate (or null or lightlike) metric with values on an open subset of a\n differentiable manifold.\n\n An instance of this class is a field of degenerate symmetric bilinear\n forms (metric field) along a differentiable manifold `... |
class MixedForm(AlgebraElement, ModuleElementWithMutability):
"\n An instance of this class is a mixed form along some differentiable map\n `\\varphi: M \\to N` between two differentiable manifolds `M` and `N`. More\n precisely, a mixed form `a` along `\\varphi: M \\to N` can be considered as a\n diff... |
class MixedFormAlgebra(Parent, UniqueRepresentation):
"\n An instance of this class represents the graded algebra of mixed forms.\n That is, if `\\varphi: M \\to N` is a differentiable map between two\n differentiable manifolds `M` and `N`, the *graded algebra of mixed forms*\n `\\Omega^*(M,\\varphi)`... |
class MultivectorModule(UniqueRepresentation, Parent):
"\n Module of multivector fields of a given degree `p` (`p`-vector\n fields) along a differentiable manifold `U` with values on a\n differentiable manifold `M`.\n\n Given a differentiable manifold `U` and a differentiable map\n `\\Phi: U \\righ... |
class MultivectorFreeModule(ExtPowerFreeModule):
"\n Free module of multivector fields of a given degree `p` (`p`-vector\n fields) along a differentiable manifold `U` with values on a\n parallelizable manifold `M`.\n\n Given a differentiable manifold `U` and a differentiable map\n `\\Phi:\\; U \\ri... |
class MultivectorField(TensorField):
"\n Multivector field with values on a generic (i.e. a priori not\n parallelizable) differentiable manifold.\n\n Given a differentiable manifold `U`, a differentiable map\n `\\Phi: U \\rightarrow M` to a differentiable manifold `M` and a positive\n integer `p`, ... |
class MultivectorFieldParal(AlternatingContrTensor, TensorFieldParal):
"\n Multivector field with values on a parallelizable manifold.\n\n Given a differentiable manifold `U`, a differentiable map\n `\\Phi: U \\rightarrow M` to a parallelizable manifold `M` and a positive\n integer `p`, a *multivecto... |
class PoissonTensorField(MultivectorField):
"\n A Poisson bivector field `\\varpi` on a differentiable manifold.\n\n That is, at each point `m \\in M`, `\\varpi_m` is a bilinear map of the type:\n\n .. MATH::\n\n \\varpi_m:\\ T^*_m M \\times T^*_m M \\to \\RR\n\n where `T^*_m M` stands for the... |
class PoissonTensorFieldParal(PoissonTensorField, MultivectorFieldParal):
"\n A Poisson bivector field `\\varpi` on a parallelizable manifold.\n\n INPUT:\n\n - ``manifold`` -- module `\\mathfrak{X}(M)` of vector fields on the\n manifold `M`, or the manifold `M` itself\n - ``name`` -- (default: ``... |
class PseudoRiemannianManifold(DifferentiableManifold):
'\n PseudoRiemannian manifold.\n\n A *pseudo-Riemannian manifold* is a pair `(M,g)` where `M` is a real\n differentiable manifold `M` (see\n :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`)\n and `g` is a field of non-d... |
class DiffScalarField(ScalarField):
"\n Differentiable scalar field on a differentiable manifold.\n\n Given a differentiable manifold `M` of class `C^k` over a topological field\n `K` (in most applications, `K = \\RR` or `K = \\CC`), a *differentiable\n scalar field* defined on `M` is a map\n\n .. ... |
class DiffScalarFieldAlgebra(ScalarFieldAlgebra):
"\n Commutative algebra of differentiable scalar fields on a differentiable\n manifold.\n\n If `M` is a differentiable manifold of class `C^k` over a topological\n field `K`, the *commutative algebra of scalar fields on* `M` is the set\n `C^k(M)` of... |
class SymplecticForm(DiffForm):
"\n A symplectic form on a differentiable manifold.\n\n An instance of this class is a closed nondegenerate differential `2`-form `\\omega`\n on a differentiable manifold `M` over `\\RR`.\n\n In particular, at each point `m \\in M`, `\\omega_m` is a bilinear map of the ... |
class SymplecticFormParal(SymplecticForm, DiffFormParal):
'\n A symplectic form on a parallelizable manifold.\n\n .. NOTE::\n\n If `M` is not parallelizable, the class :class:`SymplecticForm`\n should be used instead.\n\n INPUT:\n\n - ``manifold`` -- module `\\mathfrak{X}(M)` of vector f... |
class TestGenericSymplecticForm():
@pytest.fixture
def omega(self):
M = Manifold(6, 'M')
return SymplecticForm(M, 'omega')
def test_repr(self, omega: SymplecticForm):
assert (str(omega) == 'Symplectic form omega on the 6-dimensional differentiable manifold M')
def test_new_i... |
class TestCoherenceOfFormulas():
'\n Test correctness of the implementation, by checking that equivalent formulas give the correct result.\n We check it for the examples of `\\R^2` and `S^2`, which should be enough.\n '
@pytest.fixture(params=['R2', 'S2'])
def M(self, request: FixtureRequest):
... |
def generic_scalar_field(M: DifferentiableManifold, name: str) -> DiffScalarField:
chart_functions = {chart: function(name)(*chart[:]) for chart in M.atlas()}
return M.scalar_field(chart_functions, name=name)
|
class TestR2VectorSpace():
@pytest.fixture
def M(self):
return StandardSymplecticSpace(2, 'R2', symplectic_name='omega')
@pytest.fixture
def omega(self, M):
return M.symplectic_form()
def test_display(self, omega: SymplecticForm):
assert (str(omega.display()) == 'omega =... |
class TangentSpace(FiniteRankFreeModule):
"\n Tangent space to a differentiable manifold at a given point.\n\n Let `M` be a differentiable manifold of dimension `n` over a\n topological field `K` and `p \\in M`. The tangent space `T_p M` is an\n `n`-dimensional vector space over `K` (without a disting... |
class TangentVector(FiniteRankFreeModuleElement):
"\n Tangent vector to a differentiable manifold at a given point.\n\n INPUT:\n\n - ``parent`` --\n :class:`~sage.manifolds.differentiable.tangent_space.TangentSpace`;\n the tangent space to which the vector belongs\n - ``name`` -- (default: `... |
class TensorFieldModule(UniqueRepresentation, ReflexiveModule_tensor):
"\n Module of tensor fields of a given type `(k,l)` along a differentiable\n manifold `U` with values on a differentiable manifold `M`, via a\n differentiable map `U \\rightarrow M`.\n\n Given two non-negative integers `k` and `l` ... |
class TensorFieldFreeModule(TensorFreeModule):
"\n Free module of tensor fields of a given type `(k,l)` along a\n differentiable manifold `U` with values on a parallelizable manifold `M`,\n via a differentiable map `U \\rightarrow M`.\n\n Given two non-negative integers `k` and `l` and a differentiabl... |
class TensorFieldParal(FreeModuleTensor, TensorField):
"\n Tensor field along a differentiable manifold, with values on a\n parallelizable manifold.\n\n An instance of this class is a tensor field along a differentiable\n manifold `U` with values on a parallelizable manifold `M`, via a\n differenti... |
class TestR3VectorSpace():
@pytest.fixture
def manifold(self):
return EuclideanSpace(3)
def test_trace_using_metric_works(self, manifold: DifferentiableManifold):
metric = manifold.metric('g')
assert (metric.trace(using=metric) == manifold.scalar_field(3))
|
class DifferentiableVectorBundle(TopologicalVectorBundle):
"\n An instance of this class represents a differentiable vector bundle\n `E \\to M`\n\n INPUT:\n\n - ``rank`` -- positive integer; rank of the vector bundle\n - ``name`` -- string representation given to the total space\n - ``base_space... |
class TensorBundle(DifferentiableVectorBundle):
"\n Tensor bundle over a differentiable manifold along a differentiable map.\n\n An instance of this class represents the pullback tensor bundle\n `\\Phi^* T^{(k,l)}N` along a differentiable map (called *destination map*)\n\n .. MATH::\n\n \\Phi: ... |
class VectorField(MultivectorField):
"\n Vector field along a differentiable manifold.\n\n An instance of this class is a vector field along a differentiable\n manifold `U` with values on a differentiable manifold `M`, via a\n differentiable map `U \\rightarrow M`. More precisely, given a\n differe... |
class VectorFieldParal(FiniteRankFreeModuleElement, MultivectorFieldParal, VectorField):
"\n Vector field along a differentiable manifold, with values on a\n parallelizable manifold.\n\n An instance of this class is a vector field along a differentiable\n manifold `U` with values on a parallelizable m... |
class VectorFieldModule(UniqueRepresentation, ReflexiveModule_base):
"\n Module of vector fields along a differentiable manifold `U`\n with values on a differentiable manifold `M`, via a differentiable\n map `U \\rightarrow M`.\n\n Given a differentiable map\n\n .. MATH::\n\n \\Phi:\\ U \\l... |
class VectorFieldFreeModule(FiniteRankFreeModule):
"\n Free module of vector fields along a differentiable manifold `U` with\n values on a parallelizable manifold `M`, via a differentiable map\n `U \\rightarrow M`.\n\n Given a differentiable map\n\n .. MATH::\n\n \\Phi:\\ U \\longrightarrow ... |
class CoFrame(FreeModuleCoBasis):
"\n Coframe on a differentiable manifold.\n\n By *coframe*, it is meant a field `f` on some differentiable manifold `U`\n endowed with a differentiable map `\\Phi: U \\rightarrow M` to a\n differentiable manifold `M` such that for each `p\\in U`, `f(p)` is a basis\n ... |
class VectorFrame(FreeModuleBasis):
'\n Vector frame on a differentiable manifold.\n\n By *vector frame*, it is meant a field `e` on some\n differentiable manifold `U` endowed with a differentiable map\n `\\Phi: U\\rightarrow M` to a differentiable manifold `M` such that for\n each `p\\in U`, `e(p)... |
class CoordCoFrame(CoFrame):
"\n Coordinate coframe on a differentiable manifold.\n\n By *coordinate coframe*, it is meant the `n`-tuple of the\n differentials of the coordinates of some chart on the manifold,\n with `n` being the manifold's dimension.\n\n INPUT:\n\n - ``coord_frame`` -- coordin... |
class CoordFrame(VectorFrame):
"\n Coordinate frame on a differentiable manifold.\n\n By *coordinate frame*, it is meant a vector frame on a differentiable\n manifold `M` that is associated to a coordinate chart on `M`.\n\n INPUT:\n\n - ``chart`` -- the chart defining the coordinates\n\n EXAMPLE... |
@total_ordering
class ManifoldObjectFiniteFamily(FiniteFamily):
"\n Finite family of manifold objects, indexed by their names.\n\n The class :class:`ManifoldObjectFiniteFamily` inherits from\n :class:`FiniteFamily`. Therefore it is an associative container.\n\n It provides specialized ``__repr__`` an... |
class ManifoldSubsetFiniteFamily(ManifoldObjectFiniteFamily):
"\n Finite family of subsets of a topological manifold, indexed by their names.\n\n The class :class:`ManifoldSubsetFiniteFamily` inherits from\n :class:`ManifoldObjectFiniteFamily`. It provides an associative\n container with specialized ... |
class LocalCoFrame(FreeModuleCoBasis):
"\n Local coframe on a vector bundle.\n\n A *local coframe* on a vector bundle `E \\to M` of class `C^k` is a\n local section `e^*: U \\to E^n` of class `C^k` on some subset `U` of the base\n space `M`, such that `e^*(p)` is a basis of the fiber `E^*_p` of the du... |
class LocalFrame(FreeModuleBasis):
"\n Local frame on a vector bundle.\n\n A *local frame* on a vector bundle `E \\to M` of class `C^k` is a local\n section `(e_1,\\dots,e_n):U \\to E^n` of class `C^k` defined on some subset `U`\n of the base space `M`, such that `e(p)` is a basis of the fiber `E_p` f... |
class TrivializationCoFrame(LocalCoFrame):
"\n Trivialization coframe on a vector bundle.\n\n A *trivialization coframe* is the coframe of the trivialization frame\n induced by a trivialization (see: :class:`~sage.manifolds.local_frame.TrivializationFrame`).\n\n More precisely, a *trivialization frame... |
class TrivializationFrame(LocalFrame):
"\n Trivialization frame on a topological vector bundle.\n\n A *trivialization frame* on a topological vector bundle `E \\to M` of rank\n `n` over the topological field `K` and over a topological manifold `M` is a\n local frame induced by a local trivialization `... |
class TopologicalManifold(ManifoldSubset):
'\n Topological manifold over a topological field `K`.\n\n Given a topological field `K` (in most applications, `K = \\RR` or\n `K = \\CC`) and a non-negative integer `n`, a *topological manifold of\n dimension* `n` *over K* is a topological space `M` such th... |
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