Multivector Field Modules¶
The set \(A^p(U, \Phi)\) of \(p\)-vector fields along a differentiable manifold \(U\) with values on a differentiable manifold \(M\) via a differentiable map \(\Phi:\ U \rightarrow M\) (possibly \(U = M\) and \(\Phi = \mathrm{Id}_M\)) is a module over the algebra \(C^k(U)\) of differentiable scalar fields on \(U\). It is a free module if and only if \(M\) is parallelizable. Accordingly, two classes implement \(A^p(U,\Phi)\):
MultivectorModule
for \(p\)-vector fields with values on a generic (in practice, not parallelizable) differentiable manifold \(M\)MultivectorFreeModule
for \(p\)-vector fields with values on a parallelizable manifold \(M\)
AUTHORS:
Eric Gourgoulhon (2017): initial version
REFERENCES:
- class sage.manifolds.differentiable.multivector_module.MultivectorFreeModule(vector_field_module, degree)[source]¶
Bases:
ExtPowerFreeModule
Free module of multivector fields of a given degree \(p\) (\(p\)-vector fields) along a differentiable manifold \(U\) with values on a parallelizable manifold \(M\).
Given a differentiable manifold \(U\) and a differentiable map \(\Phi:\; U \rightarrow M\) to a parallelizable manifold \(M\) of dimension \(n\), the set \(A^p(U, \Phi)\) of \(p\)-vector fields (i.e. alternating tensor fields of type \((p,0)\)) along \(U\) with values on \(M\) is a free module of rank \(\binom{n}{p}\) over \(C^k(U)\), the commutative algebra of differentiable scalar fields on \(U\) (see
DiffScalarFieldAlgebra
). The standard case of \(p\)-vector fields on a differentiable manifold \(M\) corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).Note
This class implements \(A^p(U, \Phi)\) in the case where \(M\) is parallelizable; \(A^p(U, \Phi)\) is then a free module. If \(M\) is not parallelizable, the class
MultivectorModule
must be used instead.INPUT:
vector_field_module
– free module \(\mathfrak{X}(U,\Phi)\) of vector fields along \(U\) associated with the map \(\Phi: U \rightarrow V\)degree
– positive integer; the degree \(p\) of the multivector fields
EXAMPLES:
Free module of 2-vector fields on a parallelizable 3-dimensional manifold:
sage: M = Manifold(3, 'M') sage: X.<x,y,z> = M.chart() sage: XM = M.vector_field_module() ; XM Free module X(M) of vector fields on the 3-dimensional differentiable manifold M sage: A = M.multivector_module(2) ; A Free module A^2(M) of 2-vector fields on the 3-dimensional differentiable manifold M sage: latex(A) A^{2}\left(M\right)
>>> from sage.all import * >>> M = Manifold(Integer(3), 'M') >>> X = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = X._first_ngens(3) >>> XM = M.vector_field_module() ; XM Free module X(M) of vector fields on the 3-dimensional differentiable manifold M >>> A = M.multivector_module(Integer(2)) ; A Free module A^2(M) of 2-vector fields on the 3-dimensional differentiable manifold M >>> latex(A) A^{2}\left(M\right)
A
is nothing but the second exterior power ofXM
, i.e. we have \(A^{2}(M) = \Lambda^2(\mathfrak{X}(M))\) (seeExtPowerFreeModule
):sage: A is XM.exterior_power(2) True
>>> from sage.all import * >>> A is XM.exterior_power(Integer(2)) True
\(A^{2}(M)\) is a module over the algebra \(C^k(M)\) of (differentiable) scalar fields on \(M\):
sage: A.category() Category of finite dimensional modules over Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold M sage: CM = M.scalar_field_algebra() ; CM Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold M sage: A in Modules(CM) True sage: A.base_ring() Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold M sage: A.base_module() Free module X(M) of vector fields on the 3-dimensional differentiable manifold M sage: A.base_module() is XM True sage: A.rank() 3
>>> from sage.all import * >>> A.category() Category of finite dimensional modules over Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold M >>> CM = M.scalar_field_algebra() ; CM Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold M >>> A in Modules(CM) True >>> A.base_ring() Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold M >>> A.base_module() Free module X(M) of vector fields on the 3-dimensional differentiable manifold M >>> A.base_module() is XM True >>> A.rank() 3
Elements can be constructed from \(A\). In particular,
0
yields the zero element of \(A\):sage: A(0) 2-vector field zero on the 3-dimensional differentiable manifold M sage: A(0) is A.zero() True
>>> from sage.all import * >>> A(Integer(0)) 2-vector field zero on the 3-dimensional differentiable manifold M >>> A(Integer(0)) is A.zero() True
while nonzero elements are constructed by providing their components in a given vector frame:
sage: comp = [[0,3*x,-z],[-3*x,0,4],[z,-4,0]] sage: a = A(comp, frame=X.frame(), name='a') ; a 2-vector field a on the 3-dimensional differentiable manifold M sage: a.display() a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z
>>> from sage.all import * >>> comp = [[Integer(0),Integer(3)*x,-z],[-Integer(3)*x,Integer(0),Integer(4)],[z,-Integer(4),Integer(0)]] >>> a = A(comp, frame=X.frame(), name='a') ; a 2-vector field a on the 3-dimensional differentiable manifold M >>> a.display() a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z
An alternative is to construct the 2-vector field from an empty list of components and to set the nonzero nonredundant components afterwards:
sage: a = A([], name='a') sage: a[0,1] = 3*x # component in the manifold's default frame sage: a[0,2] = -z sage: a[1,2] = 4 sage: a.display() a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z
>>> from sage.all import * >>> a = A([], name='a') >>> a[Integer(0),Integer(1)] = Integer(3)*x # component in the manifold's default frame >>> a[Integer(0),Integer(2)] = -z >>> a[Integer(1),Integer(2)] = Integer(4) >>> a.display() a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z
The module \(A^1(M)\) is nothing but \(\mathfrak{X}(M)\) (the free module of vector fields on \(M\)):
sage: A1 = M.multivector_module(1) ; A1 Free module X(M) of vector fields on the 3-dimensional differentiable manifold M sage: A1 is XM True
>>> from sage.all import * >>> A1 = M.multivector_module(Integer(1)) ; A1 Free module X(M) of vector fields on the 3-dimensional differentiable manifold M >>> A1 is XM True
There is a coercion map \(A^p(M) \rightarrow T^{(p,0)}(M)\):
sage: T20 = M.tensor_field_module((2,0)); T20 Free module T^(2,0)(M) of type-(2,0) tensors fields on the 3-dimensional differentiable manifold M sage: T20.has_coerce_map_from(A) True
>>> from sage.all import * >>> T20 = M.tensor_field_module((Integer(2),Integer(0))); T20 Free module T^(2,0)(M) of type-(2,0) tensors fields on the 3-dimensional differentiable manifold M >>> T20.has_coerce_map_from(A) True
but of course not in the reverse direction, since not all contravariant tensor field is alternating:
sage: A.has_coerce_map_from(T20) False
>>> from sage.all import * >>> A.has_coerce_map_from(T20) False
The coercion map \(A^2(M) \rightarrow T^{(2,0)}(M)\) in action:
sage: T20 = M.tensor_field_module((2,0)) ; T20 Free module T^(2,0)(M) of type-(2,0) tensors fields on the 3-dimensional differentiable manifold M sage: ta = T20(a) ; ta Tensor field a of type (2,0) on the 3-dimensional differentiable manifold M sage: ta.display() a = 3*x ∂/∂x⊗∂/∂y - z ∂/∂x⊗∂/∂z - 3*x ∂/∂y⊗∂/∂x + 4 ∂/∂y⊗∂/∂z + z ∂/∂z⊗∂/∂x - 4 ∂/∂z⊗∂/∂y sage: a.display() a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z sage: ta.symmetries() # the antisymmetry is preserved no symmetry; antisymmetry: (0, 1)
>>> from sage.all import * >>> T20 = M.tensor_field_module((Integer(2),Integer(0))) ; T20 Free module T^(2,0)(M) of type-(2,0) tensors fields on the 3-dimensional differentiable manifold M >>> ta = T20(a) ; ta Tensor field a of type (2,0) on the 3-dimensional differentiable manifold M >>> ta.display() a = 3*x ∂/∂x⊗∂/∂y - z ∂/∂x⊗∂/∂z - 3*x ∂/∂y⊗∂/∂x + 4 ∂/∂y⊗∂/∂z + z ∂/∂z⊗∂/∂x - 4 ∂/∂z⊗∂/∂y >>> a.display() a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z >>> ta.symmetries() # the antisymmetry is preserved no symmetry; antisymmetry: (0, 1)
There is also coercion to subdomains, which is nothing but the restriction of the multivector field to some subset of its domain:
sage: U = M.open_subset('U', coord_def={X: x^2+y^2<1}) sage: B = U.multivector_module(2) ; B Free module A^2(U) of 2-vector fields on the Open subset U of the 3-dimensional differentiable manifold M sage: B.has_coerce_map_from(A) True sage: a_U = B(a) ; a_U 2-vector field a on the Open subset U of the 3-dimensional differentiable manifold M sage: a_U.display() a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z
>>> from sage.all import * >>> U = M.open_subset('U', coord_def={X: x**Integer(2)+y**Integer(2)<Integer(1)}) >>> B = U.multivector_module(Integer(2)) ; B Free module A^2(U) of 2-vector fields on the Open subset U of the 3-dimensional differentiable manifold M >>> B.has_coerce_map_from(A) True >>> a_U = B(a) ; a_U 2-vector field a on the Open subset U of the 3-dimensional differentiable manifold M >>> a_U.display() a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z
- Element[source]¶
alias of
MultivectorFieldParal
- class sage.manifolds.differentiable.multivector_module.MultivectorModule(vector_field_module, degree)[source]¶
Bases:
UniqueRepresentation
,Parent
Module of multivector fields of a given degree \(p\) (\(p\)-vector fields) along a differentiable manifold \(U\) with values on a differentiable manifold \(M\).
Given a differentiable manifold \(U\) and a differentiable map \(\Phi: U \rightarrow M\) to a differentiable manifold \(M\), the set \(A^p(U, \Phi)\) of \(p\)-vector fields (i.e. alternating tensor fields of type \((p,0)\)) along \(U\) with values on \(M\) is a module over \(C^k(U)\), the commutative algebra of differentiable scalar fields on \(U\) (see
DiffScalarFieldAlgebra
). The standard case of \(p\)-vector fields on a differentiable manifold \(M\) corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).Note
This class implements \(A^p(U,\Phi)\) in the case where \(M\) is not assumed to be parallelizable; the module \(A^p(U, \Phi)\) is then not necessarily free. If \(M\) is parallelizable, the class
MultivectorFreeModule
must be used instead.INPUT:
vector_field_module
– module \(\mathfrak{X}(U, \Phi)\) of vector fields along \(U\) with values on \(M\) via the map \(\Phi: U \rightarrow M\)degree
– positive integer; the degree \(p\) of the multivector fields
EXAMPLES:
Module of 2-vector fields on a non-parallelizable 2-dimensional manifold:
sage: M = Manifold(2, 'M') sage: U = M.open_subset('U') ; V = M.open_subset('V') sage: M.declare_union(U,V) # M is the union of U and V sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart() sage: transf = c_xy.transition_map(c_uv, (x+y, x-y), ....: intersection_name='W', restrictions1= x>0, ....: restrictions2= u+v>0) sage: inv = transf.inverse() sage: W = U.intersection(V) sage: eU = c_xy.frame() ; eV = c_uv.frame() sage: XM = M.vector_field_module() ; XM Module X(M) of vector fields on the 2-dimensional differentiable manifold M sage: A = M.multivector_module(2) ; A Module A^2(M) of 2-vector fields on the 2-dimensional differentiable manifold M sage: latex(A) A^{2}\left(M\right)
>>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> U = M.open_subset('U') ; V = M.open_subset('V') >>> M.declare_union(U,V) # M is the union of U and V >>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2); c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2) >>> transf = c_xy.transition_map(c_uv, (x+y, x-y), ... intersection_name='W', restrictions1= x>Integer(0), ... restrictions2= u+v>Integer(0)) >>> inv = transf.inverse() >>> W = U.intersection(V) >>> eU = c_xy.frame() ; eV = c_uv.frame() >>> XM = M.vector_field_module() ; XM Module X(M) of vector fields on the 2-dimensional differentiable manifold M >>> A = M.multivector_module(Integer(2)) ; A Module A^2(M) of 2-vector fields on the 2-dimensional differentiable manifold M >>> latex(A) A^{2}\left(M\right)
A
is nothing but the second exterior power ofXM
, i.e. we have \(A^{2}(M) = \Lambda^2(\mathfrak{X}(M))\):sage: A is XM.exterior_power(2) True
>>> from sage.all import * >>> A is XM.exterior_power(Integer(2)) True
Modules of multivector fields are unique:
sage: A is M.multivector_module(2) True
>>> from sage.all import * >>> A is M.multivector_module(Integer(2)) True
\(A^2(M)\) is a module over the algebra \(C^k(M)\) of (differentiable) scalar fields on \(M\):
sage: A.category() Category of modules over Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold M sage: CM = M.scalar_field_algebra() ; CM Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold M sage: A in Modules(CM) True sage: A.base_ring() is CM True sage: A.base_module() Module X(M) of vector fields on the 2-dimensional differentiable manifold M sage: A.base_module() is XM True
>>> from sage.all import * >>> A.category() Category of modules over Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold M >>> CM = M.scalar_field_algebra() ; CM Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold M >>> A in Modules(CM) True >>> A.base_ring() is CM True >>> A.base_module() Module X(M) of vector fields on the 2-dimensional differentiable manifold M >>> A.base_module() is XM True
Elements can be constructed from
A()
. In particular,0
yields the zero element ofA
:sage: z = A(0) ; z 2-vector field zero on the 2-dimensional differentiable manifold M sage: z.display(eU) zero = 0 sage: z.display(eV) zero = 0 sage: z is A.zero() True
>>> from sage.all import * >>> z = A(Integer(0)) ; z 2-vector field zero on the 2-dimensional differentiable manifold M >>> z.display(eU) zero = 0 >>> z.display(eV) zero = 0 >>> z is A.zero() True
while nonzero elements are constructed by providing their components in a given vector frame:
sage: a = A([[0,3*x],[-3*x,0]], frame=eU, name='a') ; a 2-vector field a on the 2-dimensional differentiable manifold M sage: a.add_comp_by_continuation(eV, W, c_uv) # finishes initializ. of a sage: a.display(eU) a = 3*x ∂/∂x∧∂/∂y sage: a.display(eV) a = (-3*u - 3*v) ∂/∂u∧∂/∂v
>>> from sage.all import * >>> a = A([[Integer(0),Integer(3)*x],[-Integer(3)*x,Integer(0)]], frame=eU, name='a') ; a 2-vector field a on the 2-dimensional differentiable manifold M >>> a.add_comp_by_continuation(eV, W, c_uv) # finishes initializ. of a >>> a.display(eU) a = 3*x ∂/∂x∧∂/∂y >>> a.display(eV) a = (-3*u - 3*v) ∂/∂u∧∂/∂v
An alternative is to construct the 2-vector field from an empty list of components and to set the nonzero nonredundant components afterwards:
sage: a = A([], name='a') sage: a[eU,0,1] = 3*x sage: a.add_comp_by_continuation(eV, W, c_uv) sage: a.display(eU) a = 3*x ∂/∂x∧∂/∂y sage: a.display(eV) a = (-3*u - 3*v) ∂/∂u∧∂/∂v
>>> from sage.all import * >>> a = A([], name='a') >>> a[eU,Integer(0),Integer(1)] = Integer(3)*x >>> a.add_comp_by_continuation(eV, W, c_uv) >>> a.display(eU) a = 3*x ∂/∂x∧∂/∂y >>> a.display(eV) a = (-3*u - 3*v) ∂/∂u∧∂/∂v
The module \(A^1(M)\) is nothing but the dual of \(\mathfrak{X}(M)\) (the module of vector fields on \(M\)):
sage: A1 = M.multivector_module(1) ; A1 Module X(M) of vector fields on the 2-dimensional differentiable manifold M sage: A1 is XM True
>>> from sage.all import * >>> A1 = M.multivector_module(Integer(1)) ; A1 Module X(M) of vector fields on the 2-dimensional differentiable manifold M >>> A1 is XM True
There is a coercion map \(A^p(M)\rightarrow T^{(p,0)}(M)\):
sage: T20 = M.tensor_field_module((2,0)) ; T20 Module T^(2,0)(M) of type-(2,0) tensors fields on the 2-dimensional differentiable manifold M sage: T20.has_coerce_map_from(A) True
>>> from sage.all import * >>> T20 = M.tensor_field_module((Integer(2),Integer(0))) ; T20 Module T^(2,0)(M) of type-(2,0) tensors fields on the 2-dimensional differentiable manifold M >>> T20.has_coerce_map_from(A) True
but of course not in the reverse direction, since not all contravariant tensor field is alternating:
sage: A.has_coerce_map_from(T20) False
>>> from sage.all import * >>> A.has_coerce_map_from(T20) False
The coercion map \(A^2(M) \rightarrow T^{(2,0)}(M)\) in action:
sage: ta = T20(a) ; ta Tensor field a of type (2,0) on the 2-dimensional differentiable manifold M sage: ta.display(eU) a = 3*x ∂/∂x⊗∂/∂y - 3*x ∂/∂y⊗∂/∂x sage: a.display(eU) a = 3*x ∂/∂x∧∂/∂y sage: ta.display(eV) a = (-3*u - 3*v) ∂/∂u⊗∂/∂v + (3*u + 3*v) ∂/∂v⊗∂/∂u sage: a.display(eV) a = (-3*u - 3*v) ∂/∂u∧∂/∂v
>>> from sage.all import * >>> ta = T20(a) ; ta Tensor field a of type (2,0) on the 2-dimensional differentiable manifold M >>> ta.display(eU) a = 3*x ∂/∂x⊗∂/∂y - 3*x ∂/∂y⊗∂/∂x >>> a.display(eU) a = 3*x ∂/∂x∧∂/∂y >>> ta.display(eV) a = (-3*u - 3*v) ∂/∂u⊗∂/∂v + (3*u + 3*v) ∂/∂v⊗∂/∂u >>> a.display(eV) a = (-3*u - 3*v) ∂/∂u∧∂/∂v
There is also coercion to subdomains, which is nothing but the restriction of the multivector field to some subset of its domain:
sage: A2U = U.multivector_module(2) ; A2U Free module A^2(U) of 2-vector fields on the Open subset U of the 2-dimensional differentiable manifold M sage: A2U.has_coerce_map_from(A) True sage: a_U = A2U(a) ; a_U 2-vector field a on the Open subset U of the 2-dimensional differentiable manifold M sage: a_U.display(eU) a = 3*x ∂/∂x∧∂/∂y
>>> from sage.all import * >>> A2U = U.multivector_module(Integer(2)) ; A2U Free module A^2(U) of 2-vector fields on the Open subset U of the 2-dimensional differentiable manifold M >>> A2U.has_coerce_map_from(A) True >>> a_U = A2U(a) ; a_U 2-vector field a on the Open subset U of the 2-dimensional differentiable manifold M >>> a_U.display(eU) a = 3*x ∂/∂x∧∂/∂y
- Element[source]¶
alias of
MultivectorField
- base_module()[source]¶
Return the vector field module on which the multivector field module
self
is constructed.OUTPUT:
a
VectorFieldModule
representing the module on whichself
is defined
EXAMPLES:
sage: M = Manifold(3, 'M') sage: A2 = M.multivector_module(2) ; A2 Module A^2(M) of 2-vector fields on the 3-dimensional differentiable manifold M sage: A2.base_module() Module X(M) of vector fields on the 3-dimensional differentiable manifold M sage: A2.base_module() is M.vector_field_module() True sage: U = M.open_subset('U') sage: A2U = U.multivector_module(2) ; A2U Module A^2(U) of 2-vector fields on the Open subset U of the 3-dimensional differentiable manifold M sage: A2U.base_module() Module X(U) of vector fields on the Open subset U of the 3-dimensional differentiable manifold M
>>> from sage.all import * >>> M = Manifold(Integer(3), 'M') >>> A2 = M.multivector_module(Integer(2)) ; A2 Module A^2(M) of 2-vector fields on the 3-dimensional differentiable manifold M >>> A2.base_module() Module X(M) of vector fields on the 3-dimensional differentiable manifold M >>> A2.base_module() is M.vector_field_module() True >>> U = M.open_subset('U') >>> A2U = U.multivector_module(Integer(2)) ; A2U Module A^2(U) of 2-vector fields on the Open subset U of the 3-dimensional differentiable manifold M >>> A2U.base_module() Module X(U) of vector fields on the Open subset U of the 3-dimensional differentiable manifold M
- degree()[source]¶
Return the degree of the multivector fields in
self
.OUTPUT: integer \(p\) such that
self
is a set of \(p\)-vector fieldsEXAMPLES:
sage: M = Manifold(3, 'M') sage: M.multivector_module(2).degree() 2 sage: M.multivector_module(3).degree() 3
>>> from sage.all import * >>> M = Manifold(Integer(3), 'M') >>> M.multivector_module(Integer(2)).degree() 2 >>> M.multivector_module(Integer(3)).degree() 3
- zero()[source]¶
Return the zero of
self
.EXAMPLES:
sage: M = Manifold(3, 'M') sage: A2 = M.multivector_module(2) sage: A2.zero() 2-vector field zero on the 3-dimensional differentiable manifold M
>>> from sage.all import * >>> M = Manifold(Integer(3), 'M') >>> A2 = M.multivector_module(Integer(2)) >>> A2.zero() 2-vector field zero on the 3-dimensional differentiable manifold M