Tensor Fields#
The class TensorField
implements tensor fields on differentiable
manifolds. The derived class
TensorFieldParal
is devoted to tensor fields with values on parallelizable manifolds.
Various derived classes of TensorField
are devoted to specific tensor
fields:
VectorField
for vector fields (rank-1 contravariant tensor fields)AutomorphismField
for fields of tangent-space automorphismsDiffForm
for differential forms (fully antisymmetric covariant tensor fields)MultivectorField
for multivector fields (fully antisymmetric contravariant tensor fields)
AUTHORS:
Eric Gourgoulhon, Michal Bejger (2013-2015) : initial version
Travis Scrimshaw (2016): review tweaks
Eric Gourgoulhon (2018): operators divergence, Laplacian and d’Alembertian; method
TensorField.along()
Florentin Jaffredo (2018) : series expansion with respect to a given parameter
Michael Jung (2019): improve treatment of the zero element; add method
TensorField.copy_from()
Eric Gourgoulhon (2020): add method
TensorField.apply_map()
REFERENCES:
- class sage.manifolds.differentiable.tensorfield.TensorField(vector_field_module: VectorFieldModule, tensor_type: TensorType, name: Optional[str] = None, latex_name: Optional[str] = None, sym=None, antisym=None, parent=None)#
Bases:
ModuleElementWithMutability
Tensor field along a differentiable manifold.
An instance of this class is a tensor field along a differentiable manifold \(U\) with values on a differentiable manifold \(M\), via a differentiable map \(\Phi: U \rightarrow M\). More precisely, given two non-negative integers \(k\) and \(l\) and a differentiable map
\[\Phi:\ U \longrightarrow M,\]a tensor field of type \((k,l)\) along \(U\) with values on \(M\) is a differentiable map
\[t:\ U \longrightarrow T^{(k,l)}M\](where \(T^{(k,l)}M\) is the tensor bundle of type \((k,l)\) over \(M\)) such that
\[\forall p \in U,\ t(p) \in T^{(k,l)}(T_q M)\]i.e. \(t(p)\) is a tensor of type \((k,l)\) on the tangent space \(T_q M\) at the point \(q = \Phi(p)\), that is to say a multilinear map
\[t(p):\ \underbrace{T_q^*M\times\cdots\times T_q^*M}_{k\ \; \text{times}} \times \underbrace{T_q M\times\cdots\times T_q M}_{l\ \; \text{times}} \longrightarrow K,\]where \(T_q^* M\) is the dual vector space to \(T_q M\) and \(K\) is the topological field over which the manifold \(M\) is defined. The integer \(k+l\) is called the tensor rank.
The standard case of a tensor field on a differentiable manifold 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 \(\RR\)).
If \(M\) is parallelizable, the class
TensorFieldParal
should be used instead.This is a Sage element class, the corresponding parent class being
TensorFieldModule
.INPUT:
vector_field_module
– module \(\mathfrak{X}(U,\Phi)\) of vector fields along \(U\) associated with the map \(\Phi: U \rightarrow M\) (cf.VectorFieldModule
)tensor_type
– pair \((k,l)\) with \(k\) being the contravariant rank and \(l\) the covariant rankname
– (default:None
) name given to the tensor fieldlatex_name
– (default:None
) LaTeX symbol to denote the tensor field; if none is provided, the LaTeX symbol is set toname
sym
– (default:None
) a symmetry or a list of symmetries among the tensor arguments: each symmetry is described by a tuple containing the positions of the involved arguments, with the conventionposition = 0
for the first argument; for instance:sym = (0,1)
for a symmetry between the 1st and 2nd argumentssym = [(0,2), (1,3,4)]
for a symmetry between the 1st and 3rd arguments and a symmetry between the 2nd, 4th and 5th arguments.
antisym
– (default:None
) antisymmetry or list of antisymmetries among the arguments, with the same convention as forsym
parent
– (default:None
) some specific parent (e.g. exterior power for differential forms); ifNone
,vector_field_module.tensor_module(k,l)
is used
EXAMPLES:
Tensor field of type (0,2) on the sphere \(S^2\):
sage: M = Manifold(2, 'S^2') # the 2-dimensional sphere S^2 sage: U = M.open_subset('U') # complement of the North pole sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole sage: V = M.open_subset('V') # complement of the South pole sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole sage: M.declare_union(U,V) # S^2 is the union of U and V sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)), ....: intersection_name='W', restrictions1= x^2+y^2!=0, ....: restrictions2= u^2+v^2!=0) sage: uv_to_xy = xy_to_uv.inverse() sage: W = U.intersection(V) sage: t = M.tensor_field(0,2, name='t') ; t Tensor field t of type (0,2) on the 2-dimensional differentiable manifold S^2 sage: t.parent() Module T^(0,2)(S^2) of type-(0,2) tensors fields on the 2-dimensional differentiable manifold S^2 sage: t.parent().category() Category of tensor products of modules over Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold S^2
The parent of \(t\) is not a free module, for the sphere \(S^2\) is not parallelizable:
sage: isinstance(t.parent(), FiniteRankFreeModule) False
To fully define \(t\), we have to specify its components in some vector frames defined on subsets of \(S^2\); let us start by the open subset \(U\):
sage: eU = c_xy.frame() sage: t[eU,:] = [[1,0], [-2,3]] sage: t.display(eU) t = dx⊗dx - 2 dy⊗dx + 3 dy⊗dy
To set the components of \(t\) on \(V\) consistently, we copy the expressions of the components in the common subset \(W\):
sage: eV = c_uv.frame() sage: eVW = eV.restrict(W) sage: c_uvW = c_uv.restrict(W) sage: t[eV,0,0] = t[eVW,0,0,c_uvW].expr() # long time sage: t[eV,0,1] = t[eVW,0,1,c_uvW].expr() # long time sage: t[eV,1,0] = t[eVW,1,0,c_uvW].expr() # long time sage: t[eV,1,1] = t[eVW,1,1,c_uvW].expr() # long time
Actually, the above operation can be performed in a single line by means of the method
add_comp_by_continuation()
:sage: t.add_comp_by_continuation(eV, W, chart=c_uv) # long time
At this stage, \(t\) is fully defined, having components in frames eU and eV and the union of the domains of eU and eV being the whole manifold:
sage: t.display(eV) # long time t = (u^4 - 4*u^3*v + 10*u^2*v^2 + 4*u*v^3 + v^4)/(u^8 + 4*u^6*v^2 + 6*u^4*v^4 + 4*u^2*v^6 + v^8) du⊗du - 4*(u^3*v + 2*u^2*v^2 - u*v^3)/(u^8 + 4*u^6*v^2 + 6*u^4*v^4 + 4*u^2*v^6 + v^8) du⊗dv + 2*(u^4 - 2*u^3*v - 2*u^2*v^2 + 2*u*v^3 + v^4)/(u^8 + 4*u^6*v^2 + 6*u^4*v^4 + 4*u^2*v^6 + v^8) dv⊗du + (3*u^4 + 4*u^3*v - 2*u^2*v^2 - 4*u*v^3 + 3*v^4)/(u^8 + 4*u^6*v^2 + 6*u^4*v^4 + 4*u^2*v^6 + v^8) dv⊗dv
Let us consider two vector fields, \(a\) and \(b\), on \(S^2\):
sage: a = M.vector_field({eU: [-y, x]}, name='a') sage: a.add_comp_by_continuation(eV, W, chart=c_uv) sage: a.display(eV) a = -v ∂/∂u + u ∂/∂v sage: b = M.vector_field({eU: [y, -1]}, name='b') sage: b.add_comp_by_continuation(eV, W, chart=c_uv) sage: b.display(eV) b = ((2*u + 1)*v^3 + (2*u^3 - u^2)*v)/(u^2 + v^2) ∂/∂u - (u^4 - v^4 + 2*u*v^2)/(u^2 + v^2) ∂/∂v
As a tensor field of type \((0,2)\), \(t\) acts on the pair \((a,b)\), resulting in a scalar field:
sage: f = t(a,b); f Scalar field t(a,b) on the 2-dimensional differentiable manifold S^2 sage: f.display() # long time t(a,b): S^2 → ℝ on U: (x, y) ↦ -2*x*y - y^2 - 3*x on V: (u, v) ↦ -(3*u^3 + (3*u + 1)*v^2 + 2*u*v)/(u^4 + 2*u^2*v^2 + v^4)
The vectors can be defined only on subsets of \(S^2\), the domain of the result is then the common subset:
sage: # long time sage: s = t(a.restrict(U), b) ; s Scalar field t(a,b) on the Open subset U of the 2-dimensional differentiable manifold S^2 sage: s.display() t(a,b): U → ℝ (x, y) ↦ -2*x*y - y^2 - 3*x on W: (u, v) ↦ -(3*u^3 + (3*u + 1)*v^2 + 2*u*v)/(u^4 + 2*u^2*v^2 + v^4) sage: s = t(a.restrict(U), b.restrict(W)) ; s Scalar field t(a,b) on the Open subset W of the 2-dimensional differentiable manifold S^2 sage: s.display() t(a,b): W → ℝ (x, y) ↦ -2*x*y - y^2 - 3*x (u, v) ↦ -(3*u^3 + (3*u + 1)*v^2 + 2*u*v)/(u^4 + 2*u^2*v^2 + v^4)
The tensor itself can be defined only on some open subset of \(S^2\), yielding a result whose domain is this subset:
sage: s = t.restrict(V)(a,b); s # long time Scalar field t(a,b) on the Open subset V of the 2-dimensional differentiable manifold S^2 sage: s.display() # long time t(a,b): V → ℝ (u, v) ↦ -(3*u^3 + (3*u + 1)*v^2 + 2*u*v)/(u^4 + 2*u^2*v^2 + v^4) on W: (x, y) ↦ -2*x*y - y^2 - 3*x
Tests regarding the multiplication by a scalar field:
sage: f = M.scalar_field({c_xy: 1/(1+x^2+y^2), ....: c_uv: (u^2 + v^2)/(u^2 + v^2 + 1)}, name='f') sage: t.parent().base_ring() is f.parent() True sage: s = f*t; s # long time Tensor field f*t of type (0,2) on the 2-dimensional differentiable manifold S^2 sage: s[[0,0]] == f*t[[0,0]] # long time True sage: s.restrict(U) == f.restrict(U) * t.restrict(U) # long time True sage: s = f*t.restrict(U); s Tensor field f*t of type (0,2) on the Open subset U of the 2-dimensional differentiable manifold S^2 sage: s.restrict(U) == f.restrict(U) * t.restrict(U) True
Same examples with SymPy as the symbolic engine
From now on, we ask that all symbolic calculus on manifold \(M\) are performed by SymPy:
sage: M.set_calculus_method('sympy')
We define the tensor \(t\) as above:
sage: t = M.tensor_field(0, 2, {eU: [[1,0], [-2,3]]}, name='t') sage: t.display(eU) t = dx⊗dx - 2 dy⊗dx + 3 dy⊗dy sage: t.add_comp_by_continuation(eV, W, chart=c_uv) # long time sage: t.display(eV) # long time t = (u**4 - 4*u**3*v + 10*u**2*v**2 + 4*u*v**3 + v**4)/(u**8 + 4*u**6*v**2 + 6*u**4*v**4 + 4*u**2*v**6 + v**8) du⊗du + 4*u*v*(-u**2 - 2*u*v + v**2)/(u**8 + 4*u**6*v**2 + 6*u**4*v**4 + 4*u**2*v**6 + v**8) du⊗dv + 2*(u**4 - 2*u**3*v - 2*u**2*v**2 + 2*u*v**3 + v**4)/(u**8 + 4*u**6*v**2 + 6*u**4*v**4 + 4*u**2*v**6 + v**8) dv⊗du + (3*u**4 + 4*u**3*v - 2*u**2*v**2 - 4*u*v**3 + 3*v**4)/(u**8 + 4*u**6*v**2 + 6*u**4*v**4 + 4*u**2*v**6 + v**8) dv⊗dv
The default coordinate representations of tensor components are now SymPy objects:
sage: t[eV,1,1,c_uv].expr() # long time (3*u**4 + 4*u**3*v - 2*u**2*v**2 - 4*u*v**3 + 3*v**4)/(u**8 + 4*u**6*v**2 + 6*u**4*v**4 + 4*u**2*v**6 + v**8) sage: type(t[eV,1,1,c_uv].expr()) # long time <class 'sympy.core.mul.Mul'>
Let us consider two vector fields, \(a\) and \(b\), on \(S^2\):
sage: a = M.vector_field({eU: [-y, x]}, name='a') sage: a.add_comp_by_continuation(eV, W, chart=c_uv) sage: a.display(eV) a = -v ∂/∂u + u ∂/∂v sage: b = M.vector_field({eU: [y, -1]}, name='b') sage: b.add_comp_by_continuation(eV, W, chart=c_uv) sage: b.display(eV) b = v*(2*u**3 - u**2 + 2*u*v**2 + v**2)/(u**2 + v**2) ∂/∂u + (-u**4 - 2*u*v**2 + v**4)/(u**2 + v**2) ∂/∂v
As a tensor field of type \((0,2)\), \(t\) acts on the pair \((a,b)\), resulting in a scalar field:
sage: f = t(a,b) sage: f.display() # long time t(a,b): S^2 → ℝ on U: (x, y) ↦ -2*x*y - 3*x - y**2 on V: (u, v) ↦ (-3*u**3 - 3*u*v**2 - 2*u*v - v**2)/(u**4 + 2*u**2*v**2 + v**4)
The vectors can be defined only on subsets of \(S^2\), the domain of the result is then the common subset:
sage: s = t(a.restrict(U), b) sage: s.display() # long time t(a,b): U → ℝ (x, y) ↦ -2*x*y - 3*x - y**2 on W: (u, v) ↦ (-3*u**3 - 3*u*v**2 - 2*u*v - v**2)/(u**4 + 2*u**2*v**2 + v**4) sage: s = t(a.restrict(U), b.restrict(W)) # long time sage: s.display() # long time t(a,b): W → ℝ (x, y) ↦ -2*x*y - 3*x - y**2 (u, v) ↦ (-3*u**3 - 3*u*v**2 - 2*u*v - v**2)/(u**4 + 2*u**2*v**2 + v**4)
The tensor itself can be defined only on some open subset of \(S^2\), yielding a result whose domain is this subset:
sage: s = t.restrict(V)(a,b) # long time sage: s.display() # long time t(a,b): V → ℝ (u, v) ↦ (-3*u**3 - 3*u*v**2 - 2*u*v - v**2)/(u**4 + 2*u**2*v**2 + v**4) on W: (x, y) ↦ -2*x*y - 3*x - y**2
Tests regarding the multiplication by a scalar field:
sage: f = M.scalar_field({c_xy: 1/(1+x^2+y^2), ....: c_uv: (u^2 + v^2)/(u^2 + v^2 + 1)}, name='f') sage: s = f*t # long time sage: s[[0,0]] == f*t[[0,0]] # long time True sage: s.restrict(U) == f.restrict(U) * t.restrict(U) # long time True sage: s = f*t.restrict(U) sage: s.restrict(U) == f.restrict(U) * t.restrict(U) True
Notice that the zero tensor field is immutable, and therefore its components cannot be changed:
sage: zer = M.tensor_field_module((1, 1)).zero() sage: zer.is_immutable() True sage: zer.set_comp() Traceback (most recent call last): ... ValueError: the components of an immutable element cannot be changed
Other tensor fields can be declared immutable, too:
sage: t.is_immutable() False sage: t.set_immutable() sage: t.is_immutable() True sage: t.set_comp() Traceback (most recent call last): ... ValueError: the components of an immutable element cannot be changed sage: t.set_name('b') Traceback (most recent call last): ... ValueError: the name of an immutable element cannot be changed
- add_comp(basis=None)#
Return the components of
self
in a given vector frame for assignment.The components with respect to other frames having the same domain as the provided vector frame are kept. To delete them, use the method
set_comp()
instead.INPUT:
basis
– (default:None
) vector frame in which the components are defined; ifNone
, the components are assumed to refer to the tensor field domain’s default frame
OUTPUT:
components in the given frame, as a
Components
; if such components did not exist previously, they are created
EXAMPLES:
sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2 sage: U = M.open_subset('U') # complement of the North pole sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole sage: V = M.open_subset('V') # complement of the South pole sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole sage: M.declare_union(U,V) # S^2 is the union of U and V sage: e_uv = c_uv.frame() sage: t = M.tensor_field(1, 2, name='t') sage: t.add_comp(e_uv) 3-indices components w.r.t. Coordinate frame (V, (∂/∂u,∂/∂v)) sage: t.add_comp(e_uv)[1,0,1] = u+v sage: t.display(e_uv) t = (u + v) ∂/∂v⊗du⊗dv
Setting the components in a new frame:
sage: e = V.vector_frame('e') sage: t.add_comp(e) 3-indices components w.r.t. Vector frame (V, (e_0,e_1)) sage: t.add_comp(e)[0,1,1] = u*v sage: t.display(e) t = u*v e_0⊗e^1⊗e^1
The components with respect to
e_uv
are kept:sage: t.display(e_uv) t = (u + v) ∂/∂v⊗du⊗dv
Since zero is a special element, its components cannot be changed:
sage: z = M.tensor_field_module((1, 1)).zero() sage: z.add_comp(e_uv)[1, 1] = u^2 Traceback (most recent call last): ... ValueError: the components of an immutable element cannot be changed
- add_comp_by_continuation(frame, subdomain, chart=None)#
Set components with respect to a vector frame by continuation of the coordinate expression of the components in a subframe.
The continuation is performed by demanding that the components have the same coordinate expression as those on the restriction of the frame to a given subdomain.
INPUT:
frame
– vector frame \(e\) in which the components are to be setsubdomain
– open subset of \(e\)’s domain in which the components are known or can be evaluated from other componentschart
– (default:None
) coordinate chart on \(e\)’s domain in which the extension of the expression of the components is to be performed; ifNone
, the default’s chart of \(e\)’s domain is assumed
EXAMPLES:
Components of a vector field on the sphere \(S^2\):
sage: M = Manifold(2, 'S^2', start_index=1)
The two open subsets covered by stereographic coordinates (North and South):
sage: U = M.open_subset('U') ; V = M.open_subset('V') sage: M.declare_union(U,V) # S^2 is the union of U and V sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart() # stereographic coordinates sage: transf = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)), ....: intersection_name='W', restrictions1= x^2+y^2!=0, ....: restrictions2= u^2+v^2!=0) sage: inv = transf.inverse() sage: W = U.intersection(V) # The complement of the two poles sage: eU = c_xy.frame() ; eV = c_uv.frame() sage: a = M.vector_field({eU: [x, 2+y]}, name='a')
At this stage, the vector field has been defined only on the open subset
U
(through its components in the frameeU
):sage: a.display(eU) a = x ∂/∂x + (y + 2) ∂/∂y
The components with respect to the restriction of
eV
to the common subdomainW
, in terms of the(u,v)
coordinates, are obtained by a change-of-frame formula onW
:sage: a.display(eV.restrict(W), c_uv.restrict(W)) a = (-4*u*v - u) ∂/∂u + (2*u^2 - 2*v^2 - v) ∂/∂v
The continuation consists in extending the definition of the vector field to the whole open subset
V
by demanding that the components in the frame eV have the same coordinate expression as the above one:sage: a.add_comp_by_continuation(eV, W, chart=c_uv)
We have then:
sage: a.display(eV) a = (-4*u*v - u) ∂/∂u + (2*u^2 - 2*v^2 - v) ∂/∂v
and \(a\) is defined on the entire manifold \(S^2\).
- add_expr_from_subdomain(frame, subdomain)#
Add an expression to an existing component from a subdomain.
INPUT:
frame
– vector frame \(e\) in which the components are to be setsubdomain
– open subset of \(e\)’s domain in which the components have additional expressions.
EXAMPLES:
We are going to consider a vector field in \(\RR^3\) along the 2-sphere:
sage: M = Manifold(3, 'M', structure="Riemannian") sage: S = Manifold(2, 'S', structure="Riemannian") sage: E.<X,Y,Z> = M.chart()
Let us define
S
in terms of stereographic charts:sage: U = S.open_subset('U') sage: V = S.open_subset('V') sage: S.declare_union(U,V) sage: stereoN.<x,y> = U.chart() sage: stereoS.<xp,yp> = V.chart("xp:x' yp:y'") sage: stereoN_to_S = stereoN.transition_map(stereoS, ....: (x/(x^2+y^2), y/(x^2+y^2)), ....: intersection_name='W', ....: restrictions1= x^2+y^2!=0, ....: restrictions2= xp^2+yp^2!=0) sage: stereoS_to_N = stereoN_to_S.inverse() sage: W = U.intersection(V) sage: stereoN_W = stereoN.restrict(W) sage: stereoS_W = stereoS.restrict(W)
The embedding of \(S^2\) in \(\RR^3\):
sage: phi = S.diff_map(M, {(stereoN, E): [2*x/(1+x^2+y^2), ....: 2*y/(1+x^2+y^2), ....: (x^2+y^2-1)/(1+x^2+y^2)], ....: (stereoS, E): [2*xp/(1+xp^2+yp^2), ....: 2*yp/(1+xp^2+yp^2), ....: (1-xp^2-yp^2)/(1+xp^2+yp^2)]}, ....: name='Phi', latex_name=r'\Phi')
To define a vector field
v
alongS
taking its values inM
, we first set the components onU
:sage: v = M.vector_field(name='v').along(phi) sage: vU = v.restrict(U) sage: vU[:] = [x,y,x**2+y**2]
But because
M
is parallelizable, these components can be extended toS
itself:sage: v.add_comp_by_continuation(E.frame().along(phi), U)
One can see that
v
is not yet fully defined: the components (scalar fields) do not have values on the whole manifold:sage: sorted(v._components.values())[0]._comp[(0,)].display() S → ℝ on U: (x, y) ↦ x on W: (xp, yp) ↦ xp/(xp^2 + yp^2)
To fix that, we first extend the components from
W
toV
usingadd_comp_by_continuation()
:sage: v.add_comp_by_continuation(E.frame().along(phi).restrict(V), ....: W, stereoS)
Then, the expression on the subdomain
V
is added to the already known components onS
by:sage: v.add_expr_from_subdomain(E.frame().along(phi), V)
The definition of
v
is now complete:sage: sorted(v._components.values())[0]._comp[(2,)].display() S → ℝ on U: (x, y) ↦ x^2 + y^2 on V: (xp, yp) ↦ 1/(xp^2 + yp^2)
- along(mapping)#
Return the tensor field deduced from
self
via a differentiable map, the codomain of which is included in the domain ofself
.More precisely, if
self
is a tensor field \(t\) on \(M\) and if \(\Phi: U \rightarrow M\) is a differentiable map from some differentiable manifold \(U\) to \(M\), the returned object is a tensor field \(\tilde t\) along \(U\) with values on \(M\) such that\[\forall p \in U,\ \tilde t(p) = t(\Phi(p)).\]INPUT:
mapping
– differentiable map \(\Phi: U \rightarrow M\)
OUTPUT:
tensor field \(\tilde t\) along \(U\) defined above.
EXAMPLES:
Let us consider the 2-dimensional sphere \(S^2\):
sage: M = Manifold(2, 'S^2') # the 2-dimensional sphere S^2 sage: U = M.open_subset('U') # complement of the North pole sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole sage: V = M.open_subset('V') # complement of the South pole sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole sage: M.declare_union(U,V) # S^2 is the union of U and V sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)), ....: intersection_name='W', restrictions1= x^2+y^2!=0, ....: restrictions2= u^2+v^2!=0) sage: uv_to_xy = xy_to_uv.inverse() sage: W = U.intersection(V)
and the following map from the open interval \((0,5\pi/2)\) to \(S^2\), the image of it being the great circle \(x=0\), \(u=0\), which goes through the North and South poles:
sage: I.<t> = manifolds.OpenInterval(0, 5*pi/2) sage: J = I.open_interval(0, 3*pi/2) sage: K = I.open_interval(pi, 5*pi/2) sage: c_J = J.canonical_chart(); c_K = K.canonical_chart() sage: Phi = I.diff_map(M, {(c_J, c_xy): ....: (0, sgn(pi-t)*sqrt((1+cos(t))/(1-cos(t)))), ....: (c_K, c_uv): ....: (0, sgn(t-2*pi)*sqrt((1-cos(t))/(1+cos(t))))}, ....: name='Phi')
Let us consider a vector field on \(S^2\):
sage: eU = c_xy.frame(); eV = c_uv.frame() sage: w = M.vector_field(name='w') sage: w[eU,0] = 1 sage: w.add_comp_by_continuation(eV, W, chart=c_uv) sage: w.display(eU) w = ∂/∂x sage: w.display(eV) w = (-u^2 + v^2) ∂/∂u - 2*u*v ∂/∂v
We have then:
sage: wa = w.along(Phi); wa Vector field w along the Real interval (0, 5/2*pi) with values on the 2-dimensional differentiable manifold S^2 sage: wa.display(eU.along(Phi)) w = ∂/∂x sage: wa.display(eV.along(Phi)) w = -(cos(t) - 1)*sgn(-2*pi + t)^2/(cos(t) + 1) ∂/∂u
Some tests:
sage: p = K.an_element() sage: wa.at(p) == w.at(Phi(p)) True sage: wa.at(J(4*pi/3)) == wa.at(K(4*pi/3)) True sage: wa.at(I(4*pi/3)) == wa.at(K(4*pi/3)) True sage: wa.at(K(7*pi/4)) == eU[0].at(Phi(I(7*pi/4))) # since eU[0]=∂/∂x True
- antisymmetrize(*pos)#
Antisymmetrization over some arguments.
INPUT:
pos
– (default:None
) list of argument positions involved in the antisymmetrization (with the conventionposition=0
for the first argument); ifNone
, the antisymmetrization is performed over all the arguments
OUTPUT:
the antisymmetrized tensor field (instance of
TensorField
)
EXAMPLES:
Antisymmetrization of a type-\((0,2)\) tensor field on a 2-dimensional non-parallelizable 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: a = M.tensor_field(0,2, {eU: [[1,x], [2,y]]}, name='a') sage: a.add_comp_by_continuation(eV, W, chart=c_uv) sage: a[eV,:] [ 1/4*u + 3/4 -1/4*u + 3/4] [ 1/4*v - 1/4 -1/4*v - 1/4] sage: s = a.antisymmetrize() ; s 2-form on the 2-dimensional differentiable manifold M sage: s[eU,:] [ 0 1/2*x - 1] [-1/2*x + 1 0] sage: s[eV,:] [ 0 -1/8*u - 1/8*v + 1/2] [ 1/8*u + 1/8*v - 1/2 0] sage: s == a.antisymmetrize(0,1) # explicit positions True sage: s == a.antisymmetrize(1,0) # the order of positions does not matter True
See also
For more details and examples, see
sage.tensor.modules.free_module_tensor.FreeModuleTensor.antisymmetrize()
.
- apply_map(fun, frame=None, chart=None, keep_other_components=False)#
Apply a function to the coordinate expressions of all components of
self
in a given vector frame.This method allows operations like factorization, expansion, simplification or substitution to be performed on all components of
self
in a given vector frame (see examples below).INPUT:
fun
– function to be applied to the coordinate expressions of the componentsframe
– (default:None
) vector frame defining the components on which the operationfun
is to be performed; ifNone
, the default frame of the domain ofself
is assumedchart
– (default:None
) coordinate chart; if specified, the operationfun
is performed only on the coordinate expressions with respect tochart
of the components w.r.t.frame
; ifNone
, the operationfun
is performed on all available coordinate expressionskeep_other_components
– (default:False
) determine whether the components with respect to vector frames distinct fromframe
and having the same domain asframe
are kept. Iffun
is non-destructive,keep_other_components
can be set toTrue
; otherwise, it is advised to set it toFalse
(the default) in order to avoid any inconsistency between the various sets of components
EXAMPLES:
Factorizing all components in the default frame of a vector field:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: a, b = var('a b') sage: v = M.vector_field(x^2 - y^2, a*(b^2 - b)*x) sage: v.display() (x^2 - y^2) ∂/∂x + (b^2 - b)*a*x ∂/∂y sage: v.apply_map(factor) sage: v.display() (x + y)*(x - y) ∂/∂x + a*(b - 1)*b*x ∂/∂y
Performing a substitution in all components in the default frame:
sage: v.apply_map(lambda f: f.subs({a: 2})) sage: v.display() (x + y)*(x - y) ∂/∂x + 2*(b - 1)*b*x ∂/∂y
Specifying the vector frame via the argument
frame
:sage: P.<p, q> = M.chart() sage: X_to_P = X.transition_map(P, [x + 1, y - 1]) sage: P_to_X = X_to_P.inverse() sage: v.display(P) (p^2 - q^2 - 2*p - 2*q) ∂/∂p + (-2*b^2 + 2*(b^2 - b)*p + 2*b) ∂/∂q sage: v.apply_map(lambda f: f.subs({b: pi}), frame=P.frame()) sage: v.display(P) (p^2 - q^2 - 2*p - 2*q) ∂/∂p + (2*pi - 2*pi^2 - 2*(pi - pi^2)*p) ∂/∂q
Note that the required operation has been performed in all charts:
sage: v.display(P.frame(), P) (p^2 - q^2 - 2*p - 2*q) ∂/∂p + (2*pi - 2*pi^2 - 2*(pi - pi^2)*p) ∂/∂q sage: v.display(P.frame(), X) (x + y)*(x - y) ∂/∂p + 2*pi*(pi - 1)*x ∂/∂q
By default, the components of
v
in frames distinct from the specified one have been deleted:sage: X.frame() in v._components False
When requested, they are recomputed by change-of-frame formulas, thereby enforcing the consistency between the representations in various vector frames. In particular, we can check that the substitution of
b
bypi
, which was asked inP.frame()
, is effective inX.frame()
as well:sage: v.display(X.frame(), X) (x + y)*(x - y) ∂/∂x + 2*pi*(pi - 1)*x ∂/∂y
When the requested operation does not change the value of the tensor field, one can use the keyword argument
keep_other_components=True
, in order to avoid the recomputation of the components in other frames:sage: v.apply_map(factor, keep_other_components=True) sage: v.display() (x + y)*(x - y) ∂/∂x + 2*pi*(pi - 1)*x ∂/∂y
The components with respect to
P.frame()
have been kept:sage: P.frame() in v._components True
One can restrict the operation to expressions in a given chart, via the argument
chart
:sage: v.display(X.frame(), P) (p + q)*(p - q - 2) ∂/∂x + 2*pi*(pi - 1)*(p - 1) ∂/∂y sage: v.apply_map(expand, chart=P) sage: v.display(X.frame(), P) (p^2 - q^2 - 2*p - 2*q) ∂/∂x + (2*pi + 2*pi^2*p - 2*pi^2 - 2*pi*p) ∂/∂y sage: v.display(X.frame(), X) (x + y)*(x - y) ∂/∂x + 2*pi*(pi - 1)*x ∂/∂y
- at(point)#
Value of
self
at a point of its domain.If the current tensor field is
\[t:\ U \longrightarrow T^{(k,l)} M\]associated with the differentiable map
\[\Phi:\ U \longrightarrow M,\]where \(U\) and \(M\) are two manifolds (possibly \(U = M\) and \(\Phi = \mathrm{Id}_M\)), then for any point \(p \in U\), \(t(p)\) is a tensor on the tangent space to \(M\) at the point \(\Phi(p)\).
INPUT:
point
–ManifoldPoint
; point \(p\) in the domain of the tensor field \(U\)
OUTPUT:
FreeModuleTensor
representing the tensor \(t(p)\) on the tangent vector space \(T_{\Phi(p)} M\)
EXAMPLES:
Tensor on a tangent space of 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: a = M.tensor_field(1, 1, {eU: [[1+y,x], [0,x+y]]}, name='a') sage: a.add_comp_by_continuation(eV, W, chart=c_uv) sage: a.display(eU) a = (y + 1) ∂/∂x⊗dx + x ∂/∂x⊗dy + (x + y) ∂/∂y⊗dy sage: a.display(eV) a = (u + 1/2) ∂/∂u⊗du + (-1/2*u - 1/2*v + 1/2) ∂/∂u⊗dv + 1/2 ∂/∂v⊗du + (1/2*u - 1/2*v + 1/2) ∂/∂v⊗dv sage: p = M.point((2,3), chart=c_xy, name='p') sage: ap = a.at(p) ; ap Type-(1,1) tensor a on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: ap.parent() Free module of type-(1,1) tensors on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: ap.display(eU.at(p)) a = 4 ∂/∂x⊗dx + 2 ∂/∂x⊗dy + 5 ∂/∂y⊗dy sage: ap.display(eV.at(p)) a = 11/2 ∂/∂u⊗du - 3/2 ∂/∂u⊗dv + 1/2 ∂/∂v⊗du + 7/2 ∂/∂v⊗dv sage: p.coord(c_uv) # to check the above expression (5, -1)
- base_module()#
Return the vector field module on which
self
acts as a tensor.OUTPUT:
instance of
VectorFieldModule
EXAMPLES:
The module of vector fields on the 2-sphere as a “base module”:
sage: M = Manifold(2, 'S^2') sage: t = M.tensor_field(0,2) sage: t.base_module() Module X(S^2) of vector fields on the 2-dimensional differentiable manifold S^2 sage: t.base_module() is M.vector_field_module() True sage: XM = M.vector_field_module() sage: XM.an_element().base_module() is XM True
- comp(basis=None, from_basis=None)#
Return the components in a given vector frame.
If the components are not known already, they are computed by the tensor change-of-basis formula from components in another vector frame.
INPUT:
basis
– (default:None
) vector frame in which the components are required; if none is provided, the components are assumed to refer to the tensor field domain’s default framefrom_basis
– (default:None
) vector frame from which the required components are computed, via the tensor change-of-basis formula, if they are not known already in the basisbasis
OUTPUT:
components in the vector frame
basis
, as aComponents
EXAMPLES:
Components of a type-\((1,1)\) tensor field defined on two open subsets:
sage: M = Manifold(2, 'M') sage: U = M.open_subset('U') sage: c_xy.<x, y> = U.chart() sage: e = U.default_frame() ; e Coordinate frame (U, (∂/∂x,∂/∂y)) sage: V = M.open_subset('V') sage: c_uv.<u, v> = V.chart() sage: f = V.default_frame() ; f Coordinate frame (V, (∂/∂u,∂/∂v)) sage: M.declare_union(U,V) # M is the union of U and V sage: t = M.tensor_field(1,1, name='t') sage: t[e,0,0] = - x + y^3 sage: t[e,0,1] = 2+x sage: t[f,1,1] = - u*v sage: t.comp(e) 2-indices components w.r.t. Coordinate frame (U, (∂/∂x,∂/∂y)) sage: t.comp(e)[:] [y^3 - x x + 2] [ 0 0] sage: t.comp(f) 2-indices components w.r.t. Coordinate frame (V, (∂/∂u,∂/∂v)) sage: t.comp(f)[:] [ 0 0] [ 0 -u*v]
Since
e
isM
’s default frame, the argumente
can be omitted:sage: e is M.default_frame() True sage: t.comp() is t.comp(e) True
Example of computation of the components via a change of frame:
sage: a = V.automorphism_field() sage: a[:] = [[1+v, -u^2], [0, 1-u]] sage: h = f.new_frame(a, 'h') sage: t.comp(h) 2-indices components w.r.t. Vector frame (V, (h_0,h_1)) sage: t.comp(h)[:] [ 0 -u^3*v/(v + 1)] [ 0 -u*v]
- contract(*args)#
Contraction of
self
with another tensor field on one or more indices.INPUT:
pos1
– positions of the indices in the current tensor field involved in the contraction;pos1
must be a sequence of integers, with 0 standing for the first index position, 1 for the second one, etc.; ifpos1
is not provided, a single contraction on the last index position of the tensor field is assumedother
– the tensor field to contract withpos2
– positions of the indices inother
involved in the contraction, with the same conventions as forpos1
; ifpos2
is not provided, a single contraction on the first index position ofother
is assumed
OUTPUT:
tensor field resulting from the contraction at the positions
pos1
andpos2
of the tensor field withother
EXAMPLES:
Contractions of a type-\((1,1)\) tensor field with a type-\((2,0)\) one on a 2-dimensional non-parallelizable 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: a = M.tensor_field(1, 1, {eU: [[1, x], [0, 2]]}, name='a') sage: a.add_comp_by_continuation(eV, W, chart=c_uv) sage: b = M.tensor_field(2, 0, {eU: [[y, -1], [x+y, 2]]}, name='b') sage: b.add_comp_by_continuation(eV, W, chart=c_uv) sage: s = a.contract(b) ; s # contraction on last index of a and first one of b Tensor field of type (2,0) on the 2-dimensional differentiable manifold M
Check 1: components with respect to the manifold’s default frame (
eU
):sage: all(bool(s[i,j] == sum(a[i,k]*b[k,j] for k in M.irange())) ....: for i in M.irange() for j in M.irange()) True
Check 2: components with respect to the frame
eV
:sage: all(bool(s[eV,i,j] == sum(a[eV,i,k]*b[eV,k,j] ....: for k in M.irange())) ....: for i in M.irange() for j in M.irange()) True
Instead of the explicit call to the method
contract()
, one may use the index notation with Einstein convention (summation over repeated indices); it suffices to pass the indices as a string inside square brackets:sage: a['^i_k']*b['^kj'] == s True
Indices not involved in the contraction may be replaced by dots:
sage: a['^._k']*b['^k.'] == s True
LaTeX notation may be used:
sage: a['^{i}_{k}']*b['^{kj}'] == s True
Contraction on the last index of
a
and last index ofb
:sage: s = a.contract(b, 1) ; s Tensor field of type (2,0) on the 2-dimensional differentiable manifold M sage: a['^i_k']*b['^jk'] == s True
Contraction on the first index of
b
and the last index ofa
:sage: s = b.contract(0,a,1) ; s Tensor field of type (2,0) on the 2-dimensional differentiable manifold M sage: b['^ki']*a['^j_k'] == s True
The domain of the result is the intersection of the domains of the two tensor fields:
sage: aU = a.restrict(U) ; bV = b.restrict(V) sage: s = aU.contract(b) ; s Tensor field of type (2,0) on the Open subset U of the 2-dimensional differentiable manifold M sage: s = a.contract(bV) ; s Tensor field of type (2,0) on the Open subset V of the 2-dimensional differentiable manifold M sage: s = aU.contract(bV) ; s Tensor field of type (2,0) on the Open subset W of the 2-dimensional differentiable manifold M sage: s0 = a.contract(b) sage: s == s0.restrict(W) True
The contraction can be performed on more than one index:
c
being a type-\((2,2)\) tensor, contracting the indices in positions 2 and 3 ofc
with respectively those in positions 0 and 1 ofb
is:sage: c = a*a ; c Tensor field of type (2,2) on the 2-dimensional differentiable manifold M sage: s = c.contract(2,3, b, 0,1) ; s # long time Tensor field of type (2,0) on the 2-dimensional differentiable manifold M
The same double contraction using index notation:
sage: s == c['^.._kl']*b['^kl'] # long time True
The symmetries are either conserved or destroyed by the contraction:
sage: c = c.symmetrize(0,1).antisymmetrize(2,3) sage: c.symmetries() symmetry: (0, 1); antisymmetry: (2, 3) sage: s = b.contract(0, c, 2) ; s Tensor field of type (3,1) on the 2-dimensional differentiable manifold M sage: s.symmetries() symmetry: (1, 2); no antisymmetry
Case of a scalar field result:
sage: a = M.one_form({eU: [y, 1+x]}, name='a') sage: a.add_comp_by_continuation(eV, W, chart=c_uv) sage: b = M.vector_field({eU: [x, y^2]}, name='b') sage: b.add_comp_by_continuation(eV, W, chart=c_uv) sage: a.display(eU) a = y dx + (x + 1) dy sage: b.display(eU) b = x ∂/∂x + y^2 ∂/∂y sage: s = a.contract(b) ; s Scalar field on the 2-dimensional differentiable manifold M sage: s.display() M → ℝ on U: (x, y) ↦ (x + 1)*y^2 + x*y on V: (u, v) ↦ 1/8*u^3 - 1/8*u*v^2 + 1/8*v^3 + 1/2*u^2 - 1/8*(u^2 + 4*u)*v sage: s == a['_i']*b['^i'] # use of index notation True sage: s == b.contract(a) True
Case of a vanishing scalar field result:
sage: b = M.vector_field({eU: [1+x, -y]}, name='b') sage: b.add_comp_by_continuation(eV, W, chart=c_uv) sage: s = a.contract(b) ; s Scalar field zero on the 2-dimensional differentiable manifold M sage: s.display() zero: M → ℝ on U: (x, y) ↦ 0 on V: (u, v) ↦ 0
- copy(name=None, latex_name=None)#
Return an exact copy of
self
.INPUT:
name
– (default:None
) name given to the copylatex_name
– (default:None
) LaTeX symbol to denote the copy; if none is provided, the LaTeX symbol is set toname
Note
The name and the derived quantities are not copied.
EXAMPLES:
Copy of a type-\((1,1)\) tensor field on a 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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y), ....: intersection_name='W', restrictions1= x>0, ....: restrictions2= u+v>0) sage: uv_to_xy = xy_to_uv.inverse() sage: e_xy = c_xy.frame(); e_uv = c_uv.frame() sage: t = M.tensor_field(1, 1, name='t') sage: t[e_xy,:] = [[x+y, 0], [2, 1-y]] sage: t.add_comp_by_continuation(e_uv, U.intersection(V), c_uv) sage: s = t.copy(); s Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: s.display(e_xy) (x + y) ∂/∂x⊗dx + 2 ∂/∂y⊗dx + (-y + 1) ∂/∂y⊗dy sage: s == t True
If the original tensor field is modified, the copy is not:
sage: t[e_xy,0,0] = -1 sage: t.display(e_xy) t = -∂/∂x⊗dx + 2 ∂/∂y⊗dx + (-y + 1) ∂/∂y⊗dy sage: s.display(e_xy) (x + y) ∂/∂x⊗dx + 2 ∂/∂y⊗dx + (-y + 1) ∂/∂y⊗dy sage: s == t False
- copy_from(other)#
Make
self
a copy ofother
.INPUT:
other
– other tensor field, in the same module asself
Note
While the derived quantities are not copied, the name is kept.
Warning
All previous defined components and restrictions will be deleted!
EXAMPLES:
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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y), ....: intersection_name='W', restrictions1= x>0, ....: restrictions2= u+v>0) sage: uv_to_xy = xy_to_uv.inverse() sage: e_xy = c_xy.frame(); e_uv = c_uv.frame() sage: t = M.tensor_field(1, 1, name='t') sage: t[e_xy,:] = [[x+y, 0], [2, 1-y]] sage: t.add_comp_by_continuation(e_uv, U.intersection(V), c_uv) sage: s = M.tensor_field(1, 1, name='s') sage: s.copy_from(t) sage: s.display(e_xy) s = (x + y) ∂/∂x⊗dx + 2 ∂/∂y⊗dx + (-y + 1) ∂/∂y⊗dy sage: s == t True
While the original tensor field is modified, the copy is not:
sage: t[e_xy,0,0] = -1 sage: t.display(e_xy) t = -∂/∂x⊗dx + 2 ∂/∂y⊗dx + (-y + 1) ∂/∂y⊗dy sage: s.display(e_xy) s = (x + y) ∂/∂x⊗dx + 2 ∂/∂y⊗dx + (-y + 1) ∂/∂y⊗dy sage: s == t False
- dalembertian(metric=None)#
Return the d’Alembertian of
self
with respect to a given Lorentzian metric.The d’Alembertian of a tensor field \(t\) with respect to a Lorentzian metric \(g\) is nothing but the Laplace-Beltrami operator of \(g\) applied to \(t\) (see
laplacian()
); ifself
a tensor field \(t\) of type \((k,l)\), the d’Alembertian of \(t\) with respect to \(g\) is then the tensor field of type \((k,l)\) defined by\[(\Box t)^{a_1\ldots a_k}_{\phantom{a_1\ldots a_k}\,{b_1\ldots b_k}} = \nabla_i \nabla^i t^{a_1\ldots a_k}_{\phantom{a_1\ldots a_k}\,{b_1\ldots b_k}},\]where \(\nabla\) is the Levi-Civita connection of \(g\) (cf.
LeviCivitaConnection
) and \(\nabla^i := g^{ij} \nabla_j\).Note
If the metric \(g\) is not Lorentzian, the name d’Alembertian is not appropriate and one should use
laplacian()
instead.INPUT:
metric
– (default:None
) the Lorentzian metric \(g\) involved in the definition of the d’Alembertian; if none is provided, the domain ofself
is supposed to be endowed with a default Lorentzian metric (i.e. is supposed to be Lorentzian manifold, seePseudoRiemannianManifold
) and the latter is used to define the d’Alembertian
OUTPUT:
instance of
TensorField
representing the d’Alembertian ofself
EXAMPLES:
d’Alembertian of a vector field in Minkowski spacetime, representing the electric field of a simple plane electromagnetic wave:
sage: M = Manifold(4, 'M', structure='Lorentzian') sage: X.<t,x,y,z> = M.chart() sage: g = M.metric() sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1, 1, 1, 1 sage: e = M.vector_field(name='e') sage: e[1] = cos(t-z) sage: e.display() # plane wave propagating in the z direction e = cos(t - z) ∂/∂x sage: De = e.dalembertian(); De # long time Vector field Box(e) on the 4-dimensional Lorentzian manifold M
The function
dalembertian()
from theoperators
module can be used instead of the methoddalembertian()
:sage: from sage.manifolds.operators import dalembertian sage: dalembertian(e) == De # long time True
We check that the electric field obeys the wave equation:
sage: De.display() # long time Box(e) = 0
- disp(frame=None, chart=None)#
Display the tensor field in terms of its expansion with respect to a given vector frame.
The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).
INPUT:
frame
– (default:None
) vector frame with respect to which the tensor is expanded; ifframe
isNone
andchart
is notNone
, the coordinate frame associated withchart
is assumed; if bothframe
andchart
areNone
, the default frame of the domain of definition of the tensor field is assumedchart
– (default:None
) chart with respect to which the components of the tensor field in the selected frame are expressed; ifNone
, the default chart of the vector frame domain is assumed
EXAMPLES:
Display of a type-\((1,1)\) tensor field on a 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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y), ....: intersection_name='W', restrictions1= x>0, ....: restrictions2= u+v>0) sage: uv_to_xy = xy_to_uv.inverse() sage: W = U.intersection(V) sage: e_xy = c_xy.frame(); e_uv = c_uv.frame() sage: t = M.tensor_field(1,1, name='t') sage: t[e_xy,:] = [[x, 1], [y, 0]] sage: t.add_comp_by_continuation(e_uv, W, c_uv) sage: t.display(e_xy) t = x ∂/∂x⊗dx + ∂/∂x⊗dy + y ∂/∂y⊗dx sage: t.display(e_uv) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
Since
e_xy
isM
’s default frame, the argumente_xy
can be omitted:sage: e_xy is M.default_frame() True sage: t.display() t = x ∂/∂x⊗dx + ∂/∂x⊗dy + y ∂/∂y⊗dx
Similarly, since
e_uv
isV
’s default frame, the argumente_uv
can be omitted when considering the restriction oft
toV
:sage: t.restrict(V).display() t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
If the coordinate expression of the components are to be displayed in a chart distinct from the default one on the considered domain, then the chart has to be passed as the second argument of
display
. For instance, on \(W = U \cap V\), two charts are available:c_xy.restrict(W)
(the default one) andc_uv.restrict(W)
. Accordingly, one can have two views of the expansion oft
in the same vector framee_uv.restrict(W)
:sage: t.display(e_uv.restrict(W)) # W's default chart assumed t = (1/2*x + 1/2*y + 1/2) ∂/∂u⊗du + (1/2*x + 1/2*y - 1/2) ∂/∂u⊗dv + (1/2*x - 1/2*y + 1/2) ∂/∂v⊗du + (1/2*x - 1/2*y - 1/2) ∂/∂v⊗dv sage: t.display(e_uv.restrict(W), c_uv.restrict(W)) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
As a shortcut, one can pass just a chart to
display
. It is then understood that the expansion is to be performed with respect to the coordinate frame associated with this chart. Therefore the above command can be abridged to:sage: t.display(c_uv.restrict(W)) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
and one has:
sage: t.display(c_xy) t = x ∂/∂x⊗dx + ∂/∂x⊗dy + y ∂/∂y⊗dx sage: t.display(c_uv) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv sage: t.display(c_xy.restrict(W)) t = x ∂/∂x⊗dx + ∂/∂x⊗dy + y ∂/∂y⊗dx sage: t.restrict(W).display(c_uv.restrict(W)) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
One can ask for the display with respect to a frame in which
t
has not been initialized yet (this will automatically trigger the use of the change-of-frame formula for tensors):sage: a = V.automorphism_field() sage: a[:] = [[1+v, -u^2], [0, 1-u]] sage: f = e_uv.new_frame(a, 'f') sage: [f[i].display() for i in M.irange()] [f_0 = (v + 1) ∂/∂u, f_1 = -u^2 ∂/∂u + (-u + 1) ∂/∂v] sage: t.display(f) t = -1/2*(u^2*v + 1)/(u - 1) f_0⊗f^0 - 1/2*(2*u^3 - 5*u^2 - (u^4 + u^3 - u^2)*v + 3*u - 1)/((u - 1)*v + u - 1) f_0⊗f^1 - 1/2*(v^2 + 2*v + 1)/(u - 1) f_1⊗f^0 + 1/2*(u^2 + (u^2 + u - 1)*v - u + 1)/(u - 1) f_1⊗f^1
A shortcut of
display()
isdisp()
:sage: t.disp(e_uv) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
- display(frame=None, chart=None)#
Display the tensor field in terms of its expansion with respect to a given vector frame.
The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).
INPUT:
frame
– (default:None
) vector frame with respect to which the tensor is expanded; ifframe
isNone
andchart
is notNone
, the coordinate frame associated withchart
is assumed; if bothframe
andchart
areNone
, the default frame of the domain of definition of the tensor field is assumedchart
– (default:None
) chart with respect to which the components of the tensor field in the selected frame are expressed; ifNone
, the default chart of the vector frame domain is assumed
EXAMPLES:
Display of a type-\((1,1)\) tensor field on a 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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y), ....: intersection_name='W', restrictions1= x>0, ....: restrictions2= u+v>0) sage: uv_to_xy = xy_to_uv.inverse() sage: W = U.intersection(V) sage: e_xy = c_xy.frame(); e_uv = c_uv.frame() sage: t = M.tensor_field(1,1, name='t') sage: t[e_xy,:] = [[x, 1], [y, 0]] sage: t.add_comp_by_continuation(e_uv, W, c_uv) sage: t.display(e_xy) t = x ∂/∂x⊗dx + ∂/∂x⊗dy + y ∂/∂y⊗dx sage: t.display(e_uv) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
Since
e_xy
isM
’s default frame, the argumente_xy
can be omitted:sage: e_xy is M.default_frame() True sage: t.display() t = x ∂/∂x⊗dx + ∂/∂x⊗dy + y ∂/∂y⊗dx
Similarly, since
e_uv
isV
’s default frame, the argumente_uv
can be omitted when considering the restriction oft
toV
:sage: t.restrict(V).display() t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
If the coordinate expression of the components are to be displayed in a chart distinct from the default one on the considered domain, then the chart has to be passed as the second argument of
display
. For instance, on \(W = U \cap V\), two charts are available:c_xy.restrict(W)
(the default one) andc_uv.restrict(W)
. Accordingly, one can have two views of the expansion oft
in the same vector framee_uv.restrict(W)
:sage: t.display(e_uv.restrict(W)) # W's default chart assumed t = (1/2*x + 1/2*y + 1/2) ∂/∂u⊗du + (1/2*x + 1/2*y - 1/2) ∂/∂u⊗dv + (1/2*x - 1/2*y + 1/2) ∂/∂v⊗du + (1/2*x - 1/2*y - 1/2) ∂/∂v⊗dv sage: t.display(e_uv.restrict(W), c_uv.restrict(W)) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
As a shortcut, one can pass just a chart to
display
. It is then understood that the expansion is to be performed with respect to the coordinate frame associated with this chart. Therefore the above command can be abridged to:sage: t.display(c_uv.restrict(W)) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
and one has:
sage: t.display(c_xy) t = x ∂/∂x⊗dx + ∂/∂x⊗dy + y ∂/∂y⊗dx sage: t.display(c_uv) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv sage: t.display(c_xy.restrict(W)) t = x ∂/∂x⊗dx + ∂/∂x⊗dy + y ∂/∂y⊗dx sage: t.restrict(W).display(c_uv.restrict(W)) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
One can ask for the display with respect to a frame in which
t
has not been initialized yet (this will automatically trigger the use of the change-of-frame formula for tensors):sage: a = V.automorphism_field() sage: a[:] = [[1+v, -u^2], [0, 1-u]] sage: f = e_uv.new_frame(a, 'f') sage: [f[i].display() for i in M.irange()] [f_0 = (v + 1) ∂/∂u, f_1 = -u^2 ∂/∂u + (-u + 1) ∂/∂v] sage: t.display(f) t = -1/2*(u^2*v + 1)/(u - 1) f_0⊗f^0 - 1/2*(2*u^3 - 5*u^2 - (u^4 + u^3 - u^2)*v + 3*u - 1)/((u - 1)*v + u - 1) f_0⊗f^1 - 1/2*(v^2 + 2*v + 1)/(u - 1) f_1⊗f^0 + 1/2*(u^2 + (u^2 + u - 1)*v - u + 1)/(u - 1) f_1⊗f^1
A shortcut of
display()
isdisp()
:sage: t.disp(e_uv) t = (1/2*u + 1/2) ∂/∂u⊗du + (1/2*u - 1/2) ∂/∂u⊗dv + (1/2*v + 1/2) ∂/∂v⊗du + (1/2*v - 1/2) ∂/∂v⊗dv
- display_comp(frame=None, chart=None, coordinate_labels=True, only_nonzero=True, only_nonredundant=False)#
Display the tensor components with respect to a given frame, one per line.
The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).
INPUT:
frame
– (default:None
) vector frame with respect to which the tensor field components are defined; ifNone
, thenif
chart
is notNone
, the coordinate frame associated tochart
is usedotherwise, the default basis of the vector field module on which the tensor field is defined is used
chart
– (default:None
) chart specifying the coordinate expression of the components; ifNone
, the default chart of the tensor field domain is usedcoordinate_labels
– (default:True
) boolean; ifTrue
, coordinate symbols are used by default (instead of integers) as index labels wheneverframe
is a coordinate frameonly_nonzero
– (default:True
) boolean; ifTrue
, only nonzero components are displayedonly_nonredundant
– (default:False
) boolean; ifTrue
, only nonredundant components are displayed in case of symmetries
EXAMPLES:
Display of the components of a type-\((1,1)\) tensor field defined on two open subsets:
sage: M = Manifold(2, 'M') sage: U = M.open_subset('U') sage: c_xy.<x, y> = U.chart() sage: e = U.default_frame() sage: V = M.open_subset('V') sage: c_uv.<u, v> = V.chart() sage: f = V.default_frame() sage: M.declare_union(U,V) # M is the union of U and V sage: t = M.tensor_field(1,1, name='t') sage: t[e,0,0] = - x + y^3 sage: t[e,0,1] = 2+x sage: t[f,1,1] = - u*v sage: t.display_comp(e) t^x_x = y^3 - x t^x_y = x + 2 sage: t.display_comp(f) t^v_v = -u*v
Components in a chart frame:
sage: t.display_comp(chart=c_xy) t^x_x = y^3 - x t^x_y = x + 2 sage: t.display_comp(chart=c_uv) t^v_v = -u*v
See documentation of
sage.manifolds.differentiable.tensorfield_paral.TensorFieldParal.display_comp()
for more options.
- div(metric=None)#
Return the divergence of
self
(with respect to a given metric).The divergence is taken on the last index: if
self
is a tensor field \(t\) of type \((k,0)\) with \(k\geq 1\), the divergence of \(t\) with respect to the metric \(g\) is the tensor field of type \((k-1,0)\) defined by\[(\mathrm{div}\, t)^{a_1\ldots a_{k-1}} = \nabla_i t^{a_1\ldots a_{k-1} i} = (\nabla t)^{a_1\ldots a_{k-1} i}_{\phantom{a_1\ldots a_{k-1} i}\, i},\]where \(\nabla\) is the Levi-Civita connection of \(g\) (cf.
LeviCivitaConnection
).This definition is extended to tensor fields of type \((k,l)\) with \(k\geq 0\) and \(l\geq 1\), by raising the last index with the metric \(g\): \(\mathrm{div}\, t\) is then the tensor field of type \((k,l-1)\) defined by
\[(\mathrm{div}\, t)^{a_1\ldots a_k}_{\phantom{a_1\ldots a_k}\, b_1 \ldots b_{l-1}} = \nabla_i (g^{ij} t^{a_1\ldots a_k}_{\phantom{a_1 \ldots a_k}\, b_1\ldots b_{l-1} j}) = (\nabla t^\sharp)^{a_1\ldots a_k i}_{\phantom{a_1\ldots a_k i}\, b_1\ldots b_{l-1} i},\]where \(t^\sharp\) is the tensor field deduced from \(t\) by raising the last index with the metric \(g\) (see
up()
).INPUT:
metric
– (default:None
) the pseudo-Riemannian metric \(g\) involved in the definition of the divergence; if none is provided, the domain ofself
is supposed to be endowed with a default metric (i.e. is supposed to be pseudo-Riemannian manifold, seePseudoRiemannianManifold
) and the latter is used to define the divergence.
OUTPUT:
instance of either
DiffScalarField
if \((k,l)=(1,0)\) (self
is a vector field) or \((k,l)=(0,1)\) (self
is a 1-form) or ofTensorField
if \(k+l\geq 2\) representing the divergence ofself
with respect tometric
EXAMPLES:
Divergence of a vector field in the Euclidean plane:
sage: M.<x,y> = EuclideanSpace() sage: v = M.vector_field(x, y, name='v') sage: s = v.divergence(); s Scalar field div(v) on the Euclidean plane E^2 sage: s.display() div(v): E^2 → ℝ (x, y) ↦ 2
A shortcut alias of
divergence
isdiv
:sage: v.div() == s True
The function
div()
from theoperators
module can be used instead of the methoddivergence()
:sage: from sage.manifolds.operators import div sage: div(v) == s True
The divergence can be taken with respect to a metric tensor that is not the default one:
sage: h = M.lorentzian_metric('h') sage: h[1,1], h[2,2] = -1, 1/(1+x^2+y^2) sage: s = v.div(h); s Scalar field div_h(v) on the Euclidean plane E^2 sage: s.display() div_h(v): E^2 → ℝ (x, y) ↦ (x^2 + y^2 + 2)/(x^2 + y^2 + 1)
The standard formula
\[\mathrm{div}_h \, v = \frac{1}{\sqrt{|\det h|}} \frac{\partial}{\partial x^i} \left( \sqrt{|\det h|} \, v^i \right)\]is checked as follows:
sage: sqrth = h.sqrt_abs_det().expr(); sqrth 1/sqrt(x^2 + y^2 + 1) sage: s == 1/sqrth * sum( (sqrth*v[i]).diff(i) for i in M.irange()) True
A divergence-free vector:
sage: w = M.vector_field(-y, x, name='w') sage: w.div().display() div(w): E^2 → ℝ (x, y) ↦ 0 sage: w.div(h).display() div_h(w): E^2 → ℝ (x, y) ↦ 0
Divergence of a type-
(2,0)
tensor field:sage: t = v*w; t Tensor field v⊗w of type (2,0) on the Euclidean plane E^2 sage: s = t.div(); s Vector field div(v⊗w) on the Euclidean plane E^2 sage: s.display() div(v⊗w) = -y e_x + x e_y
- divergence(metric=None)#
Return the divergence of
self
(with respect to a given metric).The divergence is taken on the last index: if
self
is a tensor field \(t\) of type \((k,0)\) with \(k\geq 1\), the divergence of \(t\) with respect to the metric \(g\) is the tensor field of type \((k-1,0)\) defined by\[(\mathrm{div}\, t)^{a_1\ldots a_{k-1}} = \nabla_i t^{a_1\ldots a_{k-1} i} = (\nabla t)^{a_1\ldots a_{k-1} i}_{\phantom{a_1\ldots a_{k-1} i}\, i},\]where \(\nabla\) is the Levi-Civita connection of \(g\) (cf.
LeviCivitaConnection
).This definition is extended to tensor fields of type \((k,l)\) with \(k\geq 0\) and \(l\geq 1\), by raising the last index with the metric \(g\): \(\mathrm{div}\, t\) is then the tensor field of type \((k,l-1)\) defined by
\[(\mathrm{div}\, t)^{a_1\ldots a_k}_{\phantom{a_1\ldots a_k}\, b_1 \ldots b_{l-1}} = \nabla_i (g^{ij} t^{a_1\ldots a_k}_{\phantom{a_1 \ldots a_k}\, b_1\ldots b_{l-1} j}) = (\nabla t^\sharp)^{a_1\ldots a_k i}_{\phantom{a_1\ldots a_k i}\, b_1\ldots b_{l-1} i},\]where \(t^\sharp\) is the tensor field deduced from \(t\) by raising the last index with the metric \(g\) (see
up()
).INPUT:
metric
– (default:None
) the pseudo-Riemannian metric \(g\) involved in the definition of the divergence; if none is provided, the domain ofself
is supposed to be endowed with a default metric (i.e. is supposed to be pseudo-Riemannian manifold, seePseudoRiemannianManifold
) and the latter is used to define the divergence.
OUTPUT:
instance of either
DiffScalarField
if \((k,l)=(1,0)\) (self
is a vector field) or \((k,l)=(0,1)\) (self
is a 1-form) or ofTensorField
if \(k+l\geq 2\) representing the divergence ofself
with respect tometric
EXAMPLES:
Divergence of a vector field in the Euclidean plane:
sage: M.<x,y> = EuclideanSpace() sage: v = M.vector_field(x, y, name='v') sage: s = v.divergence(); s Scalar field div(v) on the Euclidean plane E^2 sage: s.display() div(v): E^2 → ℝ (x, y) ↦ 2
A shortcut alias of
divergence
isdiv
:sage: v.div() == s True
The function
div()
from theoperators
module can be used instead of the methoddivergence()
:sage: from sage.manifolds.operators import div sage: div(v) == s True
The divergence can be taken with respect to a metric tensor that is not the default one:
sage: h = M.lorentzian_metric('h') sage: h[1,1], h[2,2] = -1, 1/(1+x^2+y^2) sage: s = v.div(h); s Scalar field div_h(v) on the Euclidean plane E^2 sage: s.display() div_h(v): E^2 → ℝ (x, y) ↦ (x^2 + y^2 + 2)/(x^2 + y^2 + 1)
The standard formula
\[\mathrm{div}_h \, v = \frac{1}{\sqrt{|\det h|}} \frac{\partial}{\partial x^i} \left( \sqrt{|\det h|} \, v^i \right)\]is checked as follows:
sage: sqrth = h.sqrt_abs_det().expr(); sqrth 1/sqrt(x^2 + y^2 + 1) sage: s == 1/sqrth * sum( (sqrth*v[i]).diff(i) for i in M.irange()) True
A divergence-free vector:
sage: w = M.vector_field(-y, x, name='w') sage: w.div().display() div(w): E^2 → ℝ (x, y) ↦ 0 sage: w.div(h).display() div_h(w): E^2 → ℝ (x, y) ↦ 0
Divergence of a type-
(2,0)
tensor field:sage: t = v*w; t Tensor field v⊗w of type (2,0) on the Euclidean plane E^2 sage: s = t.div(); s Vector field div(v⊗w) on the Euclidean plane E^2 sage: s.display() div(v⊗w) = -y e_x + x e_y
- domain()#
Return the manifold on which
self
is defined.OUTPUT:
instance of class
DifferentiableManifold
EXAMPLES:
sage: M = Manifold(2, 'M') sage: c_xy.<x,y> = M.chart() sage: t = M.tensor_field(1,2) sage: t.domain() 2-dimensional differentiable manifold M sage: U = M.open_subset('U', coord_def={c_xy: x<0}) sage: h = t.restrict(U) sage: h.domain() Open subset U of the 2-dimensional differentiable manifold M
- down(non_degenerate_form, pos=None)#
Compute a dual of the tensor field by lowering some index with a given non-degenerate form (pseudo-Riemannian metric or symplectic form).
If \(T\) is the tensor field, \((k,l)\) its type and \(p\) the position of a contravariant index (i.e. \(0\leq p < k\)), this method called with
pos
\(=p\) yields the tensor field \(T^\flat\) of type \((k-1,l+1)\) whose components are\[(T^\flat)^{a_1\ldots a_{k-1}}_{\phantom{a_1\ldots a_{k-1}} \, b_1 \ldots b_{l+1}} = g_{i b_1} \, T^{a_1\ldots a_{p} \, i \, a_{p+1}\ldots a_{k-1}}_{\phantom{a_1 \ldots a_{p} \, i \, a_{p+1}\ldots a_{k-1}}\, b_2 \ldots b_{l+1}},\]\(g_{ab}\) being the components of the metric tensor or the symplectic form, respectively.
The reverse operation is
TensorField.up()
.INPUT:
non_degenerate_form
– non-degenerate form \(g\)pos
– (default:None
) position of the index (with the conventionpos=0
for the first index); ifNone
, the lowering is performed over all the contravariant indices, starting from the last one
OUTPUT:
the tensor field \(T^\flat\) resulting from the index lowering operation
EXAMPLES:
Lowering the index of a vector field results in a 1-form:
sage: M = Manifold(2, 'M', start_index=1) sage: c_xy.<x,y> = M.chart() sage: g = M.metric('g') sage: g[1,1], g[1,2], g[2,2] = 1+x, x*y, 1-y sage: v = M.vector_field(-1, 2) sage: w = v.down(g) ; w 1-form on the 2-dimensional differentiable manifold M sage: w.display() (2*x*y - x - 1) dx + (-(x + 2)*y + 2) dy
Using the index notation instead of
down()
:sage: w == g['_ab']*v['^b'] True
The reverse operation:
sage: v1 = w.up(g) ; v1 Vector field on the 2-dimensional differentiable manifold M sage: v1 == v True
Lowering the indices of a tensor field of type (2,0):
sage: t = M.tensor_field(2, 0, [[1,2], [3,4]]) sage: td0 = t.down(g, 0) ; td0 # lowering the first index Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: td0 == g['_ac']*t['^cb'] # the same operation in index notation True sage: td0[:] [ 3*x*y + x + 1 (x - 3)*y + 3] [4*x*y + 2*x + 2 2*(x - 2)*y + 4] sage: tdd0 = td0.down(g) ; tdd0 # the two indices have been lowered, starting from the first one Tensor field of type (0,2) on the 2-dimensional differentiable manifold M sage: tdd0 == g['_ac']*td0['^c_b'] # the same operation in index notation True sage: tdd0[:] [ 4*x^2*y^2 + x^2 + 5*(x^2 + x)*y + 2*x + 1 2*(x^2 - 2*x)*y^2 + (x^2 + 2*x - 3)*y + 3*x + 3] [(3*x^2 - 4*x)*y^2 + (x^2 + 3*x - 2)*y + 2*x + 2 (x^2 - 5*x + 4)*y^2 + (5*x - 8)*y + 4] sage: td1 = t.down(g, 1) ; td1 # lowering the second index Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: td1 == g['_ac']*t['^bc'] # the same operation in index notation True sage: td1[:] [ 2*x*y + x + 1 (x - 2)*y + 2] [4*x*y + 3*x + 3 (3*x - 4)*y + 4] sage: tdd1 = td1.down(g) ; tdd1 # the two indices have been lowered, starting from the second one Tensor field of type (0,2) on the 2-dimensional differentiable manifold M sage: tdd1 == g['_ac']*td1['^c_b'] # the same operation in index notation True sage: tdd1[:] [ 4*x^2*y^2 + x^2 + 5*(x^2 + x)*y + 2*x + 1 (3*x^2 - 4*x)*y^2 + (x^2 + 3*x - 2)*y + 2*x + 2] [2*(x^2 - 2*x)*y^2 + (x^2 + 2*x - 3)*y + 3*x + 3 (x^2 - 5*x + 4)*y^2 + (5*x - 8)*y + 4] sage: tdd1 == tdd0 # the order of index lowering is important False sage: tdd = t.down(g) ; tdd # both indices are lowered, starting from the last one Tensor field of type (0,2) on the 2-dimensional differentiable manifold M sage: tdd[:] [ 4*x^2*y^2 + x^2 + 5*(x^2 + x)*y + 2*x + 1 (3*x^2 - 4*x)*y^2 + (x^2 + 3*x - 2)*y + 2*x + 2] [2*(x^2 - 2*x)*y^2 + (x^2 + 2*x - 3)*y + 3*x + 3 (x^2 - 5*x + 4)*y^2 + (5*x - 8)*y + 4] sage: tdd0 == tdd # to get tdd0, indices have been lowered from the first one, contrary to tdd False sage: tdd1 == tdd # the same order for index lowering has been applied True sage: u0tdd = tdd.up(g, 0) ; u0tdd # the first index is raised again Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: uu0tdd = u0tdd.up(g) ; uu0tdd # the second index is then raised Tensor field of type (2,0) on the 2-dimensional differentiable manifold M sage: u1tdd = tdd.up(g, 1) ; u1tdd # raising operation, starting from the last index Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: uu1tdd = u1tdd.up(g) ; uu1tdd Tensor field of type (2,0) on the 2-dimensional differentiable manifold M sage: uutdd = tdd.up(g) ; uutdd # both indices are raised, starting from the first one Tensor field of type (2,0) on the 2-dimensional differentiable manifold M sage: uutdd == t # should be true True sage: uu0tdd == t # should be true True sage: uu1tdd == t # not true, because of the order of index raising to get uu1tdd False
- laplacian(metric=None)#
Return the Laplacian of
self
with respect to a given metric (Laplace-Beltrami operator).If
self
is a tensor field \(t\) of type \((k,l)\), the Laplacian of \(t\) with respect to the metric \(g\) is the tensor field of type \((k,l)\) defined by\[(\Delta t)^{a_1\ldots a_k}_{\phantom{a_1\ldots a_k}\,{b_1\ldots b_k}} = \nabla_i \nabla^i t^{a_1\ldots a_k}_{\phantom{a_1\ldots a_k}\,{b_1\ldots b_k}},\]where \(\nabla\) is the Levi-Civita connection of \(g\) (cf.
LeviCivitaConnection
) and \(\nabla^i := g^{ij} \nabla_j\). The operator \(\Delta = \nabla_i \nabla^i\) is called the Laplace-Beltrami operator of metric \(g\).INPUT:
metric
– (default:None
) the pseudo-Riemannian metric \(g\) involved in the definition of the Laplacian; if none is provided, the domain ofself
is supposed to be endowed with a default metric (i.e. is supposed to be pseudo-Riemannian manifold, seePseudoRiemannianManifold
) and the latter is used to define the Laplacian
OUTPUT:
instance of
TensorField
representing the Laplacian ofself
EXAMPLES:
Laplacian of a vector field in the Euclidean plane:
sage: M.<x,y> = EuclideanSpace() sage: v = M.vector_field(x^3 + y^2, x*y, name='v') sage: Dv = v.laplacian(); Dv Vector field Delta(v) on the Euclidean plane E^2 sage: Dv.display() Delta(v) = (6*x + 2) e_x
The function
laplacian()
from theoperators
module can be used instead of the methodlaplacian()
:sage: from sage.manifolds.operators import laplacian sage: laplacian(v) == Dv True
In the present case (Euclidean metric and Cartesian coordinates), the components of the Laplacian are the Laplacians of the components:
sage: all(Dv[[i]] == laplacian(v[[i]]) for i in M.irange()) True
The Laplacian can be taken with respect to a metric tensor that is not the default one:
sage: h = M.lorentzian_metric('h') sage: h[1,1], h[2,2] = -1, 1+x^2 sage: Dv = v.laplacian(h); Dv Vector field Delta_h(v) on the Euclidean plane E^2 sage: Dv.display() Delta_h(v) = -(8*x^5 - 2*x^4 - x^2*y^2 + 15*x^3 - 4*x^2 + 6*x - 2)/(x^4 + 2*x^2 + 1) e_x - 3*x^3*y/(x^4 + 2*x^2 + 1) e_y
- lie_der(vector)#
Lie derivative of
self
with respect to a vector field.INPUT:
vector
– vector field with respect to which the Lie derivative is to be taken
OUTPUT:
the tensor field that is the Lie derivative of the current tensor field with respect to
vector
EXAMPLES:
Lie derivative of a type-\((1,1)\) tensor field along a vector field 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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y), ....: intersection_name='W', restrictions1= x>0, ....: restrictions2= u+v>0) sage: uv_to_xy = xy_to_uv.inverse() sage: e_xy = c_xy.frame(); e_uv = c_uv.frame() sage: t = M.tensor_field(1, 1, {e_xy: [[x, 1], [y, 0]]}, name='t') sage: t.add_comp_by_continuation(e_uv, U.intersection(V), c_uv) sage: w = M.vector_field({e_xy: [-y, x]}, name='w') sage: w.add_comp_by_continuation(e_uv, U.intersection(V), c_uv) sage: lt = t.lie_derivative(w); lt Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: lt.display(e_xy) ∂/∂x⊗dx - x ∂/∂x⊗dy + (-y - 1) ∂/∂y⊗dy sage: lt.display(e_uv) -1/2*u ∂/∂u⊗du + (1/2*u + 1) ∂/∂u⊗dv + (-1/2*v + 1) ∂/∂v⊗du + 1/2*v ∂/∂v⊗dv
The result is cached:
sage: t.lie_derivative(w) is lt True
An alias is
lie_der
:sage: t.lie_der(w) is t.lie_derivative(w) True
Lie derivative of a vector field:
sage: a = M.vector_field({e_xy: [1-x, x-y]}, name='a') sage: a.add_comp_by_continuation(e_uv, U.intersection(V), c_uv) sage: a.lie_der(w) Vector field on the 2-dimensional differentiable manifold M sage: a.lie_der(w).display(e_xy) x ∂/∂x + (-y - 1) ∂/∂y sage: a.lie_der(w).display(e_uv) (v - 1) ∂/∂u + (u + 1) ∂/∂v
The Lie derivative is antisymmetric:
sage: a.lie_der(w) == - w.lie_der(a) True
and it coincides with the commutator of the two vector fields:
sage: f = M.scalar_field({c_xy: 3*x-1, c_uv: 3/2*(u+v)-1}) sage: a.lie_der(w)(f) == w(a(f)) - a(w(f)) # long time True
- lie_derivative(vector)#
Lie derivative of
self
with respect to a vector field.INPUT:
vector
– vector field with respect to which the Lie derivative is to be taken
OUTPUT:
the tensor field that is the Lie derivative of the current tensor field with respect to
vector
EXAMPLES:
Lie derivative of a type-\((1,1)\) tensor field along a vector field 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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y), ....: intersection_name='W', restrictions1= x>0, ....: restrictions2= u+v>0) sage: uv_to_xy = xy_to_uv.inverse() sage: e_xy = c_xy.frame(); e_uv = c_uv.frame() sage: t = M.tensor_field(1, 1, {e_xy: [[x, 1], [y, 0]]}, name='t') sage: t.add_comp_by_continuation(e_uv, U.intersection(V), c_uv) sage: w = M.vector_field({e_xy: [-y, x]}, name='w') sage: w.add_comp_by_continuation(e_uv, U.intersection(V), c_uv) sage: lt = t.lie_derivative(w); lt Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: lt.display(e_xy) ∂/∂x⊗dx - x ∂/∂x⊗dy + (-y - 1) ∂/∂y⊗dy sage: lt.display(e_uv) -1/2*u ∂/∂u⊗du + (1/2*u + 1) ∂/∂u⊗dv + (-1/2*v + 1) ∂/∂v⊗du + 1/2*v ∂/∂v⊗dv
The result is cached:
sage: t.lie_derivative(w) is lt True
An alias is
lie_der
:sage: t.lie_der(w) is t.lie_derivative(w) True
Lie derivative of a vector field:
sage: a = M.vector_field({e_xy: [1-x, x-y]}, name='a') sage: a.add_comp_by_continuation(e_uv, U.intersection(V), c_uv) sage: a.lie_der(w) Vector field on the 2-dimensional differentiable manifold M sage: a.lie_der(w).display(e_xy) x ∂/∂x + (-y - 1) ∂/∂y sage: a.lie_der(w).display(e_uv) (v - 1) ∂/∂u + (u + 1) ∂/∂v
The Lie derivative is antisymmetric:
sage: a.lie_der(w) == - w.lie_der(a) True
and it coincides with the commutator of the two vector fields:
sage: f = M.scalar_field({c_xy: 3*x-1, c_uv: 3/2*(u+v)-1}) sage: a.lie_der(w)(f) == w(a(f)) - a(w(f)) # long time True
- restrict(subdomain, dest_map=None)#
Return the restriction of
self
to some subdomain.If the restriction has not been defined yet, it is constructed here.
INPUT:
subdomain
–DifferentiableManifold
; open subset \(U\) of the tensor field domain \(S\)dest_map
–DiffMap
(default:None
); destination map \(\Psi:\ U \rightarrow V\), where \(V\) is an open subset of the manifold \(M\) where the tensor field takes it values; ifNone
, the restriction of \(\Phi\) to \(U\) is used, \(\Phi\) being the differentiable map \(S \rightarrow M\) associated with the tensor field
OUTPUT:
TensorField
representing the restriction
EXAMPLES:
Restrictions of a vector field on the 2-sphere:
sage: M = Manifold(2, 'S^2', start_index=1) sage: U = M.open_subset('U') # the complement of the North pole sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole sage: eN = stereoN.frame() # the associated vector frame sage: V = M.open_subset('V') # the complement of the South pole sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole sage: eS = stereoS.frame() # the associated vector frame sage: transf = stereoN.transition_map(stereoS, (x/(x^2+y^2), y/(x^2+y^2)), ....: intersection_name='W', restrictions1= x^2+y^2!=0, ....: restrictions2= u^2+v^2!=0) sage: inv = transf.inverse() # transformation from stereoS to stereoN sage: W = U.intersection(V) # the complement of the North and South poles sage: stereoN_W = W.atlas()[0] # restriction of stereographic coord. from North pole to W sage: stereoS_W = W.atlas()[1] # restriction of stereographic coord. from South pole to W sage: eN_W = stereoN_W.frame() ; eS_W = stereoS_W.frame() sage: v = M.vector_field({eN: [1, 0]}, name='v') sage: v.display() v = ∂/∂x sage: vU = v.restrict(U) ; vU Vector field v on the Open subset U of the 2-dimensional differentiable manifold S^2 sage: vU.display() v = ∂/∂x sage: vU == eN[1] True sage: vW = v.restrict(W) ; vW Vector field v on the Open subset W of the 2-dimensional differentiable manifold S^2 sage: vW.display() v = ∂/∂x sage: vW.display(eS_W, stereoS_W) v = (-u^2 + v^2) ∂/∂u - 2*u*v ∂/∂v sage: vW == eN_W[1] True
At this stage, defining the restriction of
v
to the open subsetV
fully specifiesv
:sage: v.restrict(V)[1] = vW[eS_W, 1, stereoS_W].expr() # note that eS is the default frame on V sage: v.restrict(V)[2] = vW[eS_W, 2, stereoS_W].expr() sage: v.display(eS, stereoS) v = (-u^2 + v^2) ∂/∂u - 2*u*v ∂/∂v sage: v.restrict(U).display() v = ∂/∂x sage: v.restrict(V).display() v = (-u^2 + v^2) ∂/∂u - 2*u*v ∂/∂v
The restriction of the vector field to its own domain is of course itself:
sage: v.restrict(M) is v True sage: vU.restrict(U) is vU True
- set_calc_order(symbol, order, truncate=False)#
Trigger a series expansion with respect to a small parameter in computations involving the tensor field.
This property is propagated by usual operations. The internal representation must be
SR
for this to take effect.If the small parameter is \(\epsilon\) and \(T\) is
self
, the power series expansion to order \(n\) is\[T = T_0 + \epsilon T_1 + \epsilon^2 T_2 + \cdots + \epsilon^n T_n + O(\epsilon^{n+1}),\]where \(T_0, T_1, \ldots, T_n\) are \(n+1\) tensor fields of the same tensor type as
self
and do not depend upon \(\epsilon\).INPUT:
symbol
– symbolic variable (the “small parameter” \(\epsilon\)) with respect to which the components ofself
are expanded in power seriesorder
– integer; the order \(n\) of the expansion, defined as the degree of the polynomial representing the truncated power series insymbol
truncate
– (default:False
) determines whether the components ofself
are replaced by their expansions to the given order
EXAMPLES:
Let us consider two vector fields depending on a small parameter \(h\) on a non-parallelizable 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: a = M.vector_field() sage: h = var('h', domain='real') sage: a[eU,:] = (cos(h*x), -y) sage: a.add_comp_by_continuation(eV, W, chart=c_uv) sage: b = M.vector_field() sage: b[eU,:] = (exp(h*x), exp(h*y)) sage: b.add_comp_by_continuation(eV, W, chart=c_uv)
If we set the calculus order on one of the vector fields, any operation involving both of them is performed to that order:
sage: a.set_calc_order(h, 2) sage: s = a + b sage: s[eU,:] [h*x + 2, 1/2*h^2*y^2 + h*y - y + 1] sage: s[eV,:] [1/8*(u^2 - 2*u*v + v^2)*h^2 + h*u - 1/2*u + 1/2*v + 3, -1/8*(u^2 - 2*u*v + v^2)*h^2 + h*v + 1/2*u - 1/2*v + 1]
Note that the components of
a
have not been affected by the above call toset_calc_order
:sage: a[eU,:] [cos(h*x), -y] sage: a[eV,:] [cos(1/2*h*u)*cos(1/2*h*v) - sin(1/2*h*u)*sin(1/2*h*v) - 1/2*u + 1/2*v, cos(1/2*h*u)*cos(1/2*h*v) - sin(1/2*h*u)*sin(1/2*h*v) + 1/2*u - 1/2*v]
To have
set_calc_order
act on them, set the optional argumenttruncate
toTrue
:sage: a.set_calc_order(h, 2, truncate=True) sage: a[eU,:] [-1/2*h^2*x^2 + 1, -y] sage: a[eV,:] [-1/8*(u^2 + 2*u*v + v^2)*h^2 - 1/2*u + 1/2*v + 1, -1/8*(u^2 + 2*u*v + v^2)*h^2 + 1/2*u - 1/2*v + 1]
- set_comp(basis=None)#
Return the components of
self
in a given vector frame for assignment.The components with respect to other frames having the same domain as the provided vector frame are deleted, in order to avoid any inconsistency. To keep them, use the method
add_comp()
instead.INPUT:
basis
– (default:None
) vector frame in which the components are defined; if none is provided, the components are assumed to refer to the tensor field domain’s default frame
OUTPUT:
components in the given frame, as a
Components
; if such components did not exist previously, they are created
EXAMPLES:
sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2 sage: U = M.open_subset('U') # complement of the North pole sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole sage: V = M.open_subset('V') # complement of the South pole sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole sage: M.declare_union(U,V) # S^2 is the union of U and V sage: e_uv = c_uv.frame() sage: t = M.tensor_field(1, 2, name='t') sage: t.set_comp(e_uv) 3-indices components w.r.t. Coordinate frame (V, (∂/∂u,∂/∂v)) sage: t.set_comp(e_uv)[1,0,1] = u+v sage: t.display(e_uv) t = (u + v) ∂/∂v⊗du⊗dv
Setting the components in a new frame (
e
):sage: e = V.vector_frame('e') sage: t.set_comp(e) 3-indices components w.r.t. Vector frame (V, (e_0,e_1)) sage: t.set_comp(e)[0,1,1] = u*v sage: t.display(e) t = u*v e_0⊗e^1⊗e^1
Since the frames
e
ande_uv
are defined on the same domain, the components w.r.t.e_uv
have been erased:sage: t.display(c_uv.frame()) Traceback (most recent call last): ... ValueError: no basis could be found for computing the components in the Coordinate frame (V, (∂/∂u,∂/∂v))
Since zero is an immutable, its components cannot be changed:
sage: z = M.tensor_field_module((1, 1)).zero() sage: z.set_comp(e)[0,1] = u*v Traceback (most recent call last): ... ValueError: the components of an immutable element cannot be changed
- set_immutable()#
Set
self
and all restrictions ofself
immutable.EXAMPLES:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: U = M.open_subset('U', coord_def={X: x^2+y^2<1}) sage: a = M.tensor_field(1, 1, [[1+y,x], [0,x+y]], name='a') sage: aU = a.restrict(U) sage: a.set_immutable() sage: aU.is_immutable() True
- set_name(name=None, latex_name=None)#
Set (or change) the text name and LaTeX name of
self
.INPUT:
name
– string (default:None
); name given to the tensor fieldlatex_name
– string (default:None
); LaTeX symbol to denote the tensor field; ifNone
whilename
is provided, the LaTeX symbol is set toname
EXAMPLES:
sage: M = Manifold(2, 'M') sage: t = M.tensor_field(1, 3); t Tensor field of type (1,3) on the 2-dimensional differentiable manifold M sage: t.set_name(name='t') sage: t Tensor field t of type (1,3) on the 2-dimensional differentiable manifold M sage: latex(t) t sage: t.set_name(latex_name=r'\tau') sage: latex(t) \tau sage: t.set_name(name='a') sage: t Tensor field a of type (1,3) on the 2-dimensional differentiable manifold M sage: latex(t) a
- set_restriction(rst)#
Define a restriction of
self
to some subdomain.INPUT:
rst
–TensorField
of the same type and symmetries as the current tensor fieldself
, defined on a subdomain of the domain ofself
EXAMPLES:
sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2 sage: U = M.open_subset('U') # complement of the North pole sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole sage: V = M.open_subset('V') # complement of the South pole sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole sage: M.declare_union(U,V) # S^2 is the union of U and V sage: t = M.tensor_field(1, 2, name='t') sage: s = U.tensor_field(1, 2) sage: s[0,0,1] = x+y sage: t.set_restriction(s) sage: t.display(c_xy.frame()) t = (x + y) ∂/∂x⊗dx⊗dy sage: t.restrict(U) == s True
If the restriction is defined on the very same domain, the tensor field becomes a copy of it (see
copy_from()
):sage: v = M.tensor_field(1, 2, name='v') sage: v.set_restriction(t) sage: v.restrict(U) == t.restrict(U) True
- symmetries()#
Print the list of symmetries and antisymmetries.
EXAMPLES:
sage: M = Manifold(2, 'S^2') sage: t = M.tensor_field(1,2) sage: t.symmetries() no symmetry; no antisymmetry sage: t = M.tensor_field(1,2, sym=(1,2)) sage: t.symmetries() symmetry: (1, 2); no antisymmetry sage: t = M.tensor_field(2,2, sym=(0,1), antisym=(2,3)) sage: t.symmetries() symmetry: (0, 1); antisymmetry: (2, 3) sage: t = M.tensor_field(2,2, antisym=[(0,1),(2,3)]) sage: t.symmetries() no symmetry; antisymmetries: [(0, 1), (2, 3)]
- symmetrize(*pos)#
Symmetrization over some arguments.
INPUT:
pos
– (default:None
) list of argument positions involved in the symmetrization (with the conventionposition=0
for the first argument); ifNone
, the symmetrization is performed over all the arguments
OUTPUT:
the symmetrized tensor field (instance of
TensorField
)
EXAMPLES:
Symmetrization of a type-\((0,2)\) tensor field on a 2-dimensional non-parallelizable 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: a = M.tensor_field(0,2, {eU: [[1,x], [2,y]]}, name='a') sage: a.add_comp_by_continuation(eV, W, chart=c_uv) sage: a[eV,:] [ 1/4*u + 3/4 -1/4*u + 3/4] [ 1/4*v - 1/4 -1/4*v - 1/4] sage: s = a.symmetrize() ; s Field of symmetric bilinear forms on the 2-dimensional differentiable manifold M sage: s[eU,:] [ 1 1/2*x + 1] [1/2*x + 1 y] sage: s[eV,:] [ 1/4*u + 3/4 -1/8*u + 1/8*v + 1/4] [-1/8*u + 1/8*v + 1/4 -1/4*v - 1/4] sage: s == a.symmetrize(0,1) # explicit positions True
See also
For more details and examples, see
sage.tensor.modules.free_module_tensor.FreeModuleTensor.symmetrize()
.
- tensor_rank()#
Return the tensor rank of
self
.OUTPUT:
integer \(k+l\), where \(k\) is the contravariant rank and \(l\) is the covariant rank
EXAMPLES:
sage: M = Manifold(2, 'S^2') sage: t = M.tensor_field(1,2) sage: t.tensor_rank() 3 sage: v = M.vector_field() sage: v.tensor_rank() 1
- tensor_type()#
Return the tensor type of
self
.OUTPUT:
pair \((k,l)\), where \(k\) is the contravariant rank and \(l\) is the covariant rank
EXAMPLES:
sage: M = Manifold(2, 'S^2') sage: t = M.tensor_field(1,2) sage: t.tensor_type() (1, 2) sage: v = M.vector_field() sage: v.tensor_type() (1, 0)
- trace(pos1=0, pos2=1, using=None)#
Trace (contraction) on two slots of the tensor field.
If a non-degenerate form is provided, the trace of a \((0,2)\) tensor field is computed by first raising the last index.
INPUT:
pos1
– (default: 0) position of the first index for the contraction, with the conventionpos1=0
for the first slotpos2
– (default: 1) position of the second index for the contraction, with the same convention as forpos1
. The variance type ofpos2
must be opposite to that ofpos1
using
– (default:None
) a non-degenerate form
OUTPUT:
tensor field resulting from the
(pos1, pos2)
contraction
EXAMPLES:
Trace of a type-\((1,1)\) tensor field on a 2-dimensional non-parallelizable 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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y), ....: intersection_name='W', restrictions1= x>0, ....: restrictions2= u+v>0) sage: uv_to_xy = xy_to_uv.inverse() sage: e_xy = c_xy.frame(); e_uv = c_uv.frame() sage: W = U.intersection(V) sage: a = M.tensor_field(1,1, name='a') sage: a[e_xy,:] = [[1,x], [2,y]] sage: a.add_comp_by_continuation(e_uv, W, chart=c_uv) sage: s = a.trace() ; s Scalar field on the 2-dimensional differentiable manifold M sage: s.display() M → ℝ on U: (x, y) ↦ y + 1 on V: (u, v) ↦ 1/2*u - 1/2*v + 1 sage: s == a.trace(0,1) # explicit mention of the positions True
The trace of a type-\((0,2)\) tensor field using a metric:
sage: g = M.metric('g') sage: g[0,0], g[0,1], g[1,1] = 1, 0, 1 sage: g.trace(using=g).display() M → ℝ on U: (x, y) ↦ 2 on W: (u, v) ↦ 2
Instead of the explicit call to the method
trace()
, one may use the index notation with Einstein convention (summation over repeated indices); it suffices to pass the indices as a string inside square brackets:sage: a['^i_i'] Scalar field on the 2-dimensional differentiable manifold M sage: a['^i_i'] == s True
Any letter can be used to denote the repeated index:
sage: a['^b_b'] == s True
Trace of a type-\((1,2)\) tensor field:
sage: b = M.tensor_field(1,2, name='b') ; b Tensor field b of type (1,2) on the 2-dimensional differentiable manifold M sage: b[e_xy,:] = [[[0,x+y], [y,0]], [[0,2], [3*x,-2]]] sage: b.add_comp_by_continuation(e_uv, W, chart=c_uv) # long time sage: s = b.trace(0,1) ; s # contraction on first and second slots 1-form on the 2-dimensional differentiable manifold M sage: s.display(e_xy) 3*x dx + (x + y - 2) dy sage: s.display(e_uv) # long time (5/4*u + 3/4*v - 1) du + (1/4*u + 3/4*v + 1) dv
Use of the index notation:
sage: b['^k_ki'] 1-form on the 2-dimensional differentiable manifold M sage: b['^k_ki'] == s # long time True
Indices not involved in the contraction may be replaced by dots:
sage: b['^k_k.'] == s # long time True
The symbol
^
may be omitted:sage: b['k_k.'] == s # long time True
LaTeX notations are allowed:
sage: b['^{k}_{ki}'] == s # long time True
Contraction on first and third slots:
sage: s = b.trace(0,2) ; s 1-form on the 2-dimensional differentiable manifold M sage: s.display(e_xy) 2 dx + (y - 2) dy sage: s.display(e_uv) # long time (1/4*u - 1/4*v) du + (-1/4*u + 1/4*v + 2) dv
Use of index notation:
sage: b['^k_.k'] == s # long time True
- up(non_degenerate_form, pos=None)#
Compute a dual of the tensor field by raising some index with the given tensor field (usually, a pseudo-Riemannian metric, a symplectic form or a Poisson tensor).
If \(T\) is the tensor field, \((k,l)\) its type and \(p\) the position of a covariant index (i.e. \(k\leq p < k+l\)), this method called with
pos
\(=p\) yields the tensor field \(T^\sharp\) of type \((k+1,l-1)\) whose components are\[(T^\sharp)^{a_1\ldots a_{k+1}}_{\phantom{a_1\ldots a_{k+1}}\, b_1 \ldots b_{l-1}} = g^{a_{k+1} i} \, T^{a_1\ldots a_k}_{\phantom{a_1\ldots a_k}\, b_1 \ldots b_{p-k} \, i \, b_{p-k+1}\ldots b_{l-1}},\]\(g^{ab}\) being the components of the inverse metric or the Poisson tensor, respectively.
The reverse operation is
TensorField.down()
.INPUT:
non_degenerate_form
– non-degenerate form \(g\), or a Poisson tensorpos
– (default:None
) position of the index (with the conventionpos=0
for the first index); ifNone
, the raising is performed over all the covariant indices, starting from the first one
OUTPUT:
the tensor field \(T^\sharp\) resulting from the index raising operation
EXAMPLES:
Raising the index of a 1-form results in a vector field:
sage: M = Manifold(2, 'M', start_index=1) sage: c_xy.<x,y> = M.chart() sage: g = M.metric('g') sage: g[1,1], g[1,2], g[2,2] = 1+x, x*y, 1-y sage: w = M.one_form(-1, 2) sage: v = w.up(g) ; v Vector field on the 2-dimensional differentiable manifold M sage: v.display() ((2*x - 1)*y + 1)/(x^2*y^2 + (x + 1)*y - x - 1) ∂/∂x - (x*y + 2*x + 2)/(x^2*y^2 + (x + 1)*y - x - 1) ∂/∂y sage: ig = g.inverse(); ig[:] [ (y - 1)/(x^2*y^2 + (x + 1)*y - x - 1) x*y/(x^2*y^2 + (x + 1)*y - x - 1)] [ x*y/(x^2*y^2 + (x + 1)*y - x - 1) -(x + 1)/(x^2*y^2 + (x + 1)*y - x - 1)]
Using the index notation instead of
up()
:sage: v == ig['^ab']*w['_b'] True
The reverse operation:
sage: w1 = v.down(g) ; w1 1-form on the 2-dimensional differentiable manifold M sage: w1.display() -dx + 2 dy sage: w1 == w True
The reverse operation in index notation:
sage: g['_ab']*v['^b'] == w True
Raising the indices of a tensor field of type (0,2):
sage: t = M.tensor_field(0, 2, [[1,2], [3,4]]) sage: tu0 = t.up(g, 0) ; tu0 # raising the first index Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: tu0[:] [ ((3*x + 1)*y - 1)/(x^2*y^2 + (x + 1)*y - x - 1) 2*((2*x + 1)*y - 1)/(x^2*y^2 + (x + 1)*y - x - 1)] [ (x*y - 3*x - 3)/(x^2*y^2 + (x + 1)*y - x - 1) 2*(x*y - 2*x - 2)/(x^2*y^2 + (x + 1)*y - x - 1)] sage: tu0 == ig['^ac']*t['_cb'] # the same operation in index notation True sage: tuu0 = tu0.up(g) ; tuu0 # the two indices have been raised, starting from the first one Tensor field of type (2,0) on the 2-dimensional differentiable manifold M sage: tuu0 == tu0['^a_c']*ig['^cb'] # the same operation in index notation True sage: tu1 = t.up(g, 1) ; tu1 # raising the second index Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: tu1 == ig['^ac']*t['_bc'] # the same operation in index notation True sage: tu1[:] [((2*x + 1)*y - 1)/(x^2*y^2 + (x + 1)*y - x - 1) ((4*x + 3)*y - 3)/(x^2*y^2 + (x + 1)*y - x - 1)] [ (x*y - 2*x - 2)/(x^2*y^2 + (x + 1)*y - x - 1) (3*x*y - 4*x - 4)/(x^2*y^2 + (x + 1)*y - x - 1)] sage: tuu1 = tu1.up(g) ; tuu1 # the two indices have been raised, starting from the second one Tensor field of type (2,0) on the 2-dimensional differentiable manifold M sage: tuu1 == tu1['^a_c']*ig['^cb'] # the same operation in index notation True sage: tuu0 == tuu1 # the order of index raising is important False sage: tuu = t.up(g) ; tuu # both indices are raised, starting from the first one Tensor field of type (2,0) on the 2-dimensional differentiable manifold M sage: tuu0 == tuu # the same order for index raising has been applied True sage: tuu1 == tuu # to get tuu1, indices have been raised from the last one, contrary to tuu False sage: d0tuu = tuu.down(g, 0) ; d0tuu # the first index is lowered again Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: dd0tuu = d0tuu.down(g) ; dd0tuu # the second index is then lowered Tensor field of type (0,2) on the 2-dimensional differentiable manifold M sage: d1tuu = tuu.down(g, 1) ; d1tuu # lowering operation, starting from the last index Tensor field of type (1,1) on the 2-dimensional differentiable manifold M sage: dd1tuu = d1tuu.down(g) ; dd1tuu Tensor field of type (0,2) on the 2-dimensional differentiable manifold M sage: ddtuu = tuu.down(g) ; ddtuu # both indices are lowered, starting from the last one Tensor field of type (0,2) on the 2-dimensional differentiable manifold M sage: ddtuu == t # should be true True sage: dd0tuu == t # not true, because of the order of index lowering to get dd0tuu False sage: dd1tuu == t # should be true True