Mixed Differential Forms¶
Let \(M\) and \(N\) be differentiable manifolds and \(\varphi : M \longrightarrow N\)
a differentiable map. A mixed differential form along \(\varphi\) is an element
of the graded algebra represented by
MixedFormAlgebra
.
Its homogeneous components consist of differential forms along \(\varphi\). Mixed
forms are useful to represent characteristic classes and perform computations
of such.
AUTHORS:
Michael Jung (2019) : initial version
-
class
sage.manifolds.differentiable.mixed_form.
MixedForm
(parent, name=None, latex_name=None)¶ Bases:
sage.structure.element.AlgebraElement
An instance of this class is a mixed form along some differentiable map \(\varphi: M \to N\) between two differentiable manifolds \(M\) and \(N\). More precisely, a mixed form \(a\) along \(\varphi: M \to N\) can be considered as a differentiable map
\[a: M \longrightarrow \bigoplus^n_{k=0} T^{(0,k)}N,\]where \(T^{(0,k)}\) denotes the tensor bundle of type \((0,k)\), \(\bigoplus\) the Whitney sum and \(n\) the dimension of \(N\), such that
\[\forall x\in M, \quad a(x) \in \bigoplus^n_{k=0} \Lambda^k\left( T_{\varphi(x)}^* N \right),\]where \(\Lambda^k(T^*_{\varphi(x)} N)\) is the \(k\)-th exterior power of the dual of the tangent space \(T_{\varphi(x)} N\).
The standard case of a mixed form on \(M\) corresponds to \(M=N\) with \(\varphi = \mathrm{Id}_M\).
INPUT:
parent
– graded algebra of mixed forms represented byMixedFormAlgebra
where the mixed formself
shall belong tocomp
– (default:None
) homogeneous components of the mixed form as a list; if none is provided, the components are set to innocent unnamed differential formsname
– (default:None
) name given to the mixed formlatex_name
– (default:None
) LaTeX symbol to denote the mixed form; if none is provided, the LaTeX symbol is set toname
EXAMPLES:
Initialize a mixed form on a 2-dimensional parallelizable differentiable manifold:
sage: M = Manifold(2, 'M') sage: c_xy.<x,y> = M.chart() sage: e_xy = c_xy.frame() sage: A = M.mixed_form(name='A'); A Mixed differential form A on the 2-dimensional differentiable manifold M sage: A.parent() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional differentiable manifold M
One convenient way to define the homogeneous components of a mixed form is to define some differential forms first:
sage: f = M.scalar_field(x, name='f'); f Scalar field f on the 2-dimensional differentiable manifold M sage: omega = M.diff_form(1, name='omega'); omega 1-form omega on the 2-dimensional differentiable manifold M sage: omega[e_xy,0] = y*x; omega.display() omega = x*y dx sage: eta = M.diff_form(2, name='eta'); eta 2-form eta on the 2-dimensional differentiable manifold M sage: eta[e_xy,0,1] = y^2*x; eta.display() eta = x*y^2 dx/\dy
The components of the mixed form
F
can be set very easily:sage: A[:] = [f, omega, eta]; A.display() # display names A = f + omega + eta sage: A.display_expansion() # display in coordinates A = [x] + [x*y dx] + [x*y^2 dx/\dy] sage: A[0] Scalar field f on the 2-dimensional differentiable manifold M sage: A[0] is f True sage: A[1] 1-form omega on the 2-dimensional differentiable manifold M sage: A[1] is omega True sage: A[2] 2-form eta on the 2-dimensional differentiable manifold M sage: A[2] is eta True
Alternatively, the components can be determined from scratch:
sage: B = M.mixed_form([f, omega, eta], name='B') sage: A == B True
Mixed forms are elements of an algebra, so they can be added, and multiplied via the wedge product:
sage: C = x*A; C Mixed differential form x/\A on the 2-dimensional differentiable manifold M sage: C.display_expansion() x/\A = [x^2] + [x^2*y dx] + [x^2*y^2 dx/\dy] sage: D = A+C; D Mixed differential form A+x/\A on the 2-dimensional differentiable manifold M sage: D.display_expansion() A+x/\A = [x^2 + x] + [(x^2 + x)*y dx] + [(x^2 + x)*y^2 dx/\dy] sage: E = A*C; E Mixed differential form A/\(x/\A) on the 2-dimensional differentiable manifold M sage: E.display_expansion() A/\(x/\A) = [x^3] + [2*x^3*y dx] + [2*x^3*y^2 dx/\dy]
Coercions are fully implemented:
sage: F = omega*A sage: F.display_expansion() omega/\A = [0] + [x^2*y dx] + [0] sage: G = omega+A sage: G.display_expansion() omega+A = [x] + [2*x*y dx] + [x*y^2 dx/\dy]
Moreover, it is possible to compute the exterior derivative of a mixed form:
sage: dA = A.exterior_derivative(); dA.display() dA = zero + df + domega sage: dA.display_expansion() dA = [0] + [dx] + [-x dx/\dy]
Initialize a mixed form on a 2-dimensional non-parallelizable differentiable 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: e_xy = c_xy.frame(); e_uv = c_uv.frame() # define frames sage: A = M.mixed_form(name='A') sage: A[0].set_name('f') sage: A[0].set_expr(x, c_xy) sage: A[0].display() f: M --> R on U: (x, y) |--> x on W: (u, v) |--> 1/2*u + 1/2*v sage: A[1].set_name('omega') sage: A[1][0] = y*x; A[1].display(e_xy) omega = x*y dx sage: A[2].set_name('eta') sage: A[2][e_uv,0,1] = u*v^2; A[2].display(e_uv) eta = u*v^2 du/\dv sage: A.add_comp_by_continuation(e_uv, W, c_uv) sage: A.display_expansion(e_uv) A = [1/2*u + 1/2*v] + [(1/8*u^2 - 1/8*v^2) du + (1/8*u^2 - 1/8*v^2) dv] + [u*v^2 du/\dv] sage: A.add_comp_by_continuation(e_xy, W, c_xy) sage: A.display_expansion(e_xy) A = [x] + [x*y dx] + [(-2*x^3 + 2*x^2*y + 2*x*y^2 - 2*y^3) dx/\dy]
Since zero and one are special elements, their components cannot be changed:
sage: z = M.mixed_form_algebra().zero() sage: z[0] = 1 Traceback (most recent call last): ... ValueError: the components of the element zero cannot be changed sage: one = M.mixed_form_algebra().one() sage: one[0] = 0 Traceback (most recent call last): ... ValueError: the components of the element one cannot be changed
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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:
Mixed form defined by differential forms with components on different parts of the 2-sphere:
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: 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: e_xy = c_xy.frame(); e_uv = c_uv.frame() sage: F = M.mixed_form(name='F') # No predefined components, here sage: F[0] = M.scalar_field(x, name='f') sage: F[1] = M.diff_form(1, {e_xy: [x,0]}, name='omega') sage: F[2].set_name(name='eta') sage: F[2][e_uv,0,1] = u*v sage: F.add_comp_by_continuation(e_uv, W, c_uv) sage: F.add_comp_by_continuation(e_xy, W, c_xy) # Now, F is fully defined sage: F.display_expansion(e_xy) F = [x] + [x dx] + [-x*y/(x^8 + 4*x^6*y^2 + 6*x^4*y^4 + 4*x^2*y^6 + y^8) dx/\dy] sage: F.display_expansion(e_uv) F = [u/(u^2 + v^2)] + [-(u^3 - u*v^2)/(u^6 + 3*u^4*v^2 + 3*u^2*v^4 + v^6) du - 2*u^2*v/(u^6 + 3*u^4*v^2 + 3*u^2*v^4 + v^6) dv] + [u*v du/\dv]
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copy
(name=None, latex_name=None)¶ Return an exact copy of
self
.Note
The name and names of the components are not copied.
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
EXAMPLES:
Initialize a 2-dimensional manifold and differential forms:
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: f = M.scalar_field(x, name='f', chart=c_xy) sage: f.add_expr_by_continuation(c_uv, W) sage: f.display() f: M --> R on U: (x, y) |--> x on V: (u, v) |--> 1/2*u + 1/2*v sage: omega = M.diff_form(1, name='omega') sage: omega[e_xy,0] = x sage: omega.add_comp_by_continuation(e_uv, W, c_uv) sage: omega.display() omega = x dx sage: A = M.mixed_form([f, omega, 0], name='A'); A.display() A = f + omega + zero sage: A.display_expansion(e_uv) A = [1/2*u + 1/2*v] + [(1/4*u + 1/4*v) du + (1/4*u + 1/4*v) dv] + [0]
An exact copy is made. The copy is an entirely new instance and has a different name, but has the very same values:
sage: B = A.copy(); B.display() (unnamed scalar field) + (unnamed 1-form) + (unnamed 2-form) sage: B.display_expansion(e_uv) [1/2*u + 1/2*v] + [(1/4*u + 1/4*v) du + (1/4*u + 1/4*v) dv] + [0] sage: A == B True sage: A is B False
Notice, that changes in the differential forms usually cause changes in the original instance. But for the copy of a mixed form, the components are copied as well:
sage: omega[e_xy,0] = y; omega.display() omega = y dx sage: A.display_expansion(e_xy) A = [x] + [y dx] + [0] sage: B.display_expansion(e_xy) [x] + [x dx] + [0]
-
disp
()¶ Display the homogeneous components of the mixed form.
The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).
EXAMPLES:
sage: M = Manifold(2, 'M') sage: f = M.scalar_field(name='f') sage: omega = M.diff_form(1, name='omega') sage: eta = M.diff_form(2, name='eta') sage: F = M.mixed_form([f, omega, eta], name='F'); F Mixed differential form F on the 2-dimensional differentiable manifold M sage: F.display() # display names of homogeneous components F = f + omega + eta
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disp_exp
(frame=None, chart=None, from_chart=None)¶ Display the expansion in a particular basis and chart of mixed forms.
The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).
INPUT:
frame
– (default:None
) vector frame with respect to which the mixed form is expanded; ifNone
, only the names of the components are displayedchart
– (default:None
) chart with respect to which the components of the mixed form in the selected frame are expressed; ifNone
, the default chart of the vector frame domain is assumed
EXAMPLES:
Display the expansion of a mixed form on a 2-dimensional non-parallelizable differentiable 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: e_xy = c_xy.frame(); e_uv = c_uv.frame() # define frames sage: omega = M.diff_form(1, name='omega') sage: omega[e_xy,0] = x; omega.display(e_xy) omega = x dx sage: omega.add_comp_by_continuation(e_uv, W, c_uv) # continuation onto M sage: eta = M.diff_form(2, name='eta') sage: eta[e_uv,0,1] = u*v; eta.display(e_uv) eta = u*v du/\dv sage: eta.add_comp_by_continuation(e_xy, W, c_xy) # continuation onto M sage: F = M.mixed_form([0, omega, eta], name='F'); F Mixed differential form F on the 2-dimensional differentiable manifold M sage: F.display() # display names of homogeneous components F = zero + omega + eta sage: F.display_expansion(e_uv) F = [0] + [(1/4*u + 1/4*v) du + (1/4*u + 1/4*v) dv] + [u*v du/\dv] sage: F.display_expansion(e_xy) F = [0] + [x dx] + [(2*x^2 - 2*y^2) dx/\dy]
-
display
()¶ Display the homogeneous components of the mixed form.
The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).
EXAMPLES:
sage: M = Manifold(2, 'M') sage: f = M.scalar_field(name='f') sage: omega = M.diff_form(1, name='omega') sage: eta = M.diff_form(2, name='eta') sage: F = M.mixed_form([f, omega, eta], name='F'); F Mixed differential form F on the 2-dimensional differentiable manifold M sage: F.display() # display names of homogeneous components F = f + omega + eta
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display_exp
(frame=None, chart=None, from_chart=None)¶ Display the expansion in a particular basis and chart of mixed forms.
The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).
INPUT:
frame
– (default:None
) vector frame with respect to which the mixed form is expanded; ifNone
, only the names of the components are displayedchart
– (default:None
) chart with respect to which the components of the mixed form in the selected frame are expressed; ifNone
, the default chart of the vector frame domain is assumed
EXAMPLES:
Display the expansion of a mixed form on a 2-dimensional non-parallelizable differentiable 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: e_xy = c_xy.frame(); e_uv = c_uv.frame() # define frames sage: omega = M.diff_form(1, name='omega') sage: omega[e_xy,0] = x; omega.display(e_xy) omega = x dx sage: omega.add_comp_by_continuation(e_uv, W, c_uv) # continuation onto M sage: eta = M.diff_form(2, name='eta') sage: eta[e_uv,0,1] = u*v; eta.display(e_uv) eta = u*v du/\dv sage: eta.add_comp_by_continuation(e_xy, W, c_xy) # continuation onto M sage: F = M.mixed_form([0, omega, eta], name='F'); F Mixed differential form F on the 2-dimensional differentiable manifold M sage: F.display() # display names of homogeneous components F = zero + omega + eta sage: F.display_expansion(e_uv) F = [0] + [(1/4*u + 1/4*v) du + (1/4*u + 1/4*v) dv] + [u*v du/\dv] sage: F.display_expansion(e_xy) F = [0] + [x dx] + [(2*x^2 - 2*y^2) dx/\dy]
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display_expansion
(frame=None, chart=None, from_chart=None)¶ Display the expansion in a particular basis and chart of mixed forms.
The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).
INPUT:
frame
– (default:None
) vector frame with respect to which the mixed form is expanded; ifNone
, only the names of the components are displayedchart
– (default:None
) chart with respect to which the components of the mixed form in the selected frame are expressed; ifNone
, the default chart of the vector frame domain is assumed
EXAMPLES:
Display the expansion of a mixed form on a 2-dimensional non-parallelizable differentiable 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: e_xy = c_xy.frame(); e_uv = c_uv.frame() # define frames sage: omega = M.diff_form(1, name='omega') sage: omega[e_xy,0] = x; omega.display(e_xy) omega = x dx sage: omega.add_comp_by_continuation(e_uv, W, c_uv) # continuation onto M sage: eta = M.diff_form(2, name='eta') sage: eta[e_uv,0,1] = u*v; eta.display(e_uv) eta = u*v du/\dv sage: eta.add_comp_by_continuation(e_xy, W, c_xy) # continuation onto M sage: F = M.mixed_form([0, omega, eta], name='F'); F Mixed differential form F on the 2-dimensional differentiable manifold M sage: F.display() # display names of homogeneous components F = zero + omega + eta sage: F.display_expansion(e_uv) F = [0] + [(1/4*u + 1/4*v) du + (1/4*u + 1/4*v) dv] + [u*v du/\dv] sage: F.display_expansion(e_xy) F = [0] + [x dx] + [(2*x^2 - 2*y^2) dx/\dy]
-
exterior_derivative
()¶ Compute the exterior derivative of
self
.More precisely, the exterior derivative on \(\Omega^k(M,\varphi)\) is a linear map
\[\mathrm{d}_{k} : \Omega^k(M,\varphi) \to \Omega^{k+1}(M,\varphi),\]where \(\Omega^k(M,\varphi)\) denotes the space of differential forms of degree \(k\) along \(\varphi\) (see
exterior_derivative()
for further information). By linear extension, this induces a map on \(\Omega^*(M,\varphi)\):\[\mathrm{d}: \Omega^*(M,\varphi) \to \Omega^*(M,\varphi).\]OUTPUT:
a
MixedForm
representing the exterior derivative of the mixed form
EXAMPLES:
Exterior derivative of a mixed form on a 3-dimensional manifold:
sage: M = Manifold(3, 'M', start_index=1) sage: c_xyz.<x,y,z> = M.chart() sage: f = M.scalar_field(z^2, name='f') sage: f.disp() f: M --> R (x, y, z) |--> z^2 sage: a = M.diff_form(2, 'a') sage: a[1,2], a[1,3], a[2,3] = z+y^2, z+x, x^2 sage: a.disp() a = (y^2 + z) dx/\dy + (x + z) dx/\dz + x^2 dy/\dz sage: F = M.mixed_form([f, 0, a, 0], name='F'); F.display() F = f + zero + a + zero sage: dF = F.exterior_derivative() sage: dF.display() dF = zero + df + dzero + da sage: dF = F.exterior_derivative() sage: dF.display_expansion() dF = [0] + [2*z dz] + [0] + [(2*x + 1) dx/\dy/\dz]
Due to long calculation times, the result is cached:
sage: F.exterior_derivative() is dF True
-
irange
(start=None)¶ Single index generator.
INPUT:
start
– (default:None
) initial value \(i_0\) of the index between 0 and \(n\), where \(n\) is the manifold’s dimension; if none is provided, the value 0 is assumed
OUTPUT:
an iterable index, starting from \(i_0\) and ending at \(n\), where \(n\) is the manifold’s dimension
EXAMPLES:
sage: M = Manifold(3, 'M') sage: a = M.mixed_form(name='a') sage: list(a.irange()) [0, 1, 2, 3] sage: list(a.irange(2)) [2, 3]
-
restrict
(subdomain, dest_map=None)¶ Return the restriction of
self
to some subdomain.INPUT:
subdomain
–DifferentiableManifold
; open subset \(U\) of the domain ofself
dest_map
–DiffMap
(default:None
); destination map \(\Psi:\ U \rightarrow V\), where \(V\) is an open subset of the manifold \(N\) where the mixed form takes it values; ifNone
, the restriction of \(\Phi\) to \(U\) is used, \(\Phi\) being the differentiable map \(S \rightarrow M\) associated with the mixed form
OUTPUT:
MixedForm
representing the restriction
EXAMPLES:
Initialize the 2-sphere:
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: 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: e_xy = c_xy.frame(); e_uv = c_uv.frame()
And predefine some forms:
sage: f = M.scalar_field(x^2, name='f', chart=c_xy) sage: f.add_expr_by_continuation(c_uv, W) sage: omega = M.diff_form(1, name='omega') sage: omega[e_xy,0] = y^2 sage: omega.add_comp_by_continuation(e_uv, W, c_uv) sage: eta = M.diff_form(2, name='eta') sage: eta[e_xy,0,1] = x^2*y^2 sage: eta.add_comp_by_continuation(e_uv, W, c_uv)
Now, a mixed form can be restricted to some subdomain:
sage: F = M.mixed_form([f, omega, eta], name='F') sage: FV = F.restrict(V); FV Mixed differential form F on the Open subset V of the 2-dimensional differentiable manifold M sage: FV[:] [Scalar field f on the Open subset V of the 2-dimensional differentiable manifold M, 1-form omega on the Open subset V of the 2-dimensional differentiable manifold M, 2-form eta on the Open subset V of the 2-dimensional differentiable manifold M] sage: FV.display_expansion(e_uv) F = [u^2/(u^4 + 2*u^2*v^2 + v^4)] + [-(u^2*v^2 - v^4)/(u^8 + 4*u^6*v^2 + 6*u^4*v^4 + 4*u^2*v^6 + v^8) du - 2*u*v^3/(u^8 + 4*u^6*v^2 + 6*u^4*v^4 + 4*u^2*v^6 + v^8) dv] + [-u^2*v^2/(u^12 + 6*u^10*v^2 + 15*u^8*v^4 + 20*u^6*v^6 + 15*u^4*v^8 + 6*u^2*v^10 + v^12) du/\dv]
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set_name
(name=None, latex_name=None)¶ Redefine the string and LaTeX representation of the object.
INPUT:
name
– (default:None
) name given to the mixed formlatex_name
– (default:None
) LaTeX symbol to denote the mixed form; if none is provided, the LaTeX symbol is set toname
EXAMPLES:
sage: M = Manifold(4, 'M') sage: F = M.mixed_form(name='dummy', latex_name=r'\ugly'); F Mixed differential form dummy on the 4-dimensional differentiable manifold M sage: latex(F) \ugly sage: F.set_name(name='fancy', latex_name=r'\eta'); F Mixed differential form fancy on the 4-dimensional differentiable manifold M sage: latex(F) \eta
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set_restriction
(rst)¶ Set a (component-wise) restriction of
self
to some subdomain.INPUT:
rst
–MixedForm
of the same type asself
, defined on a subdomain of the domain ofself
EXAMPLES:
Initialize the 2-sphere:
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: 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: e_xy = c_xy.frame(); e_uv = c_uv.frame()
And define some forms on the subset
U
:sage: f = U.scalar_field(x, name='f', chart=c_xy) sage: omega = U.diff_form(1, name='omega') sage: omega[e_xy,0] = y sage: AU = U.mixed_form([f, omega, 0], name='A'); AU Mixed differential form A on the Open subset U of the 2-dimensional differentiable manifold M sage: AU.display_expansion(e_xy) A = [x] + [y dx] + [0]
A mixed form on
M
can be specified by some mixed form on a subset:sage: A = M.mixed_form(name='A'); A Mixed differential form A on the 2-dimensional differentiable manifold M sage: A.set_restriction(AU) sage: A.display_expansion(e_xy) A = [x] + [y dx] + [0] sage: A.add_comp_by_continuation(e_uv, W, c_uv) sage: A.display_expansion(e_uv) A = [u/(u^2 + v^2)] + [-(u^2*v - v^3)/(u^6 + 3*u^4*v^2 + 3*u^2*v^4 + v^6) du - 2*u*v^2/(u^6 + 3*u^4*v^2 + 3*u^2*v^4 + v^6) dv] + [0] sage: A.restrict(U) == AU True
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wedge
(other)¶ Wedge product on the graded algebra of mixed forms.
More precisely, the wedge product is a bilinear map
\[\wedge: \Omega^k(M,\varphi) \times \Omega^l(M,\varphi) \to \Omega^{k+l}(M,\varphi),\]where \(\Omega^k(M,\varphi)\) denotes the space of differential forms of degree \(k\) along \(\varphi\). By bilinear extension, this induces a map
\[\wedge: \Omega^*(M,\varphi) \times \Omega^*(M,\varphi) \to \Omega^*(M,\varphi) ``\]and equips \(\Omega^*(M,\varphi)\) with a multiplication such that it becomes a graded algebra.
INPUT:
other
– mixed form in the same algebra asself
OUTPUT:
the mixed form resulting from the wedge product of
self
withother
EXAMPLES:
Initialize a mixed form on a 3-dimensional manifold:
sage: M = Manifold(3, 'M') sage: c_xyz.<x,y,z> = M.chart() sage: f = M.scalar_field(x, name='f') sage: f.display() f: M --> R (x, y, z) |--> x sage: g = M.scalar_field(y, name='g') sage: g.display() g: M --> R (x, y, z) |--> y sage: omega = M.diff_form(1, name='omega') sage: omega[0] = x sage: omega.display() omega = x dx sage: eta = M.diff_form(1, name='eta') sage: eta[1] = y sage: eta.display() eta = y dy sage: mu = M.diff_form(2, name='mu') sage: mu[0,2] = z sage: mu.display() mu = z dx/\dz sage: A = M.mixed_form([f, omega, mu, 0], name='A') sage: A.display_expansion() A = [x] + [x dx] + [z dx/\dz] + [0] sage: B = M.mixed_form([g, eta, mu, 0], name='B') sage: B.display_expansion() B = [y] + [y dy] + [z dx/\dz] + [0]
The wedge product of
A
andB
yields:sage: C = A.wedge(B); C Mixed differential form A/\B on the 3-dimensional differentiable manifold M sage: C.display_expansion() A/\B = [x*y] + [x*y dx + x*y dy] + [x*y dx/\dy + (x + y)*z dx/\dz] + [-y*z dx/\dy/\dz] sage: D = B.wedge(A); D # Don't even try, it's not commutative! Mixed differential form B/\A on the 3-dimensional differentiable manifold M sage: D.display_expansion() # I told you so! B/\A = [x*y] + [x*y dx + x*y dy] + [-x*y dx/\dy + (x + y)*z dx/\dz] + [-y*z dx/\dy/\dz]
Alternatively, the multiplication symbol can be used:
sage: A*B Mixed differential form A/\B on the 3-dimensional differentiable manifold M sage: A*B == C True
Yet, the multiplication includes coercions:
sage: E = x*A; E.display_expansion() x/\A = [x^2] + [x^2 dx] + [x*z dx/\dz] + [0] sage: F = A*eta; F.display_expansion() A/\eta = [0] + [x*y dy] + [x*y dx/\dy] + [-y*z dx/\dy/\dz]