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: AlgebraElement, ModuleElementWithMutability

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 \(T_{\varphi(x)} N\). Thus, a mixed differential form \(a\) consists of homogeneous components \(a_i\), \(i=0,1, \dots, n\), where the \(i\)-th homogeneous component represents a differential form of degree \(i\).

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 by MixedFormAlgebra where the mixed form self shall belong to

• comp – (default: None) homogeneous components of the mixed form as a list; if none is provided, the components are set to innocent unnamed differential forms

• name – (default: None) name given to the mixed form

• latex_name – (default: None) LaTeX symbol to denote the mixed form; if none is provided, the LaTeX symbol is set to name

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

The default way to specify the \(i\)-th homogeneous component of a mixed form is by accessing it via A[i] or using set_comp():

sage: A = M.mixed_form(name='A')
sage: A[0].set_expr(x) # scalar field
sage: A.set_comp(1)[0] = y*x
sage: A.set_comp(2)[0,1] = y^2*x
sage: A.display() # display names
A = A_0 + A_1 + A_2
sage: A.display_expansion() # display expansion in basis
A = x + x*y dx + x*y^2 dx∧dy

Another way to define the homogeneous components is using predefined differential forms:

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 a mixed form B can then be set as follows:

sage: B = M.mixed_form(name='B')
sage: B[:] = [f, omega, eta]; B.display() # display names
B = f + omega + eta
sage: B.display_expansion() # display in coordinates
B = x + x*y dx + x*y^2 dx∧dy
sage: B[0]
Scalar field f on the 2-dimensional differentiable manifold M
sage: B[1]
1-form omega on the 2-dimensional differentiable manifold M
sage: B[2]
2-form eta on the 2-dimensional differentiable manifold M

As we can see, the names are applied. However note that the differential forms are different instances:

sage: f is B[0]
False

Alternatively, the components can be determined from scratch:

sage: B = M.mixed_form([f, omega, eta], name='B')
sage: B.display()
B = f + omega + eta

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 = x^2*y dx
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 + dA_0 + dA_1
sage: dA.display_expansion()
dA = 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_expr(x, c_xy)
sage: A[0].display()
A_0: M → ℝ
on U: (x, y) ↦ x
on W: (u, v) ↦ 1/2*u + 1/2*v
sage: A[1][0] = y*x; A[1].display(e_xy)
A_1 = x*y dx
sage: A[2][e_uv,0,1] = u*v^2; A[2].display(e_uv)
A_2 = 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 an immutable element cannot be changed
sage: one = M.mixed_form_algebra().one()
sage: one[0] = 0
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 set

• subdomain – open subset of \(e\)’s domain in which the components are known or can be evaluated from other components

• chart – (default: None) coordinate chart on \(e\)’s domain in which the extension of the expression of the components is to be performed; if None, 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

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 copy

• latex_name – (default: None) LaTeX symbol to denote the copy; if none is provided, the LaTeX symbol is set to name

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

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
sage: A == B
True
sage: A is B
False

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 = 2*z dz + (2*x + 1) dx∧dy∧dz

Due to long calculation times, the result is cached:

sage: F.exterior_derivative() is dF
True

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

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; if None, only the names of the components are displayed

• chart – (default: None) chart with respect to which the components of the mixed form in the selected frame are expressed; if None, 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 = (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 = 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

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; if None, only the names of the components are displayed

• chart – (default: None) chart with respect to which the components of the mixed form in the selected frame are expressed; if None, 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 = (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 = x dx + (2*x^2 - 2*y^2) dx∧dy

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; if None, only the names of the components are displayed

• chart – (default: None) chart with respect to which the components of the mixed form in the selected frame are expressed; if None, 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 = (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 = 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 = 2*z dz + (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 of self

• 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; if None, 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

set_comp(i)

Return the \(i\)-th homogeneous component for assignment.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: A = M.mixed_form(name='A')
sage: A.set_comp(0).set_expr(x^2) # scalar field
sage: A.set_comp(1)[:] = [-y, x]
sage: A.set_comp(2)[0,1] = x-y
sage: A.display()
A = A_0 + A_1 + A_2
sage: A.display_expansion()
A = x^2 - y dx + x dy + (x - y) dx∧dy

set_immutable()

Set self and homogeneous components of self immutable.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: f = M.scalar_field(x^2, name='f')
sage: A = M.mixed_form([f, 0, 0], name='A')
sage: A.set_immutable()
sage: A.is_immutable()
True
sage: A[0].is_immutable()
True
sage: f.is_immutable()
False

set_name(name=None, latex_name=None, apply_to_comp=True)

Redefine the string and LaTeX representation of the object.

INPUT:

• name – (default: None) name given to the mixed form

• latex_name – (default: None) LaTeX symbol to denote the mixed form; if none is provided, the LaTeX symbol is set to name

• apply_to_comp – (default: True) if True all homogeneous components will be renamed accordingly; if False only the mixed form will be renamed

EXAMPLES:

Rename a mixed form:

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='F', latex_name=r'\mathcal{F}'); F
Mixed differential form F on the 4-dimensional differentiable
 manifold M
sage: latex(F)
\mathcal{F}

If not stated otherwise, all homogeneous components are renamed accordingly:

sage: F.display()
F = F_0 + F_1 + F_2 + F_3 + F_4

Setting the argument set_all to False prevents the renaming in the homogeneous components:

sage: F.set_name(name='eta', latex_name=r'\eta', apply_to_comp=False)
sage: F.display()
eta = F_0 + F_1 + F_2 + F_3 + F_4

To rename a homogeneous component individually, we simply access the homogeneous component and use its set_name() method:

sage: F[0].set_name(name='g'); F.display()
eta = g + F_1 + F_2 + F_3 + F_4

set_restriction(rst)

Set a (component-wise) restriction of self to some subdomain.

INPUT:

• rst – MixedForm of the same type as self, defined on a subdomain of the domain of self

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

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
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
sage: A.restrict(U) == AU
True

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 as self

OUTPUT:

• the mixed form resulting from the wedge product of self with other

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
sage: B = M.mixed_form([g, eta, mu, 0], name='B')
sage: B.display_expansion()
B = y + y dy + z dx∧dz

The wedge product of A and B 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
sage: F = A*eta; F.display_expansion()
A∧eta = x*y dy + x*y dx∧dy - y*z dx∧dy∧dz