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, comp=None, 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 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: F = M.mixed_form(name='F'); F
Mixed differential form F on the 2-dimensional differentiable manifold M
sage: F.parent()
Graded algebra Omega^*(M) of mixed differential forms on the
 2-dimensional differentiable manifold M

To define the homogenous components of a mixed form, it is convenient 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', latex_name=r'\omega'); omega
1-form omega on the 2-dimensional differentiable manifold M
sage: omega[c_xy.frame(),0] = y*x; omega.disp()
omega = x*y dx
sage: eta = M.diff_form(2, name='eta', latex_name=r'\eta'); eta
2-form eta on the 2-dimensional differentiable manifold M
sage: eta[c_xy.frame(),0,1] = y^2*x; eta.disp()
eta = x*y^2 dx/\dy

The components of the mixed form F can be manipulated very easily:

sage: F[:] = [f, omega, eta]; F.disp() # display names
F = f + omega + eta
sage: F.disp(c_xy.frame()) # display in coordinates
F = [x] + [x*y dx] + [x*y^2 dx/\dy]
sage: F[0]
Scalar field f on the 2-dimensional differentiable manifold M
sage: F[0] is f
True
sage: F[1]
1-form omega on the 2-dimensional differentiable manifold M
sage: F[1] is omega
True
sage: F[2]
2-form eta on the 2-dimensional differentiable manifold M
sage: F[2] is eta
True

Alternatively, the components can be determined from scratch:

sage: G = M.mixed_form(name='G', comp=[f, omega, eta])
sage: G == F
True

Mixed forms are elements of an algebra, so they can be added, and multiplied via the wedge product:

sage: xF = x*F; xF
Mixed differential form x/\F on the 2-dimensional differentiable
 manifold M
sage: xF.disp(c_xy.frame())
x/\F = [x^2] + [x^2*y dx] + [x^2*y^2 dx/\dy]
sage: FpxF = F+xF; FpxF
Mixed differential form F+x/\F on the 2-dimensional differentiable
 manifold M
sage: FpxF.disp(c_xy.frame())
F+x/\F = [x^2 + x] + [(x^2 + x)*y dx] + [(x^2 + x)*y^2 dx/\dy]
sage: FxF = F*xF; FxF
Mixed differential form F/\(x/\F) on the 2-dimensional differentiable
 manifold M
sage: FxF.disp(c_xy.frame())
F/\(x/\F) = [x^3] + [2*x^3*y dx] + [2*x^3*y^2 dx/\dy]

Coercions are fully implemented:

sage: omegaF = omega*F
sage: omegaF.disp(c_xy.frame())
omega/\F = [0] + [x^2*y dx] + [0]
sage: omegapF = omega+F
sage: omegapF.disp(c_xy.frame())
omega+F = [x] + [2*x*y dx] + [x*y^2 dx/\dy]

Moreover, it is possible to compute the exterior derivative of a mixed form:

sage: dF = F.exterior_derivative(); dF.disp()
dF = zero + df + domega
sage: dF.disp(c_xy.frame())
dF = [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: e_xy = c_xy.frame(); e_uv = c_uv.frame() # define frames
sage: omega = M.diff_form(1, name='omega', latex_name=r'\omega')
sage: omega[c_xy.frame(),0] = y*x; omega.disp(e_xy)
omega = x*y dx
sage: eta = M.diff_form(2, name='eta', latex_name=r'\eta')
sage: eta[c_uv.frame(),0,1] = u*v^2; eta.disp(e_uv)
eta = u*v^2 du/\dv
sage: F = M.mixed_form(name='F', comp=[x, omega, eta]); F
Mixed differential form F on the 2-dimensional differentiable manifold M
sage: F.add_comp_by_continuation(e_uv, V.intersection(U), c_uv)
sage: F.disp(e_uv)
F = [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: F.add_comp_by_continuation(e_xy, V.intersection(U), c_xy)
sage: F.disp(e_xy)
F = [x] + [x*y dx] + [(-2*x^3 + 2*x^2*y + 2*x*y^2 - 2*y^3) dx/\dy]
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: e_xy = c_xy.frame(); e_uv = c_uv.frame()
sage: f = M.scalar_field(x, name='f')
sage: omega = M.diff_form(1, name='omega')
sage: omega[e_xy,0] = x
sage: eta = M.diff_form(2, name='eta')
sage: eta[e_uv,0,1] = u*v
sage: F = M.mixed_form(name='F', comp=[f, omega, eta])
sage: F.add_comp_by_continuation(e_xy, U.intersection(V), c_xy)
sage: F.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: F.disp(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.disp(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()

Return an exact copy of self.

Note

The name and names of the components are not copied.

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: e_xy = c_xy.frame(); e_uv = c_uv.frame()
sage: f = M.scalar_field(x, name='f', chart=c_xy); f.disp()
f: M --> R
    on U: (x, y) |--> x
sage: omega = M.diff_form(1, name='omega')
sage: omega[e_xy,0] = x; omega.disp()
omega = x dx
sage: A = M.mixed_form(name='A', comp=[f, omega, 0]); A.disp()
A = f + omega + zero
sage: A.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: A.disp(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.disp()
f + (unnamed 1-form) + (unnamed 2-form)
sage: B.disp(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.disp()
omega = y dx
sage: A.disp(e_xy)
A = [x] + [y dx] + [0]
sage: B.disp(e_xy)
[x] + [x dx] + [0]
disp(basis=None, chart=None, from_chart=None)

Display the components of mixed forms.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

INPUT:

  • basis – (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 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: e_xy = c_xy.frame(); e_uv = c_uv.frame() # define frames
sage: omega = M.diff_form(1, name='omega', latex_name=r'\omega')
sage: omega[c_xy.frame(),0] = x; omega.disp(e_xy)
omega = x dx
sage: eta = M.diff_form(2, name='eta', latex_name=r'\eta')
sage: eta[c_uv.frame(),0,1] = u*v; eta.disp(e_uv)
eta = u*v du/\dv
sage: F = M.mixed_form(name='F', comp=[0, omega, eta]); F
Mixed differential form F on the 2-dimensional differentiable
 manifold M
sage: F.disp() # display names of homogenous components
F = zero + omega + eta
sage: F.add_comp_by_continuation(e_uv, V.intersection(U), c_uv)
sage: F.disp(e_uv)
F = [0] + [(1/4*u + 1/4*v) du + (1/4*u + 1/4*v) dv] + [u*v du/\dv]
sage: F.add_comp_by_continuation(e_xy, V.intersection(U), c_xy)
sage: F.disp(e_xy)
F = [0] + [x dx] + [(2*x^2 - 2*y^2) dx/\dy]
display(basis=None, chart=None, from_chart=None)

Display the components of mixed forms.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

INPUT:

  • basis – (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 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: e_xy = c_xy.frame(); e_uv = c_uv.frame() # define frames
sage: omega = M.diff_form(1, name='omega', latex_name=r'\omega')
sage: omega[c_xy.frame(),0] = x; omega.disp(e_xy)
omega = x dx
sage: eta = M.diff_form(2, name='eta', latex_name=r'\eta')
sage: eta[c_uv.frame(),0,1] = u*v; eta.disp(e_uv)
eta = u*v du/\dv
sage: F = M.mixed_form(name='F', comp=[0, omega, eta]); F
Mixed differential form F on the 2-dimensional differentiable
 manifold M
sage: F.disp() # display names of homogenous components
F = zero + omega + eta
sage: F.add_comp_by_continuation(e_uv, V.intersection(U), c_uv)
sage: F.disp(e_uv)
F = [0] + [(1/4*u + 1/4*v) du + (1/4*u + 1/4*v) dv] + [u*v du/\dv]
sage: F.add_comp_by_continuation(e_xy, V.intersection(U), c_xy)
sage: F.disp(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(name='F', comp=[f, 0, a, 0]); F.disp()
F = f + zero + a + zero
sage: dF = F.exterior_derivative()
sage: dF.disp()
dF = zero + df + dzero + da
sage: dF = F.exterior_derivative()
sage: dF.disp(c_xyz.frame())
dF = [0] + [2*z dz] + [0] + [(2*x + 1) dx/\dy/\dz]

Due to long calculation times, the result is cached, i.e. is not recomputed unless F is changed:

sage: F.exterior_derivative() is dF
True
restrict(subdomain, dest_map=None)

Return the restriction of self to some subdomain.

INPUT:

  • subdomainDifferentiableManifold; open subset \(U\) of the domain of self
  • dest_mapDiffMap (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:

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: 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: omega = M.diff_form(1, name='omega')
sage: omega[e_xy,0] = y^2
sage: eta = M.diff_form(2, name='eta')
sage: eta[e_xy,0,1] = x^2*y^2

Now, a mixed form can be restricted to some subdomain:

sage: F = M.mixed_form(name='F', comp=[f, omega, eta])
sage: F.add_comp_by_continuation(e_uv, V.intersection(U), c_uv)
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.disp(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_name(name=None, latex_name=None)

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

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
set_restriction(rst)

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

INPUT:

  • rstMixedForm 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: 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(name='A', comp=[f, omega, 0]); AU
Mixed differential form A on the Open subset U of the 2-dimensional
 differentiable manifold M
sage: AU.disp(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.disp(e_xy)
A = [x] + [y dx] + [0]
sage: A.add_comp_by_continuation(e_uv, V.intersection(U), c_uv)
sage: A.disp(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
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.disp()
f: M --> R
   (x, y, z) |--> x
sage: g = M.scalar_field(y, name='g')
sage: g.disp()
g: M --> R
   (x, y, z) |--> y
sage: omega = M.diff_form(1, name='omega')
sage: omega[c_xyz.frame(),0] = x
sage: omega.disp()
omega = x dx
sage: eta = M.diff_form(1, name='eta')
sage: eta[c_xyz.frame(),1] = y
sage: eta.disp()
eta = y dy
sage: mu = M.diff_form(2, name='mu')
sage: mu[c_xyz.frame(),[0,2]] = z
sage: mu.disp()
mu = z dx/\dz
sage: A = M.mixed_form(name='A', comp=[f, omega, mu, 0])
sage: A.disp(c_xyz.frame())
A = [x] + [x dx] + [z dx/\dz] + [0]
sage: B = M.mixed_form(name='B', comp=[g, eta, mu, 0])
sage: B.disp(c_xyz.frame())
B = [y] + [y dy] + [z dx/\dz] + [0]

The wedge product of A and B yields:

sage: AwB = A.wedge(B); AwB
Mixed differential form A/\B on the 3-dimensional differentiable
 manifold M
sage: AwB.disp(c_xyz.frame())
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: BwA = B.wedge(A); BwA # Don't even try, it's not commutative!
Mixed differential form B/\A on the 3-dimensional differentiable
 manifold M
sage: BwA.disp(c_xyz.frame()) # 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]

The multiplication symbol may be used instead as well:

sage: A*B
Mixed differential form A/\B on the 3-dimensional differentiable
 manifold M
sage: A*B == AwB
True

Yet, the multiplication includes coercions:

sage: xA = x*A; xA.disp(c_xyz.frame())
x/\A = [x^2] + [x^2 dx] + [x*z dx/\dz] + [0]
sage: Aeta = A*eta; Aeta.disp(c_xyz.frame())
A/\eta = [0] + [x*y dy] + [x*y dx/\dy] + [-y*z dx/\dy/\dz]