# 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)#

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.set_expr(x) # scalar field
sage: A.set_comp(1) = 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
Scalar field f on the 2-dimensional differentiable manifold M
sage: B
1-form omega on the 2-dimensional differentiable manifold M
sage: B
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
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.set_expr(x, c_xy)
sage: A.display()
A_0: M → ℝ
on U: (x, y) ↦ x
on W: (u, v) ↦ 1/2*u + 1/2*v
sage: A = y*x; A.display(e_xy)
A_1 = x*y dx
sage: A[e_uv,0,1] = u*v^2; A.display(e_uv)
A_2 = u*v^2 du∧dv
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.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 = 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
Traceback (most recent call last):
...
ValueError: the components of an immutable element cannot be changed


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 = M.scalar_field(x, name='f')
sage: F = M.diff_form(1, {e_xy: [x,0]}, name='omega')
sage: F.set_name(name='eta')
sage: F[e_uv,0,1] = u*v
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.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.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:

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:

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:

• 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: 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: 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([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.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.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:

• 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: 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.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 = x
sage: omega.display()
omega = x dx
sage: eta = M.diff_form(1, name='eta')
sage: eta = 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