def hodge_star(self, metric):
r"""
Compute the Hodge dual of the differential form.
If ``self`` is a `p`-form `A`, its Hodge dual is the `(n-p)`-form
`*A` defined by (`n` being the manifold's dimension)
.. MATH::
*A_{i_1\ldots i_{n-p}} = \frac{1}{p!} A_{k_1\ldots k_p}
\epsilon^{k_1\ldots k_p}_{\qquad\ i_1\ldots i_{n-p}}
where $\epsilon$ is the volume form associated with some
pseudo-Riemannian metric `g` on the manifold, and the indices
`k_1,\ldots, k_p` are raised with `g`.
INPUT:
- ``metric``: the pseudo-Riemannian metric `g` defining the Hodge dual,
via the volume form `\epsilon`; must be an instance of :class:`Metric`
OUTPUT:
- the `(n-p)`-form `*A`
EXAMPLES:
Hodge star of a 1-form in the Euclidean space `R^3`::
sage: m = Manifold(3, 'M', start_index=1)
sage: X.<x,y,z> = m.chart('x y z', 'xyz')
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, 1, 1
sage: a = OneForm(m, 'A')
sage: var('Ax Ay Az')
(Ax, Ay, Az)
sage: a[:] = (Ax, Ay, Az)
sage: sa = a.hodge_star(g) ; sa
2-form '*A' on the 3-dimensional manifold 'M'
sage: sa.view()
*A = Az dx/\dy - Ay dx/\dz + Ax dy/\dz
sage: ssa = sa.hodge_star(g) ; ssa
1-form '**A' on the 3-dimensional manifold 'M'
sage: ssa.view()
**A = Ax dx + Ay dy + Az dz
sage: ssa == a # must hold for a Riemannian metric in dimension 3
True
Hodge star of a 0-form (scalar field) in `R^3`::
sage: f = ScalarField(m, function('F',x,y,z), name='f')
sage: sf = f.hodge_star(g) ; sf
3-form '*f' on the 3-dimensional manifold 'M'
sage: sf.view()
*f = F(x, y, z) dx/\dy/\dz
sage: ssf = sf.hodge_star(g) ; ssf
scalar field '**f' on the 3-dimensional manifold 'M'
sage: ssf.view()
**f: (x, y, z) |--> F(x, y, z)
sage: ssf == f # must hold for a Riemannian metric
True
Hodge star of a 0-form in Minkowksi spacetime::
sage: m = Manifold(4, 'M')
sage: X = m.chart('t x y z', 'txyz')
sage: g = Metric(m, 'g', signature=2)
sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1, 1, 1, 1
sage: g.view() # Minkowski metric
g = -dt*dt + dx*dx + dy*dy + dz*dz
sage: var('f0')
f0
sage: f = ScalarField(m, f0, name='f')
sage: sf = f.hodge_star(g) ; sf
4-form '*f' on the 4-dimensional manifold 'M'
sage: sf.view()
*f = f0 dt/\dx/\dy/\dz
sage: ssf = sf.hodge_star(g) ; ssf
scalar field '**f' on the 4-dimensional manifold 'M'
sage: ssf.view()
**f: (t, x, y, z) |--> -f0
sage: ssf == -f # must hold for a Lorentzian metric
True
Hodge star of a 1-form in Minkowksi spacetime::
sage: a = OneForm(m, 'A')
sage: var('At Ax Ay Az')
(At, Ax, Ay, Az)
sage: a[:] = (At, Ax, Ay, Az)
sage: a.view()
A = At dt + Ax dx + Ay dy + Az dz
sage: sa = a.hodge_star(g) ; sa
3-form '*A' on the 4-dimensional manifold 'M'
sage: sa.view()
*A = -Az dt/\dx/\dy + Ay dt/\dx/\dz - Ax dt/\dy/\dz - At dx/\dy/\dz
sage: ssa = sa.hodge_star(g) ; ssa
1-form '**A' on the 4-dimensional manifold 'M'
sage: ssa.view()
**A = At dt + Ax dx + Ay dy + Az dz
sage: ssa == a # must hold for a Lorentzian metric in dimension 4
True
Hodge star of a 2-form in Minkowksi spacetime::
sage: F = DiffForm(m, 2, 'F')
sage: var('Ex Ey Ez Bx By Bz')
(Ex, Ey, Ez, Bx, By, Bz)
sage: F[0,1], F[0,2], F[0,3] = -Ex, -Ey, -Ez
sage: F[1,2], F[1,3], F[2,3] = Bz, -By, Bx
sage: F[:]
[ 0 -Ex -Ey -Ez]
[ Ex 0 Bz -By]
[ Ey -Bz 0 Bx]
[ Ez By -Bx 0]
sage: sF = F.hodge_star(g) ; sF
2-form '*F' on the 4-dimensional manifold 'M'
sage: sF[:]
[ 0 Bx By Bz]
[-Bx 0 Ez -Ey]
[-By -Ez 0 Ex]
[-Bz Ey -Ex 0]
sage: ssF = sF.hodge_star(g) ; ssF
2-form '**F' on the 4-dimensional manifold 'M'
sage: ssF[:]
[ 0 Ex Ey Ez]
[-Ex 0 -Bz By]
[-Ey Bz 0 -Bx]
[-Ez -By Bx 0]
sage: ssF.view()
**F = Ex dt/\dx + Ey dt/\dy + Ez dt/\dz - Bz dx/\dy + By dx/\dz - Bx dy/\dz
sage: F.view()
F = -Ex dt/\dx - Ey dt/\dy - Ez dt/\dz + Bz dx/\dy - By dx/\dz + Bx dy/\dz
sage: ssF == -F # must hold for a Lorentzian metric in dimension 4
True
Test of the standard identity
.. MATH::
*(A\wedge B) = \epsilon(A^\sharp, B^\sharp, ., .)
where `A` and `B` are any 1-forms and `A^\sharp` and `B^\sharp` the
vectors associated to them by the metric `g` (index raising)::
sage: b = OneForm(m, 'B')
sage: var('Bt Bx By Bz')
(Bt, Bx, By, Bz)
sage: b[:] = (Bt, Bx, By, Bz) ; b.view()
B = Bt dt + Bx dx + By dy + Bz dz
sage: epsilon = g.volume_form()
sage: (a.wedge(b)).hodge_star(g) == epsilon.contract(0, a.up(g), 0).contract(0, b.up(g), 0)
True
"""
from sage.functions.other import factorial
from utilities import format_unop_txt, format_unop_latex
p = self.rank
eps = metric.volume_form(p)
if p == 0:
resu = self * eps
else:
resu = self.contract(0, eps, 0)
for j in range(1, p):
resu = resu.self_contract(0, p-j)
if p > 1:
resu = resu / factorial(p)
# Name and LaTeX name of the result:
resu.name = format_unop_txt('*', self.name)
resu.latex_name = format_unop_latex(r'\star ', self.latex_name)
return resu
def hodge_star(self, metric):
r"""
Compute the Hodge dual of the differential form.
If ``self`` is a `p`-form `A`, its Hodge dual is the `(n-p)`-form
`*A` defined by (`n` being the manifold's dimension)
.. MATH::
*A_{i_1\ldots i_{n-p}} = \frac{1}{p!} A_{k_1\ldots k_p}
\epsilon^{k_1\ldots k_p}_{\qquad\ i_1\ldots i_{n-p}}
where $\epsilon$ is the volume form associated with some
pseudo-Riemannian metric `g` on the manifold, and the indices
`k_1,\ldots, k_p` are raised with `g`.
INPUT:
- ``metric``: the pseudo-Riemannian metric `g` defining the Hodge dual,
via the volume form `\epsilon`; must be an instance of :class:`Metric`
OUTPUT:
- the `(n-p)`-form `*A`
EXAMPLES:
Hodge star of a 1-form in the Euclidean space `R^3`::
sage: m = Manifold(3, 'M', start_index=1)
sage: X = Chart(m, 'x y z', 'xyz')
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, 1, 1
sage: a = OneForm(m, 'A')
sage: var('Ax Ay Az')
(Ax, Ay, Az)
sage: a[:] = (Ax, Ay, Az)
sage: sa = a.hodge_star(g) ; sa
2-form '*A' on the 3-dimensional manifold 'M'
sage: sa.show()
*A = Az dx/\dy - Ay dx/\dz + Ax dy/\dz
sage: ssa = sa.hodge_star(g) ; ssa
1-form '**A' on the 3-dimensional manifold 'M'
sage: ssa.show()
**A = Ax dx + Ay dy + Az dz
sage: ssa == a # must hold for a Riemannian metric in dimension 3
True
Hodge star of a 0-form (scalar field) in `R^3`::
sage: f = ScalarField(m, function('F',x,y,z), name='f')
sage: sf = f.hodge_star(g) ; sf
3-form '*f' on the 3-dimensional manifold 'M'
sage: sf.show()
*f = F(x, y, z) dx/\dy/\dz
sage: ssf = sf.hodge_star(g) ; ssf
scalar field '**f' on the 3-dimensional manifold 'M'
sage: ssf.show()
**f: (x, y, z) |--> F(x, y, z)
sage: ssf == f # must hold for a Riemannian metric
True
Hodge star of a 0-form in Minkowksi spacetime::
sage: m = Manifold(4, 'M')
sage: X = Chart(m, 't x y z', 'txyz')
sage: g = Metric(m, 'g', signature=2)
sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1, 1, 1, 1
sage: g.show() # Minkowski metric
g = -dt*dt + dx*dx + dy*dy + dz*dz
sage: var('f0')
f0
sage: f = ScalarField(m, f0, name='f')
sage: sf = f.hodge_star(g) ; sf
4-form '*f' on the 4-dimensional manifold 'M'
sage: sf.show()
*f = f0 dt/\dx/\dy/\dz
sage: ssf = sf.hodge_star(g) ; ssf
scalar field '**f' on the 4-dimensional manifold 'M'
sage: ssf.show()
**f: (t, x, y, z) |--> -f0
sage: ssf == -f # must hold for a Lorentzian metric
True
Hodge star of a 1-form in Minkowksi spacetime::
sage: a = OneForm(m, 'A')
sage: var('At Ax Ay Az')
(At, Ax, Ay, Az)
sage: a[:] = (At, Ax, Ay, Az)
sage: a.show()
A = At dt + Ax dx + Ay dy + Az dz
sage: sa = a.hodge_star(g) ; sa
3-form '*A' on the 4-dimensional manifold 'M'
sage: sa.show()
*A = -Az dt/\dx/\dy + Ay dt/\dx/\dz - Ax dt/\dy/\dz - At dx/\dy/\dz
sage: ssa = sa.hodge_star(g) ; ssa
1-form '**A' on the 4-dimensional manifold 'M'
sage: ssa.show()
**A = At dt + Ax dx + Ay dy + Az dz
sage: ssa == a # must hold for a Lorentzian metric in dimension 4
True
Hodge star of a 2-form in Minkowksi spacetime::
sage: F = DiffForm(m, 2, 'F')
sage: var('Ex Ey Ez Bx By Bz')
(Ex, Ey, Ez, Bx, By, Bz)
sage: F[0,1], F[0,2], F[0,3] = -Ex, -Ey, -Ez
sage: F[1,2], F[1,3], F[2,3] = Bz, -By, Bx
sage: F[:]
[ 0 -Ex -Ey -Ez]
[ Ex 0 Bz -By]
[ Ey -Bz 0 Bx]
[ Ez By -Bx 0]
sage: sF = F.hodge_star(g) ; sF
2-form '*F' on the 4-dimensional manifold 'M'
sage: sF[:]
[ 0 Bx By Bz]
[-Bx 0 Ez -Ey]
[-By -Ez 0 Ex]
[-Bz Ey -Ex 0]
sage: ssF = sF.hodge_star(g) ; ssF
2-form '**F' on the 4-dimensional manifold 'M'
sage: ssF[:]
[ 0 Ex Ey Ez]
[-Ex 0 -Bz By]
[-Ey Bz 0 -Bx]
[-Ez -By Bx 0]
sage: ssF.show()
**F = Ex dt/\dx + Ey dt/\dy + Ez dt/\dz - Bz dx/\dy + By dx/\dz - Bx dy/\dz
sage: F.show()
F = -Ex dt/\dx - Ey dt/\dy - Ez dt/\dz + Bz dx/\dy - By dx/\dz + Bx dy/\dz
sage: ssF == -F # must hold for a Lorentzian metric in dimension 4
True
Test of the standard identity
.. MATH::
*(A\wedge B) = \epsilon(A^\sharp, B^\sharp, ., .)
where `A` and `B` are any 1-forms and `A^\sharp` and `B^\sharp` the
vectors associated to them by the metric `g` (index raising)::
sage: b = OneForm(m, 'B')
sage: var('Bt Bx By Bz')
(Bt, Bx, By, Bz)
sage: b[:] = (Bt, Bx, By, Bz) ; b.show()
B = Bt dt + Bx dx + By dy + Bz dz
sage: epsilon = g.volume_form()
sage: (a.wedge(b)).hodge_star(g) == epsilon.contract(0, a.up(g), 0).contract(0, b.up(g), 0)
True
"""
from sage.functions.other import factorial
from utilities import format_unop_txt, format_unop_latex
p = self.rank
eps = metric.volume_form(p)
if p == 0:
resu = self * eps
else:
resu = self.contract(0, eps, 0)
for j in range(1, p):
resu = resu.self_contract(0, p - j)
if p > 1:
resu = resu / factorial(p)
# Name and LaTeX name of the result:
resu.name = format_unop_txt('*', self.name)
resu.latex_name = format_unop_latex(r'\star ', self.latex_name)
return resu
def exterior_der(self, chart_name=None):
r"""
Compute the exterior derivative of the differential form.
INPUT:
- ``chart_name`` -- (default: None): name of the chart used for the
computation; if none is provided, the computation is performed in a
coordinate basis where the components are known, or computable by
a component transformation, privileging the domain's default chart.
OUTPUT:
- the exterior derivative of ``self``.
EXAMPLE:
Exterior derivative of a 1-form on a 4-dimensional manifold::
sage: m = Manifold(4, 'M')
sage: c_txyz.<t,x,y,z> = m.chart('t x y z', 'coord_txyz')
sage: a = OneForm(m, 'A')
sage: a[:] = (t*x*y*z, z*y**2, x*z**2, x**2 + y**2)
sage: da = a.exterior_der() ; da
2-form 'dA' on the 4-dimensional manifold 'M'
sage: da.view()
dA = -t*y*z dt/\dx - t*x*z dt/\dy - t*x*y dt/\dz + (-2*y*z + z^2) dx/\dy + (-y^2 + 2*x) dx/\dz + (-2*x*z + 2*y) dy/\dz
sage: latex(da)
\mathrm{d}A
The exterior derivative is nilpotent::
sage: dda = da.exterior_der() ; dda
3-form 'ddA' on the 4-dimensional manifold 'M'
sage: dda.view()
ddA = 0
sage: dda == 0
True
"""
from sage.calculus.functional import diff
from utilities import format_unop_txt, format_unop_latex
if self._exterior_derivative is None:
# A new computation is necessary:
if chart_name is None:
frame_name = self.pick_a_coord_basis()
if frame_name is None:
raise ValueError("No coordinate basis could be found for " +
"the differential form components.")
chart_name = frame_name[:-2]
chart = self.domain.atlas[chart_name]
else:
chart = self.domain.atlas[chart_name]
frame_name = chart.frame.name
if frame_name not in self.components:
raise ValueError("Components in the frame " + frame_name +
" have not been defined.")
n = self.manifold.dim
si = self.manifold.sindex
sc = self.components[frame_name]
dc = CompFullyAntiSym(self.domain.frames[frame_name], self.rank+1)
for ind, val in sc._comp.items():
for i in range(n):
ind_d = (i+si,) + ind
if len(ind_d) == len(set(ind_d)): # all indices are different
dc[[ind_d]] += val.function_chart(chart_name).diff(chart.xx[i])
dc._del_zeros()
# Name and LaTeX name of the result (rname and rlname):
rname = format_unop_txt('d', self.name)
rlname = format_unop_latex(r'\mathrm{d}', self.latex_name)
# Final result
self._exterior_derivative = DiffForm(self.domain, self.rank+1,
rname, rlname)
self._exterior_derivative.components[frame_name] = dc
return self._exterior_derivative
def exterior_der(self, chart_name=None):
r"""
Compute the exterior derivative of the differential form.
INPUT:
- ``chart_name`` -- (default: None): name of the chart used for the
computation; if none is provided, the computation is performed in a
coordinate basis where the components are known, or computable by
a component transformation, privileging the manifold's default chart.
OUTPUT:
- the exterior derivative of ``self``.
EXAMPLE:
Exterior derivative of a 1-form on a 4-dimensional manifold::
sage: m = Manifold(4, 'M')
sage: c_txyz = Chart(m, 't x y z', 'coord_txyz')
sage: a = OneForm(m, 'A')
sage: a[:] = (t*x*y*z, z*y**2, x*z**2, x**2 + y**2)
sage: da = a.exterior_der() ; da
2-form 'dA' on the 4-dimensional manifold 'M'
sage: da.show()
dA = -t*y*z dt/\dx - t*x*z dt/\dy - t*x*y dt/\dz + (-2*y*z + z^2) dx/\dy + (-y^2 + 2*x) dx/\dz + (-2*x*z + 2*y) dy/\dz
sage: latex(da)
\mathrm{d}A
The exterior derivative is nilpotent::
sage: dda = da.exterior_der() ; dda
3-form 'ddA' on the 4-dimensional manifold 'M'
sage: dda.show()
ddA = 0
sage: dda == 0
True
"""
from sage.calculus.functional import diff
from utilities import format_unop_txt, format_unop_latex
if self._exterior_derivative is None:
# A new computation is necessary:
if chart_name is None:
frame_name = self.pick_a_coord_basis()
if frame_name is None:
raise ValueError(
"No coordinate basis could be found for " +
"the differential form components.")
chart_name = frame_name[:-2]
chart = self.manifold.atlas[chart_name]
else:
chart = self.manifold.atlas[chart_name]
frame_name = chart.frame.name
if frame_name not in self.components:
raise ValueError("Components in the frame " + frame_name +
" have not been defined.")
n = self.manifold.dim
si = self.manifold.sindex
sc = self.components[frame_name]
dc = CompFullyAntiSym(self.manifold, self.rank + 1, frame_name)
for ind, val in sc._comp.items():
for i in range(n):
ind_d = (i + si, ) + ind
if len(ind_d) == len(
set(ind_d)): # all indices are different
dc[[ind_d]] += val.function_chart(chart_name).diff(
chart.xx[i])
dc._del_zeros()
# Name and LaTeX name of the result (rname and rlname):
rname = format_unop_txt('d', self.name)
rlname = format_unop_latex(r'\mathrm{d}', self.latex_name)
# Final result
self._exterior_derivative = DiffForm(self.manifold, self.rank + 1,
rname, rlname)
self._exterior_derivative.components[frame_name] = dc
return self._exterior_derivative