def __init__(self,
vector_field_module,
degree,
name=None,
latex_name=None):
r"""
Construct a differential form.
TEST::
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: eU = c_xy.frame() ; eV = c_uv.frame()
sage: a = M.diff_form(2, name='a') ; a
2-form 'a' on the 2-dimensional manifold 'M'
sage: a[eU,0,1] = x*y^2 + 2*x
sage: a.add_comp_by_continuation(eV, W, c_uv)
"""
TensorField.__init__(
self,
vector_field_module, (0, degree),
name=name,
latex_name=latex_name,
antisym=range(degree),
parent=vector_field_module.dual_exterior_power(degree))
self._init_derived() # initialization of derived quantities
def __init__(self, vector_field_module, name=None, latex_name=None):
TensorField.__init__(self, vector_field_module, (1,0), name=name,
latex_name=latex_name)
# Initialization of derived quantities:
TensorField._init_derived(self)
# Initialization of list of quantities depending on self:
self._init_dependencies()
def __init__(self,
vector_field_module,
name=None,
latex_name=None,
is_identity=False):
if is_identity:
if name is None:
name = 'Id'
if latex_name is None and name == 'Id':
latex_name = r'\mathrm{Id}'
TensorField.__init__(self,
vector_field_module, (1, 1),
name=name,
latex_name=latex_name,
parent=vector_field_module.general_linear_group())
self._is_identity = is_identity
self._init_derived() # initialization of derived quantities
# Specific initializations for the field of identity maps:
if self._is_identity:
self._inverse = self
for dom in self._domain._subsets:
if dom.is_manifestly_parallelizable():
fmodule = dom.vector_field_module()
self._restrictions[dom] = fmodule.identity_map(
name=name, latex_name=latex_name)
def __init__(self, domain, p, name=None, latex_name=None):
TensorField.__init__(self,
domain,
0,
p,
name,
latex_name,
antisym=range(p))
DiffForm._init_derived(self) # initialization of derived quantities
def __init__(self, vector_field_module, name=None, latex_name=None):
TensorField.__init__(self,
vector_field_module, (1, 0),
name=name,
latex_name=latex_name)
# Initialization of derived quantities:
TensorField._init_derived(self)
# Initialization of list of quantities depending on self:
self._init_dependencies()
def __init__(self, manifold, p, name=None, latex_name=None):
TensorField.__init__(self,
manifold,
0,
p,
name,
latex_name,
antisym=range(p))
self._init_derived() # initialization of derived quantities
def __init__(self, vector_field_module, name=None, latex_name=None,
is_identity=False):
if is_identity:
if name is None:
name = 'Id'
if latex_name is None and name == 'Id':
latex_name = r'\mathrm{Id}'
TensorField.__init__(self, vector_field_module, (1,1), name=name,
latex_name=latex_name,
parent=vector_field_module.general_linear_group())
self._is_identity = is_identity
self._init_derived() # initialization of derived quantities
# Specific initializations for the field of identity maps:
if self._is_identity:
self._inverse = self
for dom in self._domain._subsets:
if dom.is_manifestly_parallelizable():
fmodule = dom.vector_field_module()
self._restrictions[dom] = fmodule.identity_map(name=name,
latex_name=latex_name)
def __init__(self, domain, name=None, latex_name=None):
TensorField.__init__(self, domain, 1, 1, name, latex_name)
def __init__(self, domain, p, name=None, latex_name=None):
TensorField.__init__(self, domain, 0, p, name, latex_name,
antisym=range(p))
DiffForm._init_derived(self) # initialization of derived quantities
def __init__(self, manifold, name=None, latex_name=None):
TensorField.__init__(self, manifold, 0, 2, name, latex_name, sym=(0,1))
def __init__(self, domain, name=None, latex_name=None):
TensorField.__init__(self, domain, 1, 1, name, latex_name)
def __init__(self, domain, name=None, latex_name=None):
TensorField.__init__(self, domain, 0, 2, name, latex_name, sym=(0, 1))
def __init__(self, manifold, name=None, latex_name=None) :
TensorField.__init__(self, manifold, 1, 0, name, latex_name)
self._init_dependencies()
def __init__(self, domain, name=None, latex_name=None) :
TensorField.__init__(self, domain, 1, 0, name, latex_name)
self._init_dependencies()
def __init__(self, manifold, p, name=None, latex_name=None):
TensorField.__init__(self, manifold, 0, p, name, latex_name,
antisym=range(p))
self._init_derived() # initialization of derived quantities
def __init__(self, domain, name=None, latex_name=None):
TensorField.__init__(self, domain, 0, 2, name, latex_name, sym=(0,1))
def __init__(self, manifold, name=None, latex_name=None):
TensorField.__init__(self, manifold, 1, 0, name, latex_name)
self._init_dependencies()
def __init__(self, manifold, name=None, latex_name=None):
TensorField.__init__(self, manifold, 1, 1, name, latex_name)