Operators for vector calculus#

This module defines the following operators for scalar, vector and tensor fields on any pseudo-Riemannian manifold (see pseudo_riemannian), and in particular on Euclidean spaces (see euclidean):

  • grad(): gradient of a scalar field

  • div(): divergence of a vector field, and more generally of a tensor field

  • curl(): curl of a vector field (3-dimensional case only)

  • laplacian(): Laplace-Beltrami operator acting on a scalar field, a vector field, or more generally a tensor field

  • dalembertian(): d’Alembert operator acting on a scalar field, a vector field, or more generally a tensor field, on a Lorentzian manifold

All these operators are implemented as functions that call the appropriate method on their argument. The purpose is to allow one to use standard mathematical notations, e.g. to write curl(v) instead of v.curl().

Note that the norm() operator is defined in the module functional.

See also

Examples 1 and 2 in euclidean for examples involving these operators in the Euclidean plane and in the Euclidean 3-space.

AUTHORS:

  • Eric Gourgoulhon (2018): initial version

sage.manifolds.operators.curl(vector)[source]#

Curl operator.

The curl of a vector field \(v\) on an orientable pseudo-Riemannian manifold \((M,g)\) of dimension 3 is the vector field defined by

\[\mathrm{curl}\, v = (*(\mathrm{d} v^\flat))^\sharp\]

where \(v^\flat\) is the 1-form associated to \(v\) by the metric \(g\) (see down()), \(*(\mathrm{d} v^\flat)\) is the Hodge dual with respect to \(g\) of the 2-form \(\mathrm{d} v^\flat\) (exterior derivative of \(v^\flat\)) (see hodge_dual()) and \((*(\mathrm{d} v^\flat))^\sharp\) is corresponding vector field by \(g\)-duality (see up()).

An alternative expression of the curl is

\[(\mathrm{curl}\, v)^i = \epsilon^{ijk} \nabla_j v_k\]

where \(\nabla\) is the Levi-Civita connection of \(g\) (cf. LeviCivitaConnection) and \(\epsilon\) the volume 3-form (Levi-Civita tensor) of \(g\) (cf. volume_form())

INPUT:

  • vector – vector field on an orientable 3-dimensional pseudo-Riemannian manifold, as an instance of VectorField

OUTPUT:

  • instance of VectorField representing the curl of vector

EXAMPLES:

Curl of a vector field in the Euclidean 3-space:

sage: E.<x,y,z> = EuclideanSpace()
sage: v = E.vector_field(sin(y), sin(x), 0, name='v')
sage: v.display()
v = sin(y) e_x + sin(x) e_y
sage: from sage.manifolds.operators import curl
sage: s = curl(v); s
Vector field curl(v) on the Euclidean space E^3
sage: s.display()
curl(v) = (cos(x) - cos(y)) e_z
sage: s[:]
[0, 0, cos(x) - cos(y)]

>>> from sage.all import *
>>> E = EuclideanSpace(names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3)
>>> v = E.vector_field(sin(y), sin(x), Integer(0), name='v')
>>> v.display()
v = sin(y) e_x + sin(x) e_y
>>> from sage.manifolds.operators import curl
>>> s = curl(v); s
Vector field curl(v) on the Euclidean space E^3
>>> s.display()
curl(v) = (cos(x) - cos(y)) e_z
>>> s[:]
[0, 0, cos(x) - cos(y)]

See the method curl() of VectorField for more details and examples.

sage.manifolds.operators.dalembertian(field)[source]#

d’Alembert operator.

The d’Alembert operator or d’Alembertian on a Lorentzian manifold \((M,g)\) is nothing but the Laplace-Beltrami operator:

\[\Box = \nabla_i \nabla^i = g^{ij} \nabla_i \nabla_j\]

where \(\nabla\) is the Levi-Civita connection of the metric \(g\) (cf. LeviCivitaConnection) and \(\nabla^i := g^{ij} \nabla_j\)

INPUT:

  • field – a scalar field \(f\) (instance of DiffScalarField) or a tensor field \(f\) (instance of TensorField) on a pseudo-Riemannian manifold

OUTPUT:

EXAMPLES:

d’Alembertian of a scalar field in the 2-dimensional Minkowski spacetime:

sage: M = Manifold(2, 'M', structure='Lorentzian')
sage: X.<t,x> = M.chart()
sage: g = M.metric()
sage: g[0,0], g[1,1] = -1, 1
sage: f = M.scalar_field((x-t)^3 + (x+t)^2, name='f')
sage: from sage.manifolds.operators import dalembertian
sage: Df = dalembertian(f); Df
Scalar field Box(f) on the 2-dimensional Lorentzian manifold M
sage: Df.display()
Box(f): M → ℝ
   (t, x) ↦ 0

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='Lorentzian')
>>> X = M.chart(names=('t', 'x',)); (t, x,) = X._first_ngens(2)
>>> g = M.metric()
>>> g[Integer(0),Integer(0)], g[Integer(1),Integer(1)] = -Integer(1), Integer(1)
>>> f = M.scalar_field((x-t)**Integer(3) + (x+t)**Integer(2), name='f')
>>> from sage.manifolds.operators import dalembertian
>>> Df = dalembertian(f); Df
Scalar field Box(f) on the 2-dimensional Lorentzian manifold M
>>> Df.display()
Box(f): M → ℝ
   (t, x) ↦ 0

See the method dalembertian() of DiffScalarField and the method dalembertian() of TensorField for more details and examples.

sage.manifolds.operators.div(tensor)[source]#

Divergence operator.

Let \(t\) be a tensor field of type \((k,0)\) with \(k\geq 1\) on a pseudo-Riemannian manifold \((M, g)\). The divergence of \(t\) is the tensor field of type \((k-1,0)\) defined by

\[(\mathrm{div}\, t)^{a_1\ldots a_{k-1}} = \nabla_i t^{a_1\ldots a_{k-1} i} = (\nabla t)^{a_1\ldots a_{k-1} i}_{\phantom{a_1\ldots a_{k-1} i}\, i}\]

where \(\nabla\) is the Levi-Civita connection of \(g\) (cf. LeviCivitaConnection).

Note that the divergence is taken on the last index of the tensor field. This definition is extended to tensor fields of type \((k,l)\) with \(k\geq 0\) and \(l\geq 1\), by raising the last index with the metric \(g\): \(\mathrm{div}\, t\) is then the tensor field of type \((k,l-1)\) defined by

\[(\mathrm{div}\, t)^{a_1\ldots a_k}_{\phantom{a_1\ldots a_k}\, b_1\ldots b_{l-1}} = \nabla_i (g^{ij} t^{a_1\ldots a_k}_{\phantom{a_1\ldots a_k}\, b_1\ldots b_{l-1} j}) = (\nabla t^\sharp)^{a_1\ldots a_k i}_{\phantom{a_1\ldots a_k i}\, b_1\ldots b_{l-1} i}\]

where \(t^\sharp\) is the tensor field deduced from \(t\) by raising the last index with the metric \(g\) (see up()).

INPUT:

  • tensor – tensor field \(t\) on a pseudo-Riemannian manifold \((M,g)\), as an instance of TensorField (possibly via one of its derived classes, like VectorField)

OUTPUT:

  • the divergence of tensor as an instance of either DiffScalarField if \((k,l)=(1,0)\) (tensor is a vector field) or \((k,l)=(0,1)\) (tensor is a 1-form) or of TensorField if \(k+l\geq 2\)

EXAMPLES:

Divergence of a vector field in the Euclidean plane:

sage: E.<x,y> = EuclideanSpace()
sage: v = E.vector_field(cos(x*y), sin(x*y), name='v')
sage: v.display()
v = cos(x*y) e_x + sin(x*y) e_y
sage: from sage.manifolds.operators import div
sage: s = div(v); s
Scalar field div(v) on the Euclidean plane E^2
sage: s.display()
div(v): E^2 → ℝ
   (x, y) ↦ x*cos(x*y) - y*sin(x*y)
sage: s.expr()
x*cos(x*y) - y*sin(x*y)

>>> from sage.all import *
>>> E = EuclideanSpace(names=('x', 'y',)); (x, y,) = E._first_ngens(2)
>>> v = E.vector_field(cos(x*y), sin(x*y), name='v')
>>> v.display()
v = cos(x*y) e_x + sin(x*y) e_y
>>> from sage.manifolds.operators import div
>>> s = div(v); s
Scalar field div(v) on the Euclidean plane E^2
>>> s.display()
div(v): E^2 → ℝ
   (x, y) ↦ x*cos(x*y) - y*sin(x*y)
>>> s.expr()
x*cos(x*y) - y*sin(x*y)

See the method divergence() of TensorField for more details and examples.

sage.manifolds.operators.grad(scalar)[source]#

Gradient operator.

The gradient of a scalar field \(f\) on a pseudo-Riemannian manifold \((M,g)\) is the vector field \(\mathrm{grad}\, f\) whose components in any coordinate frame are

\[(\mathrm{grad}\, f)^i = g^{ij} \frac{\partial F}{\partial x^j}\]

where the \(x^j\)’s are the coordinates with respect to which the frame is defined and \(F\) is the chart function representing \(f\) in these coordinates: \(f(p) = F(x^1(p),\ldots,x^n(p))\) for any point \(p\) in the chart domain. In other words, the gradient of \(f\) is the vector field that is the \(g\)-dual of the differential of \(f\).

INPUT:

OUTPUT:

  • instance of VectorField representing \(\mathrm{grad}\, f\)

EXAMPLES:

Gradient of a scalar field in the Euclidean plane:

sage: E.<x,y> = EuclideanSpace()
sage: f = E.scalar_field(sin(x*y), name='f')
sage: from sage.manifolds.operators import grad
sage: grad(f)
Vector field grad(f) on the Euclidean plane E^2
sage: grad(f).display()
grad(f) = y*cos(x*y) e_x + x*cos(x*y) e_y
sage: grad(f)[:]
[y*cos(x*y), x*cos(x*y)]

>>> from sage.all import *
>>> E = EuclideanSpace(names=('x', 'y',)); (x, y,) = E._first_ngens(2)
>>> f = E.scalar_field(sin(x*y), name='f')
>>> from sage.manifolds.operators import grad
>>> grad(f)
Vector field grad(f) on the Euclidean plane E^2
>>> grad(f).display()
grad(f) = y*cos(x*y) e_x + x*cos(x*y) e_y
>>> grad(f)[:]
[y*cos(x*y), x*cos(x*y)]

See the method gradient() of DiffScalarField for more details and examples.

sage.manifolds.operators.laplacian(field)[source]#

Laplace-Beltrami operator.

The Laplace-Beltrami operator on a pseudo-Riemannian manifold \((M,g)\) is the operator

\[\Delta = \nabla_i \nabla^i = g^{ij} \nabla_i \nabla_j\]

where \(\nabla\) is the Levi-Civita connection of the metric \(g\) (cf. LeviCivitaConnection) and \(\nabla^i := g^{ij} \nabla_j\)

INPUT:

  • field – a scalar field \(f\) (instance of DiffScalarField) or a tensor field \(f\) (instance of TensorField) on a pseudo-Riemannian manifold

OUTPUT:

EXAMPLES:

Laplacian of a scalar field on the Euclidean plane:

sage: E.<x,y> = EuclideanSpace()
sage: f = E.scalar_field(sin(x*y), name='f')
sage: from sage.manifolds.operators import laplacian
sage: Df = laplacian(f); Df
Scalar field Delta(f) on the Euclidean plane E^2
sage: Df.display()
Delta(f): E^2 → ℝ
   (x, y) ↦ -(x^2 + y^2)*sin(x*y)
sage: Df.expr()
-(x^2 + y^2)*sin(x*y)

>>> from sage.all import *
>>> E = EuclideanSpace(names=('x', 'y',)); (x, y,) = E._first_ngens(2)
>>> f = E.scalar_field(sin(x*y), name='f')
>>> from sage.manifolds.operators import laplacian
>>> Df = laplacian(f); Df
Scalar field Delta(f) on the Euclidean plane E^2
>>> Df.display()
Delta(f): E^2 → ℝ
   (x, y) ↦ -(x^2 + y^2)*sin(x*y)
>>> Df.expr()
-(x^2 + y^2)*sin(x*y)

The Laplacian of a scalar field is the divergence of its gradient:

sage: from sage.manifolds.operators import div, grad
sage: Df == div(grad(f))
True

>>> from sage.all import *
>>> from sage.manifolds.operators import div, grad
>>> Df == div(grad(f))
True

See the method laplacian() of DiffScalarField and the method laplacian() of TensorField for more details and examples.