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r""" Operators for vector calculus
This module defines the following operators for scalar, vector and tensor fields on any pseudo-Riemannian manifold (see :mod:`~sage.manifolds.differentiable.pseudo_riemannian`):
- :func:`grad`: gradient of a scalar field - :func:`div`: divergence of a vector field, and more generally of a tensor field - :func:`curl`: curl of a vector field (3-dimensional case only) - :func:`laplacian`: Laplace-Beltrami operator acting on a scalar field, a vector field, or more generally a tensor field - :func:`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 :func:`~sage.misc.functional.norm` operator is defined in the module :mod:`~sage.misc.functional`.
AUTHORS:
- Eric Gourgoulhon (2018): initial version
"""
#***************************************************************************** # Copyright (C) 2018 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
r""" 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
.. MATH::
(\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:
- ``scalar`` -- scalar field `f`, as an instance of :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField`
OUTPUT:
- instance of :class:`~sage.manifolds.differentiable.vectorfield.VectorField` representing `\mathrm{grad}\, f`
EXAMPLES:
Gradient of a scalar field in the Euclidean plane::
sage: M = Manifold(2, 'M', structure='Riemannian') sage: X.<x,y> = M.chart() sage: g = M.metric() sage: g[0,0], g[1,1] = 1, 1 sage: f = M.scalar_field(sin(x*y), name='f') sage: from sage.manifolds.operators import grad sage: grad(f) Vector field grad(f) on the 2-dimensional Riemannian manifold M sage: grad(f).display() grad(f) = y*cos(x*y) d/dx + x*cos(x*y) d/dy
See the method :meth:`~sage.manifolds.differentiable.scalarfield.DiffScalarField.gradient` of :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField` for more details and examples.
"""
r""" 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
.. MATH::
(\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. :class:`~sage.manifolds.differentiable.levi_civita_connection.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
.. MATH::
(\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 :meth:`~sage.manifolds.differentiable.tensorfield.TensorField.up`).
INPUT:
- ``tensor`` -- tensor field `t` on a pseudo-Riemannian manifold `(M,g)`, as an instance of :class:`~sage.manifolds.differentiable.tensorfield.TensorField` (possibly via one of its derived classes, like :class:`~sage.manifolds.differentiable.vectorfield.VectorField`)
OUTPUT:
- the divergence of ``tensor`` as an instance of either :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField` if `(k,l)=(1,0)` (``tensor`` is a vector field) or `(k,l)=(0,1)` (``tensor`` is a 1-form) or of :class:`~sage.manifolds.differentiable.tensorfield.TensorField` if `k+l\geq 2`
EXAMPLES:
Divergence of a vector field in the Euclidean plane::
sage: M = Manifold(2, 'M', structure='Riemannian') sage: X.<x,y> = M.chart() sage: g = M.metric() sage: g[0,0], g[1,1] = 1, 1 sage: v = M.vector_field('v') sage: v[:] = cos(x*y), sin(x*y) sage: v.display() v = cos(x*y) d/dx + sin(x*y) d/dy sage: from sage.manifolds.operators import div sage: s = div(v); s Scalar field div(v) on the 2-dimensional Riemannian manifold M sage: s.display() div(v): M --> R (x, y) |--> x*cos(x*y) - y*sin(x*y)
See the method :meth:`~sage.manifolds.differentiable.tensorfield.TensorField.divergence` of :class:`~sage.manifolds.differentiable.tensorfield.TensorField` for more details and examples.
"""
r""" 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
.. MATH::
\mathrm{curl}\, v = (*(\mathrm{d} v^\flat))^\sharp
where `v^\flat` is the 1-form associated to `v` by the metric `g` (see :meth:`~sage.manifolds.differentiable.tensorfield.TensorField.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 :meth:`~sage.manifolds.differentiable.diff_form.DiffForm.hodge_dual`) and `(*(\mathrm{d} v^\flat))^\sharp` is corresponding vector field by `g`-duality (see :meth:`~sage.manifolds.differentiable.tensorfield.TensorField.up`).
An alternative expression of the curl is
.. MATH::
(\mathrm{curl}\, v)^i = \epsilon^{ijk} \nabla_j v_k
where `\nabla` is the Levi-Civita connection of `g` (cf. :class:`~sage.manifolds.differentiable.levi_civita_connection.LeviCivitaConnection`) and `\epsilon` the volume 3-form (Levi-Civita tensor) of `g` (cf. :meth:`~sage.manifolds.differentiable.metric.PseudoRiemannianMetric.volume_form`)
INPUT:
- ``vector`` -- vector field on an orientable 3-dimensional pseudo-Riemannian manifold, as an instance of :class:`~sage.manifolds.differentiable.vectorfield.VectorField`
OUTPUT:
- instance of :class:`~sage.manifolds.differentiable.vectorfield.VectorField` representing the curl of ``vector``
EXAMPLES:
Curl of a vector field in the Euclidean 3-space::
sage: M = Manifold(3, 'M', structure='Riemannian') sage: X.<x,y,z> = M.chart() sage: g = M.metric() sage: g[0,0], g[1,1], g[2,2] = 1, 1, 1 sage: v = M.vector_field(name='v') sage: v[0], v[1] = sin(y), sin(x) sage: v.display() v = sin(y) d/dx + sin(x) d/dy sage: from sage.manifolds.operators import curl sage: s = curl(v); s Vector field curl(v) on the 3-dimensional Riemannian manifold M sage: s.display() curl(v) = (cos(x) - cos(y)) d/dz
See the method :meth:`~sage.manifolds.differentiable.vectorfield.VectorField.curl` of :class:`~sage.manifolds.differentiable.vectorfield.VectorField` for more details and examples.
"""
r""" Laplace-Beltrami operator.
The *Laplace-Beltrami operator* on a pseudo-Riemannian manifold `(M,g)` is the operator
.. MATH::
\Delta = \nabla_i \nabla^i = g^{ij} \nabla_i \nabla_j
where `\nabla` is the Levi-Civita connection of the metric `g` (cf. :class:`~sage.manifolds.differentiable.levi_civita_connection.LeviCivitaConnection`) and `\nabla^i := g^{ij} \nabla_j`
INPUT:
- ``field`` -- a scalar field `f` (instance of :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField`) or a tensor field `f` (instance of :class:`~sage.manifolds.differentiable.tensorfield.TensorField`) on a pseudo-Riemannian manifold
OUTPUT:
- `\Delta f`, as an instance of :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField` or of :class:`~sage.manifolds.differentiable.tensorfield.TensorField`
EXAMPLES:
Laplacian of a scalar field on the Euclidean plane::
sage: M = Manifold(2, 'M', structure='Riemannian') sage: X.<x,y> = M.chart() sage: g = M.metric() sage: g[0,0], g[1,1] = 1, 1 sage: f = M.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 2-dimensional Riemannian manifold M sage: Df.display() Delta(f): M --> R (x, y) |--> -(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
See the method :meth:`~sage.manifolds.differentiable.scalarfield.DiffScalarField.laplacian` of :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField` and the method :meth:`~sage.manifolds.differentiable.tensorfield.TensorField.laplacian` of :class:`~sage.manifolds.differentiable.tensorfield.TensorField` for more details and examples.
"""
r""" d'Alembert operator.
The *d'Alembert operator* or *d'Alembertian* on a Lorentzian manifold `(M,g)` is nothing but the Laplace-Beltrami operator:
.. MATH::
\Box = \nabla_i \nabla^i = g^{ij} \nabla_i \nabla_j
where `\nabla` is the Levi-Civita connection of the metric `g` (cf. :class:`~sage.manifolds.differentiable.levi_civita_connection.LeviCivitaConnection`) and `\nabla^i := g^{ij} \nabla_j`
INPUT:
- ``field`` -- a scalar field `f` (instance of :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField`) or a tensor field `f` (instance of :class:`~sage.manifolds.differentiable.tensorfield.TensorField`) on a pseudo-Riemannian manifold
OUTPUT:
- `\Box f`, as an instance of :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField` or of :class:`~sage.manifolds.differentiable.tensorfield.TensorField`
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
See the method :meth:`~sage.manifolds.differentiable.scalarfield.DiffScalarField.dalembertian` of :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField` and the method :meth:`~sage.manifolds.differentiable.tensorfield.TensorField.dalembertian` of :class:`~sage.manifolds.differentiable.tensorfield.TensorField` for more details and examples.
"""
# NB: norm() is already defined in src/sage/misc/functional.py |