How to perform vector calculus in curvilinear coordinates¶
This tutorial introduces some vector calculus capabilities of SageMath within the 3-dimensional Euclidean space. The corresponding tools have been developed via the SageManifolds project.
The tutorial is also available as a Jupyter notebook, either
passive (nbviewer)
or interactive (binder).
Using spherical coordinates¶
To use spherical coordinates \((r,\theta,\phi)\) within the Euclidean 3-space
\(\mathbb{E}^3\), it suffices to declare the latter with the keyword
coordinates='spherical':
sage: E.<r,th,ph> = EuclideanSpace(coordinates='spherical')
sage: E
Euclidean space E^3
>>> from sage.all import *
>>> E = EuclideanSpace(coordinates='spherical', names=('r', 'th', 'ph',)); (r, th, ph,) = E._first_ngens(3)
>>> E
Euclidean space E^3
E.<r,th,ph> = EuclideanSpace(coordinates='spherical') E
Thanks to the notation <r,th,ph> in the above declaration, the coordinates
\((r,\theta,\phi)\) are immediately available as three symbolic variables r,
th and ph (there is no need to declare them via var(), i.e. to
type r, th, ph = var('r th ph')):
sage: r is E.spherical_coordinates()[1]
True
sage: (r, th, ph) == E.spherical_coordinates()[:]
True
sage: type(r)
<class 'sage.symbolic.expression.Expression'>
>>> from sage.all import *
>>> r is E.spherical_coordinates()[Integer(1)]
True
>>> (r, th, ph) == E.spherical_coordinates()[:]
True
>>> type(r)
<class 'sage.symbolic.expression.Expression'>
r is E.spherical_coordinates()[1] (r, th, ph) == E.spherical_coordinates()[:] type(r)
Moreover, the coordinate LaTeX symbols are already set:
sage: latex(th)
{\theta}
>>> from sage.all import *
>>> latex(th)
{\theta}
latex(th)
The coordinate ranges are:
sage: E.spherical_coordinates().coord_range()
r: (0, +oo); th: (0, pi); ph: [0, 2*pi] (periodic)
>>> from sage.all import *
>>> E.spherical_coordinates().coord_range()
r: (0, +oo); th: (0, pi); ph: [0, 2*pi] (periodic)
E.spherical_coordinates().coord_range()
\(\mathbb{E}^3\) is endowed with the orthonormal vector frame \((e_r, e_\theta, e_\phi)\) associated with spherical coordinates:
sage: E.frames()
[Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph)),
Vector frame (E^3, (e_r,e_th,e_ph))]
>>> from sage.all import *
>>> E.frames()
[Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph)),
Vector frame (E^3, (e_r,e_th,e_ph))]
E.frames()
In the above output, (∂/∂r,∂/∂th,∂/∂ph) =
\(\left(\frac{\partial}{\partial r}, \frac{\partial}{\partial\theta}, \frac{\partial}{\partial \phi}\right)\)
is the coordinate frame associated with \((r,\theta,\phi)\); it is
not an orthonormal frame and will not be used below. The default frame is
the orthonormal one:
sage: E.default_frame()
Vector frame (E^3, (e_r,e_th,e_ph))
>>> from sage.all import *
>>> E.default_frame()
Vector frame (E^3, (e_r,e_th,e_ph))
E.default_frame()
Defining a vector field¶
We define a vector field on \(\mathbb{E}^3\) from its components in the orthonormal vector frame \((e_r,e_\theta,e_\phi)\):
sage: v = E.vector_field(r*sin(2*ph)*sin(th)^2 + r,
....: r*sin(2*ph)*sin(th)*cos(th),
....: 2*r*cos(ph)^2*sin(th), name='v')
sage: v.display()
v = (r*sin(2*ph)*sin(th)^2 + r) e_r + r*cos(th)*sin(2*ph)*sin(th) e_th
+ 2*r*cos(ph)^2*sin(th) e_ph
>>> from sage.all import *
>>> v = E.vector_field(r*sin(Integer(2)*ph)*sin(th)**Integer(2) + r,
... r*sin(Integer(2)*ph)*sin(th)*cos(th),
... Integer(2)*r*cos(ph)**Integer(2)*sin(th), name='v')
>>> v.display()
v = (r*sin(2*ph)*sin(th)^2 + r) e_r + r*cos(th)*sin(2*ph)*sin(th) e_th
+ 2*r*cos(ph)^2*sin(th) e_ph
v = E.vector_field(r*sin(2*ph)*sin(th)^2 + r,
r*sin(2*ph)*sin(th)*cos(th),
2*r*cos(ph)^2*sin(th), name='v')
v.display()
We can access to the components of \(v\) via the square bracket operator:
sage: v[1]
r*sin(2*ph)*sin(th)^2 + r
sage: v[:]
[r*sin(2*ph)*sin(th)^2 + r, r*cos(th)*sin(2*ph)*sin(th), 2*r*cos(ph)^2*sin(th)]
>>> from sage.all import *
>>> v[Integer(1)]
r*sin(2*ph)*sin(th)^2 + r
>>> v[:]
[r*sin(2*ph)*sin(th)^2 + r, r*cos(th)*sin(2*ph)*sin(th), 2*r*cos(ph)^2*sin(th)]
v[1] v[:]
A vector field can evaluated at any point of \(\mathbb{E}^3\):
sage: p = E((1, pi/2, pi), name='p')
sage: p
Point p on the Euclidean space E^3
sage: p.coordinates()
(1, 1/2*pi, pi)
sage: vp = v.at(p)
sage: vp
Vector v at Point p on the Euclidean space E^3
sage: vp.display()
v = e_r + 2 e_ph
>>> from sage.all import *
>>> p = E((Integer(1), pi/Integer(2), pi), name='p')
>>> p
Point p on the Euclidean space E^3
>>> p.coordinates()
(1, 1/2*pi, pi)
>>> vp = v.at(p)
>>> vp
Vector v at Point p on the Euclidean space E^3
>>> vp.display()
v = e_r + 2 e_ph
p = E((1, pi/2, pi), name='p') p p.coordinates() vp = v.at(p) vp vp.display()
We may define a vector field with generic components:
sage: u = E.vector_field(function('u_r')(r,th,ph),
....: function('u_theta')(r,th,ph),
....: function('u_phi')(r,th,ph),
....: name='u')
sage: u.display()
u = u_r(r, th, ph) e_r + u_theta(r, th, ph) e_th + u_phi(r, th, ph) e_ph
sage: u[:]
[u_r(r, th, ph), u_theta(r, th, ph), u_phi(r, th, ph)]
>>> from sage.all import *
>>> u = E.vector_field(function('u_r')(r,th,ph),
... function('u_theta')(r,th,ph),
... function('u_phi')(r,th,ph),
... name='u')
>>> u.display()
u = u_r(r, th, ph) e_r + u_theta(r, th, ph) e_th + u_phi(r, th, ph) e_ph
>>> u[:]
[u_r(r, th, ph), u_theta(r, th, ph), u_phi(r, th, ph)]
u = E.vector_field(function('u_r')(r,th,ph),
function('u_theta')(r,th,ph),
function('u_phi')(r,th,ph),
name='u')
u.display()
u[:]
Its value at the point \(p\) is then:
sage: up = u.at(p)
sage: up.display()
u = u_r(1, 1/2*pi, pi) e_r + u_theta(1, 1/2*pi, pi) e_th
+ u_phi(1, 1/2*pi, pi) e_ph
>>> from sage.all import *
>>> up = u.at(p)
>>> up.display()
u = u_r(1, 1/2*pi, pi) e_r + u_theta(1, 1/2*pi, pi) e_th
+ u_phi(1, 1/2*pi, pi) e_ph
up = u.at(p) up.display()
Differential operators in spherical coordinates¶
The standard operators \(\mathrm{grad}\), \(\mathrm{div}\), \(\mathrm{curl}\), etc.
involved in vector calculus are accessible as methods on scalar fields and
vector fields (e.g. v.div()). However, to allow for standard mathematical
notations (e.g. div(v)), let us import the functions
grad(), div(),
curl() and
laplacian():
sage: from sage.manifolds.operators import *
>>> from sage.all import *
>>> from sage.manifolds.operators import *
from sage.manifolds.operators import *
Gradient¶
We first introduce a scalar field, via its expression in terms of Cartesian coordinates; in this example, we consider a unspecified function of \((r,\theta,\phi)\):
sage: F = E.scalar_field(function('f')(r,th,ph), name='F')
sage: F.display()
F: E^3 → ℝ
(r, th, ph) ↦ f(r, th, ph)
>>> from sage.all import *
>>> F = E.scalar_field(function('f')(r,th,ph), name='F')
>>> F.display()
F: E^3 → ℝ
(r, th, ph) ↦ f(r, th, ph)
F = E.scalar_field(function('f')(r,th,ph), name='F')
F.display()
The value of \(F\) at a point:
sage: F(p)
f(1, 1/2*pi, pi)
>>> from sage.all import *
>>> F(p)
f(1, 1/2*pi, pi)
F(p)
The gradient of \(F\):
sage: grad(F)
Vector field grad(F) on the Euclidean space E^3
sage: grad(F).display()
grad(F) = d(f)/dr e_r + d(f)/dth/r e_th + d(f)/dph/(r*sin(th)) e_ph
sage: norm(grad(F)).display()
|grad(F)|: E^3 → ℝ
(r, th, ph) ↦ sqrt((r^2*(d(f)/dr)^2 + (d(f)/dth)^2)*sin(th)^2
+ (d(f)/dph)^2)/(r*sin(th))
>>> from sage.all import *
>>> grad(F)
Vector field grad(F) on the Euclidean space E^3
>>> grad(F).display()
grad(F) = d(f)/dr e_r + d(f)/dth/r e_th + d(f)/dph/(r*sin(th)) e_ph
>>> norm(grad(F)).display()
|grad(F)|: E^3 → ℝ
(r, th, ph) ↦ sqrt((r^2*(d(f)/dr)^2 + (d(f)/dth)^2)*sin(th)^2
+ (d(f)/dph)^2)/(r*sin(th))
grad(F) grad(F).display() norm(grad(F)).display()
Divergence¶
The divergence of a vector field:
sage: s = div(u)
sage: s.display()
div(u): E^3 → ℝ
(r, th, ph) ↦ ((r*d(u_r)/dr + 2*u_r(r, th, ph)
+ d(u_theta)/dth)*sin(th) + cos(th)*u_theta(r, th, ph)
+ d(u_phi)/dph)/(r*sin(th))
sage: s.expr().expand()
2*u_r(r, th, ph)/r + cos(th)*u_theta(r, th, ph)/(r*sin(th))
+ diff(u_theta(r, th, ph), th)/r + diff(u_phi(r, th, ph), ph)/(r*sin(th))
+ diff(u_r(r, th, ph), r)
>>> from sage.all import *
>>> s = div(u)
>>> s.display()
div(u): E^3 → ℝ
(r, th, ph) ↦ ((r*d(u_r)/dr + 2*u_r(r, th, ph)
+ d(u_theta)/dth)*sin(th) + cos(th)*u_theta(r, th, ph)
+ d(u_phi)/dph)/(r*sin(th))
>>> s.expr().expand()
2*u_r(r, th, ph)/r + cos(th)*u_theta(r, th, ph)/(r*sin(th))
+ diff(u_theta(r, th, ph), th)/r + diff(u_phi(r, th, ph), ph)/(r*sin(th))
+ diff(u_r(r, th, ph), r)
s = div(u) s.display() s.expr().expand()
For \(v\), we have:
sage: div(v).expr()
3
>>> from sage.all import *
>>> div(v).expr()
3
div(v).expr()
Curl¶
The curl of a vector field:
sage: s = curl(u)
sage: s
Vector field curl(u) on the Euclidean space E^3
>>> from sage.all import *
>>> s = curl(u)
>>> s
Vector field curl(u) on the Euclidean space E^3
s = curl(u) s
sage: s.display()
curl(u) = (cos(th)*u_phi(r, th, ph) + sin(th)*d(u_phi)/dth
- d(u_theta)/dph)/(r*sin(th)) e_r - ((r*d(u_phi)/dr + u_phi(r, th, ph))*sin(th)
- d(u_r)/dph)/(r*sin(th)) e_th + (r*d(u_theta)/dr + u_theta(r, th, ph)
- d(u_r)/dth)/r e_ph
>>> from sage.all import *
>>> s.display()
curl(u) = (cos(th)*u_phi(r, th, ph) + sin(th)*d(u_phi)/dth
- d(u_theta)/dph)/(r*sin(th)) e_r - ((r*d(u_phi)/dr + u_phi(r, th, ph))*sin(th)
- d(u_r)/dph)/(r*sin(th)) e_th + (r*d(u_theta)/dr + u_theta(r, th, ph)
- d(u_r)/dth)/r e_ph
s.display()
For \(v\), we have:
sage: curl(v).display()
curl(v) = 2*cos(th) e_r - 2*sin(th) e_th
>>> from sage.all import *
>>> curl(v).display()
curl(v) = 2*cos(th) e_r - 2*sin(th) e_th
curl(v).display()
The curl of a gradient is always zero:
sage: curl(grad(F)).display()
curl(grad(F)) = 0
>>> from sage.all import *
>>> curl(grad(F)).display()
curl(grad(F)) = 0
curl(grad(F)).display()
The divergence of a curl is always zero:
sage: div(curl(u)).display()
div(curl(u)): E^3 → ℝ
(r, th, ph) ↦ 0
>>> from sage.all import *
>>> div(curl(u)).display()
div(curl(u)): E^3 → ℝ
(r, th, ph) ↦ 0
div(curl(u)).display()
Laplacian¶
The Laplacian of a scalar field:
sage: s = laplacian(F)
sage: s.display()
Delta(F): E^3 → ℝ
(r, th, ph) ↦ ((r^2*d^2(f)/dr^2 + 2*r*d(f)/dr
+ d^2(f)/dth^2)*sin(th)^2 + cos(th)*sin(th)*d(f)/dth
+ d^2(f)/dph^2)/(r^2*sin(th)^2)
sage: s.expr().expand()
2*diff(f(r, th, ph), r)/r + cos(th)*diff(f(r, th, ph), th)/(r^2*sin(th))
+ diff(f(r, th, ph), th, th)/r^2 + diff(f(r, th, ph), ph, ph)/(r^2*sin(th)^2)
+ diff(f(r, th, ph), r, r)
>>> from sage.all import *
>>> s = laplacian(F)
>>> s.display()
Delta(F): E^3 → ℝ
(r, th, ph) ↦ ((r^2*d^2(f)/dr^2 + 2*r*d(f)/dr
+ d^2(f)/dth^2)*sin(th)^2 + cos(th)*sin(th)*d(f)/dth
+ d^2(f)/dph^2)/(r^2*sin(th)^2)
>>> s.expr().expand()
2*diff(f(r, th, ph), r)/r + cos(th)*diff(f(r, th, ph), th)/(r^2*sin(th))
+ diff(f(r, th, ph), th, th)/r^2 + diff(f(r, th, ph), ph, ph)/(r^2*sin(th)^2)
+ diff(f(r, th, ph), r, r)
s = laplacian(F) s.display() s.expr().expand()
The Laplacian of a vector field:
sage: Du = laplacian(u)
sage: Du.display()
Delta(u) = ((r^2*d^2(u_r)/dr^2 + 2*r*d(u_r)/dr - 2*u_r(r, th, ph)
+ d^2(u_r)/dth^2 - 2*d(u_theta)/dth)*sin(th)^2 - ((2*u_theta(r, th, ph)
- d(u_r)/dth)*cos(th) + 2*d(u_phi)/dph)*sin(th) + d^2(u_r)/dph^2)/(r^2*sin(th)^2) e_r
+ ((r^2*d^2(u_theta)/dr^2 + 2*r*d(u_theta)/dr + 2*d(u_r)/dth + d^2(u_theta)/dth^2)*sin(th)^2
+ cos(th)*sin(th)*d(u_theta)/dth - 2*cos(th)*d(u_phi)/dph - u_theta(r, th, ph)
+ d^2(u_theta)/dph^2)/(r^2*sin(th)^2) e_th
+ ((r^2*d^2(u_phi)/dr^2 + 2*r*d(u_phi)/dr
+ d^2(u_phi)/dth^2)*sin(th)^2 + (cos(th)*d(u_phi)/dth + 2*d(u_r)/dph)*sin(th)
+ 2*cos(th)*d(u_theta)/dph - u_phi(r, th, ph) + d^2(u_phi)/dph^2)/(r^2*sin(th)^2) e_ph
>>> from sage.all import *
>>> Du = laplacian(u)
>>> Du.display()
Delta(u) = ((r^2*d^2(u_r)/dr^2 + 2*r*d(u_r)/dr - 2*u_r(r, th, ph)
+ d^2(u_r)/dth^2 - 2*d(u_theta)/dth)*sin(th)^2 - ((2*u_theta(r, th, ph)
- d(u_r)/dth)*cos(th) + 2*d(u_phi)/dph)*sin(th) + d^2(u_r)/dph^2)/(r^2*sin(th)^2) e_r
+ ((r^2*d^2(u_theta)/dr^2 + 2*r*d(u_theta)/dr + 2*d(u_r)/dth + d^2(u_theta)/dth^2)*sin(th)^2
+ cos(th)*sin(th)*d(u_theta)/dth - 2*cos(th)*d(u_phi)/dph - u_theta(r, th, ph)
+ d^2(u_theta)/dph^2)/(r^2*sin(th)^2) e_th
+ ((r^2*d^2(u_phi)/dr^2 + 2*r*d(u_phi)/dr
+ d^2(u_phi)/dth^2)*sin(th)^2 + (cos(th)*d(u_phi)/dth + 2*d(u_r)/dph)*sin(th)
+ 2*cos(th)*d(u_theta)/dph - u_phi(r, th, ph) + d^2(u_phi)/dph^2)/(r^2*sin(th)^2) e_ph
Du = laplacian(u) Du.display()
Since this expression is quite lengthy, we may ask for a display component by component:
sage: Du.display_comp()
Delta(u)^1 = ((r^2*d^2(u_r)/dr^2 + 2*r*d(u_r)/dr - 2*u_r(r, th, ph) + d^2(u_r)/dth^2
- 2*d(u_theta)/dth)*sin(th)^2 - ((2*u_theta(r, th, ph) - d(u_r)/dth)*cos(th)
+ 2*d(u_phi)/dph)*sin(th) + d^2(u_r)/dph^2)/(r^2*sin(th)^2)
Delta(u)^2 = ((r^2*d^2(u_theta)/dr^2 + 2*r*d(u_theta)/dr + 2*d(u_r)/dth
+ d^2(u_theta)/dth^2)*sin(th)^2 + cos(th)*sin(th)*d(u_theta)/dth
- 2*cos(th)*d(u_phi)/dph - u_theta(r, th, ph) + d^2(u_theta)/dph^2)/(r^2*sin(th)^2)
Delta(u)^3 = ((r^2*d^2(u_phi)/dr^2 + 2*r*d(u_phi)/dr + d^2(u_phi)/dth^2)*sin(th)^2
+ (cos(th)*d(u_phi)/dth + 2*d(u_r)/dph)*sin(th) + 2*cos(th)*d(u_theta)/dph
- u_phi(r, th, ph) + d^2(u_phi)/dph^2)/(r^2*sin(th)^2)
>>> from sage.all import *
>>> Du.display_comp()
Delta(u)^1 = ((r^2*d^2(u_r)/dr^2 + 2*r*d(u_r)/dr - 2*u_r(r, th, ph) + d^2(u_r)/dth^2
- 2*d(u_theta)/dth)*sin(th)^2 - ((2*u_theta(r, th, ph) - d(u_r)/dth)*cos(th)
+ 2*d(u_phi)/dph)*sin(th) + d^2(u_r)/dph^2)/(r^2*sin(th)^2)
Delta(u)^2 = ((r^2*d^2(u_theta)/dr^2 + 2*r*d(u_theta)/dr + 2*d(u_r)/dth
+ d^2(u_theta)/dth^2)*sin(th)^2 + cos(th)*sin(th)*d(u_theta)/dth
- 2*cos(th)*d(u_phi)/dph - u_theta(r, th, ph) + d^2(u_theta)/dph^2)/(r^2*sin(th)^2)
Delta(u)^3 = ((r^2*d^2(u_phi)/dr^2 + 2*r*d(u_phi)/dr + d^2(u_phi)/dth^2)*sin(th)^2
+ (cos(th)*d(u_phi)/dth + 2*d(u_r)/dph)*sin(th) + 2*cos(th)*d(u_theta)/dph
- u_phi(r, th, ph) + d^2(u_phi)/dph^2)/(r^2*sin(th)^2)
Du.display_comp()
We may expand each component:
sage: for i in E.irange():
....: s = Du[i].expand()
sage: Du.display_comp()
Delta(u)^1 = 2*d(u_r)/dr/r - 2*u_r(r, th, ph)/r^2
- 2*cos(th)*u_theta(r, th, ph)/(r^2*sin(th)) + cos(th)*d(u_r)/dth/(r^2*sin(th))
+ d^2(u_r)/dth^2/r^2 - 2*d(u_theta)/dth/r^2 - 2*d(u_phi)/dph/(r^2*sin(th))
+ d^2(u_r)/dph^2/(r^2*sin(th)^2) + d^2(u_r)/dr^2
Delta(u)^2 = 2*d(u_theta)/dr/r + 2*d(u_r)/dth/r^2 + cos(th)*d(u_theta)/dth/(r^2*sin(th))
+ d^2(u_theta)/dth^2/r^2 - 2*cos(th)*d(u_phi)/dph/(r^2*sin(th)^2)
- u_theta(r, th, ph)/(r^2*sin(th)^2) + d^2(u_theta)/dph^2/(r^2*sin(th)^2)
+ d^2(u_theta)/dr^2
Delta(u)^3 = 2*d(u_phi)/dr/r + cos(th)*d(u_phi)/dth/(r^2*sin(th))
+ d^2(u_phi)/dth^2/r^2 + 2*d(u_r)/dph/(r^2*sin(th))
+ 2*cos(th)*d(u_theta)/dph/(r^2*sin(th)^2) - u_phi(r, th, ph)/(r^2*sin(th)^2)
+ d^2(u_phi)/dph^2/(r^2*sin(th)^2) + d^2(u_phi)/dr^2
>>> from sage.all import *
>>> for i in E.irange():
... s = Du[i].expand()
>>> Du.display_comp()
Delta(u)^1 = 2*d(u_r)/dr/r - 2*u_r(r, th, ph)/r^2
- 2*cos(th)*u_theta(r, th, ph)/(r^2*sin(th)) + cos(th)*d(u_r)/dth/(r^2*sin(th))
+ d^2(u_r)/dth^2/r^2 - 2*d(u_theta)/dth/r^2 - 2*d(u_phi)/dph/(r^2*sin(th))
+ d^2(u_r)/dph^2/(r^2*sin(th)^2) + d^2(u_r)/dr^2
Delta(u)^2 = 2*d(u_theta)/dr/r + 2*d(u_r)/dth/r^2 + cos(th)*d(u_theta)/dth/(r^2*sin(th))
+ d^2(u_theta)/dth^2/r^2 - 2*cos(th)*d(u_phi)/dph/(r^2*sin(th)^2)
- u_theta(r, th, ph)/(r^2*sin(th)^2) + d^2(u_theta)/dph^2/(r^2*sin(th)^2)
+ d^2(u_theta)/dr^2
Delta(u)^3 = 2*d(u_phi)/dr/r + cos(th)*d(u_phi)/dth/(r^2*sin(th))
+ d^2(u_phi)/dth^2/r^2 + 2*d(u_r)/dph/(r^2*sin(th))
+ 2*cos(th)*d(u_theta)/dph/(r^2*sin(th)^2) - u_phi(r, th, ph)/(r^2*sin(th)^2)
+ d^2(u_phi)/dph^2/(r^2*sin(th)^2) + d^2(u_phi)/dr^2
for i in E.irange():
s = Du[i].expand()
Du.display_comp()
As a test, we may check that these formulas coincide with those of Wikipedia’s article Del in cylindrical and spherical coordinates.
Using cylindrical coordinates¶
The use of cylindrical coordinates \((\rho,\phi,z)\) in the Euclidean space \(\mathbb{E}^3\) is on the same footing as that of spherical coordinates. To start with, one has simply to declare:
sage: E.<rh,ph,z> = EuclideanSpace(coordinates='cylindrical')
>>> from sage.all import *
>>> E = EuclideanSpace(coordinates='cylindrical', names=('rh', 'ph', 'z',)); (rh, ph, z,) = E._first_ngens(3)
E.<rh,ph,z> = EuclideanSpace(coordinates='cylindrical')
The coordinate ranges are then:
sage: E.cylindrical_coordinates().coord_range()
rh: (0, +oo); ph: [0, 2*pi] (periodic); z: (-oo, +oo)
>>> from sage.all import *
>>> E.cylindrical_coordinates().coord_range()
rh: (0, +oo); ph: [0, 2*pi] (periodic); z: (-oo, +oo)
E.cylindrical_coordinates().coord_range()
The default vector frame is the orthonormal frame \((e_\rho,e_\phi,e_z)\) associated with cylindrical coordinates:
sage: E.default_frame()
Vector frame (E^3, (e_rh,e_ph,e_z))
>>> from sage.all import *
>>> E.default_frame()
Vector frame (E^3, (e_rh,e_ph,e_z))
E.default_frame()
and one may define vector fields from their components in that frame:
sage: v = E.vector_field(rh*(1+sin(2*ph)), 2*rh*cos(ph)^2, z,
....: name='v')
sage: v.display()
v = rh*(sin(2*ph) + 1) e_rh + 2*rh*cos(ph)^2 e_ph + z e_z
sage: v[:]
[rh*(sin(2*ph) + 1), 2*rh*cos(ph)^2, z]
>>> from sage.all import *
>>> v = E.vector_field(rh*(Integer(1)+sin(Integer(2)*ph)), Integer(2)*rh*cos(ph)**Integer(2), z,
... name='v')
>>> v.display()
v = rh*(sin(2*ph) + 1) e_rh + 2*rh*cos(ph)^2 e_ph + z e_z
>>> v[:]
[rh*(sin(2*ph) + 1), 2*rh*cos(ph)^2, z]
v = E.vector_field(rh*(1+sin(2*ph)), 2*rh*cos(ph)^2, z,
name='v')
v.display()
v[:]
sage: u = E.vector_field(function('u_rho')(rh,ph,z),
....: function('u_phi')(rh,ph,z),
....: function('u_z')(rh,ph,z),
....: name='u')
sage: u.display()
u = u_rho(rh, ph, z) e_rh + u_phi(rh, ph, z) e_ph + u_z(rh, ph, z) e_z
sage: u[:]
[u_rho(rh, ph, z), u_phi(rh, ph, z), u_z(rh, ph, z)]
>>> from sage.all import *
>>> u = E.vector_field(function('u_rho')(rh,ph,z),
... function('u_phi')(rh,ph,z),
... function('u_z')(rh,ph,z),
... name='u')
>>> u.display()
u = u_rho(rh, ph, z) e_rh + u_phi(rh, ph, z) e_ph + u_z(rh, ph, z) e_z
>>> u[:]
[u_rho(rh, ph, z), u_phi(rh, ph, z), u_z(rh, ph, z)]
u = E.vector_field(function('u_rho')(rh,ph,z),
function('u_phi')(rh,ph,z),
function('u_z')(rh,ph,z),
name='u')
u.display()
u[:]
Differential operators in cylindrical coordinates¶
sage: from sage.manifolds.operators import *
>>> from sage.all import *
>>> from sage.manifolds.operators import *
from sage.manifolds.operators import *
The gradient:
sage: F = E.scalar_field(function('f')(rh,ph,z), name='F')
sage: F.display()
F: E^3 → ℝ
(rh, ph, z) ↦ f(rh, ph, z)
sage: grad(F)
Vector field grad(F) on the Euclidean space E^3
sage: grad(F).display()
grad(F) = d(f)/drh e_rh + d(f)/dph/rh e_ph + d(f)/dz e_z
>>> from sage.all import *
>>> F = E.scalar_field(function('f')(rh,ph,z), name='F')
>>> F.display()
F: E^3 → ℝ
(rh, ph, z) ↦ f(rh, ph, z)
>>> grad(F)
Vector field grad(F) on the Euclidean space E^3
>>> grad(F).display()
grad(F) = d(f)/drh e_rh + d(f)/dph/rh e_ph + d(f)/dz e_z
F = E.scalar_field(function('f')(rh,ph,z), name='F')
F.display()
grad(F)
grad(F).display()
The divergence:
sage: s = div(u)
sage: s.display()
div(u): E^3 → ℝ
(rh, ph, z) ↦ (rh*d(u_rho)/drh + rh*d(u_z)/dz + u_rho(rh, ph, z) + d(u_phi)/dph)/rh
sage: s.expr().expand()
u_rho(rh, ph, z)/rh + diff(u_phi(rh, ph, z), ph)/rh + diff(u_rho(rh, ph, z), rh)
+ diff(u_z(rh, ph, z), z)
>>> from sage.all import *
>>> s = div(u)
>>> s.display()
div(u): E^3 → ℝ
(rh, ph, z) ↦ (rh*d(u_rho)/drh + rh*d(u_z)/dz + u_rho(rh, ph, z) + d(u_phi)/dph)/rh
>>> s.expr().expand()
u_rho(rh, ph, z)/rh + diff(u_phi(rh, ph, z), ph)/rh + diff(u_rho(rh, ph, z), rh)
+ diff(u_z(rh, ph, z), z)
s = div(u) s.display() s.expr().expand()
The curl:
sage: s = curl(u)
sage: s
Vector field curl(u) on the Euclidean space E^3
sage: s.display()
curl(u) = -(rh*d(u_phi)/dz - d(u_z)/dph)/rh e_rh + (d(u_rho)/dz - d(u_z)/drh) e_ph
+ (rh*d(u_phi)/drh + u_phi(rh, ph, z) - d(u_rho)/dph)/rh e_z
>>> from sage.all import *
>>> s = curl(u)
>>> s
Vector field curl(u) on the Euclidean space E^3
>>> s.display()
curl(u) = -(rh*d(u_phi)/dz - d(u_z)/dph)/rh e_rh + (d(u_rho)/dz - d(u_z)/drh) e_ph
+ (rh*d(u_phi)/drh + u_phi(rh, ph, z) - d(u_rho)/dph)/rh e_z
s = curl(u) s s.display()
The Laplacian of a scalar field:
sage: s = laplacian(F)
sage: s.display()
Delta(F): E^3 → ℝ
(rh, ph, z) ↦ (rh^2*d^2(f)/drh^2 + rh^2*d^2(f)/dz^2 + rh*d(f)/drh
+ d^2(f)/dph^2)/rh^2
sage: s.expr().expand()
diff(f(rh, ph, z), rh)/rh + diff(f(rh, ph, z), ph, ph)/rh^2
+ diff(f(rh, ph, z), rh, rh) + diff(f(rh, ph, z), z, z)
>>> from sage.all import *
>>> s = laplacian(F)
>>> s.display()
Delta(F): E^3 → ℝ
(rh, ph, z) ↦ (rh^2*d^2(f)/drh^2 + rh^2*d^2(f)/dz^2 + rh*d(f)/drh
+ d^2(f)/dph^2)/rh^2
>>> s.expr().expand()
diff(f(rh, ph, z), rh)/rh + diff(f(rh, ph, z), ph, ph)/rh^2
+ diff(f(rh, ph, z), rh, rh) + diff(f(rh, ph, z), z, z)
s = laplacian(F) s.display() s.expr().expand()
The Laplacian of a vector field:
sage: Du = laplacian(u)
sage: Du.display()
Delta(u) = (rh^2*d^2(u_rho)/drh^2 + rh^2*d^2(u_rho)/dz^2 + rh*d(u_rho)/drh
- u_rho(rh, ph, z) - 2*d(u_phi)/dph + d^2(u_rho)/dph^2)/rh^2 e_rh
+ (rh^2*d^2(u_phi)/drh^2 + rh^2*d^2(u_phi)/dz^2 + rh*d(u_phi)/drh
- u_phi(rh, ph, z) + d^2(u_phi)/dph^2 + 2*d(u_rho)/dph)/rh^2 e_ph
+ (rh^2*d^2(u_z)/drh^2 + rh^2*d^2(u_z)/dz^2 + rh*d(u_z)/drh
+ d^2(u_z)/dph^2)/rh^2 e_z
>>> from sage.all import *
>>> Du = laplacian(u)
>>> Du.display()
Delta(u) = (rh^2*d^2(u_rho)/drh^2 + rh^2*d^2(u_rho)/dz^2 + rh*d(u_rho)/drh
- u_rho(rh, ph, z) - 2*d(u_phi)/dph + d^2(u_rho)/dph^2)/rh^2 e_rh
+ (rh^2*d^2(u_phi)/drh^2 + rh^2*d^2(u_phi)/dz^2 + rh*d(u_phi)/drh
- u_phi(rh, ph, z) + d^2(u_phi)/dph^2 + 2*d(u_rho)/dph)/rh^2 e_ph
+ (rh^2*d^2(u_z)/drh^2 + rh^2*d^2(u_z)/dz^2 + rh*d(u_z)/drh
+ d^2(u_z)/dph^2)/rh^2 e_z
Du = laplacian(u) Du.display()
sage: for i in E.irange():
....: s = Du[i].expand()
sage: Du.display_comp()
Delta(u)^1 = d(u_rho)/drh/rh - u_rho(rh, ph, z)/rh^2 - 2*d(u_phi)/dph/rh^2
+ d^2(u_rho)/dph^2/rh^2 + d^2(u_rho)/drh^2 + d^2(u_rho)/dz^2
Delta(u)^2 = d(u_phi)/drh/rh - u_phi(rh, ph, z)/rh^2 + d^2(u_phi)/dph^2/rh^2
+ 2*d(u_rho)/dph/rh^2 + d^2(u_phi)/drh^2 + d^2(u_phi)/dz^2
Delta(u)^3 = d(u_z)/drh/rh + d^2(u_z)/dph^2/rh^2 + d^2(u_z)/drh^2 + d^2(u_z)/dz^2
>>> from sage.all import *
>>> for i in E.irange():
... s = Du[i].expand()
>>> Du.display_comp()
Delta(u)^1 = d(u_rho)/drh/rh - u_rho(rh, ph, z)/rh^2 - 2*d(u_phi)/dph/rh^2
+ d^2(u_rho)/dph^2/rh^2 + d^2(u_rho)/drh^2 + d^2(u_rho)/dz^2
Delta(u)^2 = d(u_phi)/drh/rh - u_phi(rh, ph, z)/rh^2 + d^2(u_phi)/dph^2/rh^2
+ 2*d(u_rho)/dph/rh^2 + d^2(u_phi)/drh^2 + d^2(u_phi)/dz^2
Delta(u)^3 = d(u_z)/drh/rh + d^2(u_z)/dph^2/rh^2 + d^2(u_z)/drh^2 + d^2(u_z)/dz^2
for i in E.irange():
s = Du[i].expand()
Du.display_comp()
Again, we may check that the above formulas coincide with those of Wikipedia’s article Del in cylindrical and spherical coordinates.
Changing coordinates¶
Given the expression of a vector field in a given coordinate system, SageMath can compute its expression in another coordinate system, see the tutorial How to change coordinates