How to compute a gradient, a divergence or a curl

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).

First stage: introduce the Euclidean 3-space

Before evaluating some vector-field operators, one needs to define the arena in which vector fields live, namely the 3-dimensional Euclidean space \(\mathbb{E}^3\). In SageMath, we declare it, along with the standard Cartesian coordinates \((x,y,z)\), via EuclideanSpace:

sage: E.<x,y,z> = EuclideanSpace()
sage: E
Euclidean space E^3

>>> from sage.all import *
>>> E = EuclideanSpace(names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3)
>>> E
Euclidean space E^3

E.<x,y,z> = EuclideanSpace()
E

Thanks to the notation <x,y,z> in the above declaration, the coordinates \((x,y,z)\) are immediately available as three symbolic variables x, y and z (there is no need to declare them via var(), i.e. to type x, y, z = var('x y z')):

sage: x is E.cartesian_coordinates()[1]
True
sage: y is E.cartesian_coordinates()[2]
True
sage: z is E.cartesian_coordinates()[3]
True
sage: type(z)
<class 'sage.symbolic.expression.Expression'>

>>> from sage.all import *
>>> x is E.cartesian_coordinates()[Integer(1)]
True
>>> y is E.cartesian_coordinates()[Integer(2)]
True
>>> z is E.cartesian_coordinates()[Integer(3)]
True
>>> type(z)
<class 'sage.symbolic.expression.Expression'>

x is E.cartesian_coordinates()[1]
y is E.cartesian_coordinates()[2]
z is E.cartesian_coordinates()[3]
type(z)

Besides, \(\mathbb{E}^3\) is endowed with the orthonormal vector frame \((e_x, e_y, e_z)\) associated with Cartesian coordinates:

sage: E.frames()
[Coordinate frame (E^3, (e_x,e_y,e_z))]

>>> from sage.all import *
>>> E.frames()
[Coordinate frame (E^3, (e_x,e_y,e_z))]

E.frames()

At this stage, this is the default vector frame on \(\mathbb{E}^3\) (being the only vector frame introduced so far):

sage: E.default_frame()
Coordinate frame (E^3, (e_x,e_y,e_z))

>>> from sage.all import *
>>> E.default_frame()
Coordinate frame (E^3, (e_x,e_y,e_z))

E.default_frame()

Defining a vector field

We define a vector field on \(\mathbb{E}^3\) from its components in the vector frame \((e_x,e_y,e_z)\):

sage: v = E.vector_field(-y, x, sin(x*y*z), name='v')
sage: v.display()
v = -y e_x + x e_y + sin(x*y*z) e_z

>>> from sage.all import *
>>> v = E.vector_field(-y, x, sin(x*y*z), name='v')
>>> v.display()
v = -y e_x + x e_y + sin(x*y*z) e_z

v = E.vector_field(-y, x, sin(x*y*z), name='v')
v.display()

We can access to the components of \(v\) via the square bracket operator:

sage: v[1]
-y
sage: v[:]
[-y, x, sin(x*y*z)]

>>> from sage.all import *
>>> v[Integer(1)]
-y
>>> v[:]
[-y, x, sin(x*y*z)]

v[1]
v[:]

A vector field can evaluated at any point of \(\mathbb{E}^3\):

sage: p = E((3,-2,1), name='p')
sage: p
Point p on the Euclidean space E^3
sage: p.coordinates()
(3, -2, 1)
sage: vp = v.at(p)
sage: vp
Vector v at Point p on the Euclidean space E^3
sage: vp.display()
v = 2 e_x + 3 e_y - sin(6) e_z

>>> from sage.all import *
>>> p = E((Integer(3),-Integer(2),Integer(1)), name='p')
>>> p
Point p on the Euclidean space E^3
>>> p.coordinates()
(3, -2, 1)
>>> vp = v.at(p)
>>> vp
Vector v at Point p on the Euclidean space E^3
>>> vp.display()
v = 2 e_x + 3 e_y - sin(6) e_z

p = E((3,-2,1), name='p')
p
p.coordinates()
vp = v.at(p)
vp
vp.display()

Vector fields can be plotted:

sage: v.plot(max_range=1.5, scale=0.5)
Graphics3d Object

>>> from sage.all import *
>>> v.plot(max_range=RealNumber('1.5'), scale=RealNumber('0.5'))
Graphics3d Object

v.plot(max_range=1.5, scale=0.5)
../_images/vector_calc_cartesian-1.svg

For customizing the plot, see the list of options in the documentation of plot(). For instance, to get a view of the orthogonal projection of \(v\) in the plane \(y=1\), do:

sage: v.plot(fixed_coords={y: 1}, ambient_coords=(x,z), max_range=1.5,
....:        scale=0.25)
Graphics object consisting of 81 graphics primitives

>>> from sage.all import *
>>> v.plot(fixed_coords={y: Integer(1)}, ambient_coords=(x,z), max_range=RealNumber('1.5'),
...        scale=RealNumber('0.25'))
Graphics object consisting of 81 graphics primitives

v.plot(fixed_coords={y: 1}, ambient_coords=(x,z), max_range=1.5,
       scale=0.25)
../_images/vector_calc_cartesian-2.svg

We may define a vector field \(u\) with generic components \((u_x, u_y, y_z)\):

sage: u = E.vector_field(function('u_x')(x,y,z),
....:                    function('u_y')(x,y,z),
....:                    function('u_z')(x,y,z),
....:                    name='u')
sage: u.display()
u = u_x(x, y, z) e_x + u_y(x, y, z) e_y + u_z(x, y, z) e_z
sage: u[:]
[u_x(x, y, z), u_y(x, y, z), u_z(x, y, z)]

>>> from sage.all import *
>>> u = E.vector_field(function('u_x')(x,y,z),
...                    function('u_y')(x,y,z),
...                    function('u_z')(x,y,z),
...                    name='u')
>>> u.display()
u = u_x(x, y, z) e_x + u_y(x, y, z) e_y + u_z(x, y, z) e_z
>>> u[:]
[u_x(x, y, z), u_y(x, y, z), u_z(x, y, z)]

u = E.vector_field(function('u_x')(x,y,z),
                   function('u_y')(x,y,z),
                   function('u_z')(x,y,z),
                   name='u')
u.display()
u[:]

Its value at the point \(p\) is then:

sage: up = u.at(p)
sage: up.display()
u = u_x(3, -2, 1) e_x + u_y(3, -2, 1) e_y + u_z(3, -2, 1) e_z

>>> from sage.all import *
>>> up = u.at(p)
>>> up.display()
u = u_x(3, -2, 1) e_x + u_y(3, -2, 1) e_y + u_z(3, -2, 1) e_z

up = u.at(p)
up.display()

How to compute various vector products

Dot product

The dot (or scalar) product \(u\cdot v\) of the vector fields \(u\) and \(v\) is obtained by the method dot_product(), which admits dot() as a shortcut alias:

sage: u.dot(v) == u[1]*v[1] + u[2]*v[2] + u[3]*v[3]
True

>>> from sage.all import *
>>> u.dot(v) == u[Integer(1)]*v[Integer(1)] + u[Integer(2)]*v[Integer(2)] + u[Integer(3)]*v[Integer(3)]
True

u.dot(v) == u[1]*v[1] + u[2]*v[2] + u[3]*v[3]

\(s= u\cdot v\) is a scalar field, i.e. a map \(\mathbb{E}^3 \to \mathbb{R}\):

sage: s = u.dot(v)
sage: s
Scalar field u.v on the Euclidean space E^3
sage: s.display()
u.v: E^3 → ℝ
   (x, y, z) ↦ -y*u_x(x, y, z) + x*u_y(x, y, z) + sin(x*y*z)*u_z(x, y, z)

>>> from sage.all import *
>>> s = u.dot(v)
>>> s
Scalar field u.v on the Euclidean space E^3
>>> s.display()
u.v: E^3 → ℝ
   (x, y, z) ↦ -y*u_x(x, y, z) + x*u_y(x, y, z) + sin(x*y*z)*u_z(x, y, z)

s = u.dot(v)
s
s.display()

It maps points of \(\mathbb{E}^3\) to real numbers:

sage: s(p)
-sin(6)*u_z(3, -2, 1) + 2*u_x(3, -2, 1) + 3*u_y(3, -2, 1)

>>> from sage.all import *
>>> s(p)
-sin(6)*u_z(3, -2, 1) + 2*u_x(3, -2, 1) + 3*u_y(3, -2, 1)

s(p)

Its coordinate expression is:

sage: s.expr()
-y*u_x(x, y, z) + x*u_y(x, y, z) + sin(x*y*z)*u_z(x, y, z)

>>> from sage.all import *
>>> s.expr()
-y*u_x(x, y, z) + x*u_y(x, y, z) + sin(x*y*z)*u_z(x, y, z)

s.expr()

Norm

The norm \(\|u\|\) of the vector field \(u\) is defined in terms of the dot product by \(\|u\| = \sqrt{u\cdot u}\):

sage: norm(u) == sqrt(u.dot(u))
True

>>> from sage.all import *
>>> norm(u) == sqrt(u.dot(u))
True

norm(u) == sqrt(u.dot(u))

It is a scalar field on \(\mathbb{E}^3\):

sage: s = norm(u)
sage: s
Scalar field |u| on the Euclidean space E^3
sage: s.display()
|u|: E^3 → ℝ
   (x, y, z) ↦ sqrt(u_x(x, y, z)^2 + u_y(x, y, z)^2 + u_z(x, y, z)^2)
sage: s.expr()
sqrt(u_x(x, y, z)^2 + u_y(x, y, z)^2 + u_z(x, y, z)^2)

>>> from sage.all import *
>>> s = norm(u)
>>> s
Scalar field |u| on the Euclidean space E^3
>>> s.display()
|u|: E^3 → ℝ
   (x, y, z) ↦ sqrt(u_x(x, y, z)^2 + u_y(x, y, z)^2 + u_z(x, y, z)^2)
>>> s.expr()
sqrt(u_x(x, y, z)^2 + u_y(x, y, z)^2 + u_z(x, y, z)^2)

s = norm(u)
s
s.display()
s.expr()

For \(v\), we have:

sage: norm(v).expr()
sqrt(x^2 + y^2 + sin(x*y*z)^2)

>>> from sage.all import *
>>> norm(v).expr()
sqrt(x^2 + y^2 + sin(x*y*z)^2)

norm(v).expr()

Cross product

The cross product \(u\times v\) is obtained by the method cross_product(), which admits cross() as a shortcut alias:

sage: s = u.cross(v)
sage: s
Vector field u x v on the Euclidean space E^3
sage: s.display()
u x v = (sin(x*y*z)*u_y(x, y, z) - x*u_z(x, y, z)) e_x
 + (-sin(x*y*z)*u_x(x, y, z) - y*u_z(x, y, z)) e_y
 + (x*u_x(x, y, z) + y*u_y(x, y, z)) e_z

>>> from sage.all import *
>>> s = u.cross(v)
>>> s
Vector field u x v on the Euclidean space E^3
>>> s.display()
u x v = (sin(x*y*z)*u_y(x, y, z) - x*u_z(x, y, z)) e_x
 + (-sin(x*y*z)*u_x(x, y, z) - y*u_z(x, y, z)) e_y
 + (x*u_x(x, y, z) + y*u_y(x, y, z)) e_z

s = u.cross(v)
s
s.display()

We can check the standard formulas expressing the cross product in terms of the components:

sage: all([s[1] == u[2]*v[3] - u[3]*v[2],
....:      s[2] == u[3]*v[1] - u[1]*v[3],
....:      s[3] == u[1]*v[2] - u[2]*v[1]])
True

>>> from sage.all import *
>>> all([s[Integer(1)] == u[Integer(2)]*v[Integer(3)] - u[Integer(3)]*v[Integer(2)],
...      s[Integer(2)] == u[Integer(3)]*v[Integer(1)] - u[Integer(1)]*v[Integer(3)],
...      s[Integer(3)] == u[Integer(1)]*v[Integer(2)] - u[Integer(2)]*v[Integer(1)]])
True

all([s[1] == u[2]*v[3] - u[3]*v[2],
     s[2] == u[3]*v[1] - u[1]*v[3],
     s[3] == u[1]*v[2] - u[2]*v[1]])

Scalar triple product

Let us introduce a third vector field, \(w\) say. As a example, we do not pass the components as arguments of vector_field(), as we did for \(u\) and \(v\); instead, we set them in a second stage, via the square bracket operator, any unset component being assumed to be zero:

sage: w = E.vector_field(name='w')
sage: w[1] = x*z
sage: w[2] = y*z
sage: w.display()
w = x*z e_x + y*z e_y

>>> from sage.all import *
>>> w = E.vector_field(name='w')
>>> w[Integer(1)] = x*z
>>> w[Integer(2)] = y*z
>>> w.display()
w = x*z e_x + y*z e_y

w = E.vector_field(name='w')
w[1] = x*z
w[2] = y*z
w.display()

The scalar triple product of the vector fields \(u\), \(v\) and \(w\) is obtained as follows:

sage: triple_product = E.scalar_triple_product()
sage: s = triple_product(u, v, w)
sage: s
Scalar field epsilon(u,v,w) on the Euclidean space E^3
sage: s.expr()
-(y*u_x(x, y, z) - x*u_y(x, y, z))*z*sin(x*y*z) - (x^2*u_z(x, y, z)
 + y^2*u_z(x, y, z))*z

>>> from sage.all import *
>>> triple_product = E.scalar_triple_product()
>>> s = triple_product(u, v, w)
>>> s
Scalar field epsilon(u,v,w) on the Euclidean space E^3
>>> s.expr()
-(y*u_x(x, y, z) - x*u_y(x, y, z))*z*sin(x*y*z) - (x^2*u_z(x, y, z)
 + y^2*u_z(x, y, z))*z

triple_product = E.scalar_triple_product()
s = triple_product(u, v, w)
s
s.expr()

Let us check that the scalar triple product of \(u\), \(v\) and \(w\) is \(u\cdot(v\times w)\):

sage: s == u.dot(v.cross(w))
True

>>> from sage.all import *
>>> s == u.dot(v.cross(w))
True

s == u.dot(v.cross(w))

How to evaluate the standard differential operators

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 some unspecified function of \((x,y,z)\):

sage: F = E.scalar_field(function('f')(x,y,z), name='F')
sage: F.display()
F: E^3 → ℝ
   (x, y, z) ↦ f(x, y, z)

>>> from sage.all import *
>>> F = E.scalar_field(function('f')(x,y,z), name='F')
>>> F.display()
F: E^3 → ℝ
   (x, y, z) ↦ f(x, y, z)

F = E.scalar_field(function('f')(x,y,z), name='F')
F.display()

The value of \(F\) at a point:

sage: F(p)
f(3, -2, 1)

>>> from sage.all import *
>>> F(p)
f(3, -2, 1)

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)/dx e_x + d(f)/dy e_y + d(f)/dz e_z
sage: norm(grad(F)).display()
|grad(F)|: E^3 → ℝ
   (x, y, z) ↦ sqrt((d(f)/dx)^2 + (d(f)/dy)^2 + (d(f)/dz)^2)

>>> from sage.all import *
>>> grad(F)
Vector field grad(F) on the Euclidean space E^3
>>> grad(F).display()
grad(F) = d(f)/dx e_x + d(f)/dy e_y + d(f)/dz e_z
>>> norm(grad(F)).display()
|grad(F)|: E^3 → ℝ
   (x, y, z) ↦ sqrt((d(f)/dx)^2 + (d(f)/dy)^2 + (d(f)/dz)^2)

grad(F)
grad(F).display()
norm(grad(F)).display()

Divergence

The divergence of the vector field \(u\):

sage: s = div(u)
sage: s.display()
div(u): E^3 → ℝ
   (x, y, z) ↦ d(u_x)/dx + d(u_y)/dy + d(u_z)/dz

>>> from sage.all import *
>>> s = div(u)
>>> s.display()
div(u): E^3 → ℝ
   (x, y, z) ↦ d(u_x)/dx + d(u_y)/dy + d(u_z)/dz

s = div(u)
s.display()

For \(v\) and \(w\), we have:

sage: div(v).expr()
x*y*cos(x*y*z)
sage: div(w).expr()
2*z

>>> from sage.all import *
>>> div(v).expr()
x*y*cos(x*y*z)
>>> div(w).expr()
2*z

div(v).expr()
div(w).expr()

An identity valid for any scalar field \(F\) and any vector field \(u\):

sage: div(F*u) == F*div(u) + u.dot(grad(F))
True

>>> from sage.all import *
>>> div(F*u) == F*div(u) + u.dot(grad(F))
True

div(F*u) == F*div(u) + u.dot(grad(F))

Curl

The curl of the vector field \(u\):

sage: s = curl(u)
sage: s
Vector field curl(u) on the Euclidean space E^3
sage: s.display()
curl(u) = (-d(u_y)/dz + d(u_z)/dy) e_x + (d(u_x)/dz - d(u_z)/dx) e_y
 + (-d(u_x)/dy + d(u_y)/dx) e_z

>>> from sage.all import *
>>> s = curl(u)
>>> s
Vector field curl(u) on the Euclidean space E^3
>>> s.display()
curl(u) = (-d(u_y)/dz + d(u_z)/dy) e_x + (d(u_x)/dz - d(u_z)/dx) e_y
 + (-d(u_x)/dy + d(u_y)/dx) e_z

s = curl(u)
s
s.display()

To use the notation rot instead of curl, simply do:

sage: rot = curl

>>> from sage.all import *
>>> rot = curl

rot = curl

An alternative is:

sage: from sage.manifolds.operators import curl as rot

>>> from sage.all import *
>>> from sage.manifolds.operators import curl as rot

from sage.manifolds.operators import curl as rot

We have then:

sage: rot(u).display()
curl(u) = (-d(u_y)/dz + d(u_z)/dy) e_x + (d(u_x)/dz - d(u_z)/dx) e_y
 + (-d(u_x)/dy + d(u_y)/dx) e_z
sage: rot(u) == curl(u)
True

>>> from sage.all import *
>>> rot(u).display()
curl(u) = (-d(u_y)/dz + d(u_z)/dy) e_x + (d(u_x)/dz - d(u_z)/dx) e_y
 + (-d(u_x)/dy + d(u_y)/dx) e_z
>>> rot(u) == curl(u)
True

rot(u).display()
rot(u) == curl(u)

For \(v\) and \(w\), we have:

sage: curl(v).display()
curl(v) = x*z*cos(x*y*z) e_x - y*z*cos(x*y*z) e_y + 2 e_z

>>> from sage.all import *
>>> curl(v).display()
curl(v) = x*z*cos(x*y*z) e_x - y*z*cos(x*y*z) e_y + 2 e_z

curl(v).display()

sage: curl(w).display()
curl(w) = -y e_x + x e_y

>>> from sage.all import *
>>> curl(w).display()
curl(w) = -y e_x + x e_y

curl(w).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
   (x, y, z) ↦ 0

>>> from sage.all import *
>>> div(curl(u)).display()
div(curl(u)): E^3 → ℝ
   (x, y, z) ↦ 0

div(curl(u)).display()

An identity valid for any scalar field \(F\) and any vector field \(u\) is

\[\mathrm{curl}(Fu) = \mathrm{grad}\, F\times u + F\, \mathrm{curl}\, u,\]

as we can check:

sage: curl(F*u) == grad(F).cross(u) + F*curl(u)
True

>>> from sage.all import *
>>> curl(F*u) == grad(F).cross(u) + F*curl(u)
True

curl(F*u) == grad(F).cross(u) + F*curl(u)

Laplacian

The Laplacian \(\Delta F\) of a scalar field \(F\) is another scalar field:

sage: s = laplacian(F)
sage: s.display()
Delta(F): E^3 → ℝ
   (x, y, z) ↦ d^2(f)/dx^2 + d^2(f)/dy^2 + d^2(f)/dz^2

>>> from sage.all import *
>>> s = laplacian(F)
>>> s.display()
Delta(F): E^3 → ℝ
   (x, y, z) ↦ d^2(f)/dx^2 + d^2(f)/dy^2 + d^2(f)/dz^2

s = laplacian(F)
s.display()

The following identity holds:

\[\Delta F = \mathrm{div}\left(\mathrm{grad}\, F\right),\]

as we can check:

sage: laplacian(F) == div(grad(F))
True

>>> from sage.all import *
>>> laplacian(F) == div(grad(F))
True

laplacian(F) == div(grad(F))

The Laplacian \(\Delta u\) of a vector field \(u\) is another vector field:

sage: Du = laplacian(u)
sage: Du
Vector field Delta(u) on the Euclidean space E^3

>>> from sage.all import *
>>> Du = laplacian(u)
>>> Du
Vector field Delta(u) on the Euclidean space E^3

Du = laplacian(u)
Du

whose components are:

sage: Du.display()
Delta(u) = (d^2(u_x)/dx^2 + d^2(u_x)/dy^2 + d^2(u_x)/dz^2) e_x
 + (d^2(u_y)/dx^2 + d^2(u_y)/dy^2 + d^2(u_y)/dz^2) e_y
 + (d^2(u_z)/dx^2 + d^2(u_z)/dy^2 + d^2(u_z)/dz^2) e_z

>>> from sage.all import *
>>> Du.display()
Delta(u) = (d^2(u_x)/dx^2 + d^2(u_x)/dy^2 + d^2(u_x)/dz^2) e_x
 + (d^2(u_y)/dx^2 + d^2(u_y)/dy^2 + d^2(u_y)/dz^2) e_y
 + (d^2(u_z)/dx^2 + d^2(u_z)/dy^2 + d^2(u_z)/dz^2) e_z

Du.display()

In the Cartesian frame, the components of \(\Delta u\) are nothing but the (scalar) Laplacians of the components of \(u\), as we can check:

sage: e = E.cartesian_frame()
sage: Du == sum(laplacian(u[[i]])*e[i] for i in E.irange())
True

>>> from sage.all import *
>>> e = E.cartesian_frame()
>>> Du == sum(laplacian(u[[i]])*e[i] for i in E.irange())
True

e = E.cartesian_frame()
Du == sum(laplacian(u[[i]])*e[i] for i in E.irange())

In the above formula, u[[i]] return the \(i\)-th component of \(u\) as a scalar field, while u[i] would have returned the coordinate expression of this scalar field; besides, e is the Cartesian frame:

sage: e[:]
(Vector field e_x on the Euclidean space E^3,
 Vector field e_y on the Euclidean space E^3,
 Vector field e_z on the Euclidean space E^3)

>>> from sage.all import *
>>> e[:]
(Vector field e_x on the Euclidean space E^3,
 Vector field e_y on the Euclidean space E^3,
 Vector field e_z on the Euclidean space E^3)

e[:]

For the vector fields \(v\) and \(w\), we have:

sage: laplacian(v).display()
Delta(v) = -(x^2*y^2 + (x^2 + y^2)*z^2)*sin(x*y*z) e_z
sage: laplacian(w).display()
Delta(w) = 0

>>> from sage.all import *
>>> laplacian(v).display()
Delta(v) = -(x^2*y^2 + (x^2 + y^2)*z^2)*sin(x*y*z) e_z
>>> laplacian(w).display()
Delta(w) = 0

laplacian(v).display()
laplacian(w).display()

We have:

sage: curl(curl(u)).display()
curl(curl(u)) = (-d^2(u_x)/dy^2 - d^2(u_x)/dz^2 + d^2(u_y)/dxdy
 + d^2(u_z)/dxdz) e_x + (d^2(u_x)/dxdy - d^2(u_y)/dx^2 - d^2(u_y)/dz^2
 + d^2(u_z)/dydz) e_y + (d^2(u_x)/dxdz + d^2(u_y)/dydz - d^2(u_z)/dx^2
 - d^2(u_z)/dy^2) e_z
sage: grad(div(u)).display()
grad(div(u)) = (d^2(u_x)/dx^2 + d^2(u_y)/dxdy + d^2(u_z)/dxdz) e_x
 + (d^2(u_x)/dxdy + d^2(u_y)/dy^2 + d^2(u_z)/dydz) e_y
 + (d^2(u_x)/dxdz + d^2(u_y)/dydz + d^2(u_z)/dz^2) e_z

>>> from sage.all import *
>>> curl(curl(u)).display()
curl(curl(u)) = (-d^2(u_x)/dy^2 - d^2(u_x)/dz^2 + d^2(u_y)/dxdy
 + d^2(u_z)/dxdz) e_x + (d^2(u_x)/dxdy - d^2(u_y)/dx^2 - d^2(u_y)/dz^2
 + d^2(u_z)/dydz) e_y + (d^2(u_x)/dxdz + d^2(u_y)/dydz - d^2(u_z)/dx^2
 - d^2(u_z)/dy^2) e_z
>>> grad(div(u)).display()
grad(div(u)) = (d^2(u_x)/dx^2 + d^2(u_y)/dxdy + d^2(u_z)/dxdz) e_x
 + (d^2(u_x)/dxdy + d^2(u_y)/dy^2 + d^2(u_z)/dydz) e_y
 + (d^2(u_x)/dxdz + d^2(u_y)/dydz + d^2(u_z)/dz^2) e_z

curl(curl(u)).display()
grad(div(u)).display()

A famous identity is

\[\mathrm{curl}\left(\mathrm{curl}\, u\right) = \mathrm{grad}\left(\mathrm{div}\, u\right) - \Delta u .\]

Let us check it:

sage: curl(curl(u)) == grad(div(u)) - laplacian(u)
True

>>> from sage.all import *
>>> curl(curl(u)) == grad(div(u)) - laplacian(u)
True

curl(curl(u)) == grad(div(u)) - laplacian(u)

How to customize various symbols

Customizing the symbols of the orthonormal frame vectors

By default, the vectors of the orthonormal frame associated with Cartesian coordinates are denoted \((e_x,e_y,e_z)\):

sage: frame = E.cartesian_frame()
sage: frame
Coordinate frame (E^3, (e_x,e_y,e_z))

>>> from sage.all import *
>>> frame = E.cartesian_frame()
>>> frame
Coordinate frame (E^3, (e_x,e_y,e_z))

frame = E.cartesian_frame()
frame

But this can be changed, thanks to the method set_name():

sage: frame.set_name('a', indices=('x', 'y', 'z'))
sage: frame
Coordinate frame (E^3, (a_x,a_y,a_z))
sage: v.display()
v = -y a_x + x a_y + sin(x*y*z) a_z

>>> from sage.all import *
>>> frame.set_name('a', indices=('x', 'y', 'z'))
>>> frame
Coordinate frame (E^3, (a_x,a_y,a_z))
>>> v.display()
v = -y a_x + x a_y + sin(x*y*z) a_z

frame.set_name('a', indices=('x', 'y', 'z'))
frame
v.display()

sage: frame.set_name(('hx', 'hy', 'hz'),
....:                latex_symbol=(r'\mathbf{\hat{x}}', r'\mathbf{\hat{y}}',
....:                              r'\mathbf{\hat{z}}'))
sage: frame
Coordinate frame (E^3, (hx,hy,hz))
sage: v.display()
v = -y hx + x hy + sin(x*y*z) hz

>>> from sage.all import *
>>> frame.set_name(('hx', 'hy', 'hz'),
...                latex_symbol=(r'\mathbf{\hat{x}}', r'\mathbf{\hat{y}}',
...                              r'\mathbf{\hat{z}}'))
>>> frame
Coordinate frame (E^3, (hx,hy,hz))
>>> v.display()
v = -y hx + x hy + sin(x*y*z) hz

frame.set_name(('hx', 'hy', 'hz'),
               latex_symbol=(r'\mathbf{\hat{x}}', r'\mathbf{\hat{y}}',
                             r'\mathbf{\hat{z}}'))
frame
v.display()

Customizing the coordinate symbols

The coordinates symbols are defined within the angle brackets <...> at the construction of the Euclidean space. Above we did:

sage: E.<x,y,z> = EuclideanSpace()

>>> from sage.all import *
>>> E = EuclideanSpace(names=('x', 'y', 'z',)); (x, y, z,) = E._first_ngens(3)

E.<x,y,z> = EuclideanSpace()

which resulted in the coordinate symbols \((x,y,z)\) and in the corresponding Python variables x, y and z (SageMath symbolic expressions). To use other symbols, for instance \((X,Y,Z)\), it suffices to create E as:

sage: E.<X,Y,Z> = EuclideanSpace()

>>> from sage.all import *
>>> E = EuclideanSpace(names=('X', 'Y', 'Z',)); (X, Y, Z,) = E._first_ngens(3)

E.<X,Y,Z> = EuclideanSpace()

We have then:

sage: E.atlas()
[Chart (E^3, (X, Y, Z))]
sage: E.cartesian_frame()
Coordinate frame (E^3, (e_X,e_Y,e_Z))
sage: v = E.vector_field(-Y, X, sin(X*Y*Z), name='v')
sage: v.display()
v = -Y e_X + X e_Y + sin(X*Y*Z) e_Z

>>> from sage.all import *
>>> E.atlas()
[Chart (E^3, (X, Y, Z))]
>>> E.cartesian_frame()
Coordinate frame (E^3, (e_X,e_Y,e_Z))
>>> v = E.vector_field(-Y, X, sin(X*Y*Z), name='v')
>>> v.display()
v = -Y e_X + X e_Y + sin(X*Y*Z) e_Z

E.atlas()
E.cartesian_frame()
v = E.vector_field(-Y, X, sin(X*Y*Z), name='v')
v.display()

By default the LaTeX symbols of the coordinate coincide with the letters given within the angle brackets. But this can be adjusted through the optional argument symbols of the function EuclideanSpace, which has to be a string, usually prefixed by r (for raw string, in order to allow for the backslash character of LaTeX expressions). This string contains the coordinate fields separated by a blank space; each field contains the coordinate’s text symbol and possibly the coordinate’s LaTeX symbol (when the latter is different from the text symbol), both symbols being separated by a colon (:):

sage: E.<xi,et,ze> = EuclideanSpace(symbols=r"xi:\xi et:\eta ze:\zeta")
sage: E.atlas()
[Chart (E^3, (xi, et, ze))]
sage: v = E.vector_field(-et, xi, sin(xi*et*ze), name='v')
sage: v.display()
v = -et e_xi + xi e_et + sin(et*xi*ze) e_ze

>>> from sage.all import *
>>> E = EuclideanSpace(symbols=r"xi:\xi et:\eta ze:\zeta", names=('xi', 'et', 'ze',)); (xi, et, ze,) = E._first_ngens(3)
>>> E.atlas()
[Chart (E^3, (xi, et, ze))]
>>> v = E.vector_field(-et, xi, sin(xi*et*ze), name='v')
>>> v.display()
v = -et e_xi + xi e_et + sin(et*xi*ze) e_ze

E.<xi,et,ze> = EuclideanSpace(symbols=r"xi:\xi et:\eta ze:\zeta")
E.atlas()
v = E.vector_field(-et, xi, sin(xi*et*ze), name='v')
v.display()