version()
'SageMath version 8.2.rc4, Release Date: 2018-04-20'
%display latex
from sage.manifolds.operators import * # to get the operators grad, div, curl, etc.
We start by declaring the 3-dimensional Euclidean space $\mathbb{E}^3$, with $(x,y,z)$ as Cartesian coordinates:
E.<x,y,z> = EuclideanSpace()
print(E)
E
Euclidean space E^3
$\mathbb{E}^3$ is endowed with the chart of Cartesian coordinates:
E.atlas()
as well as with the associated orthonormal vector frame:
E.frames()
We define a vector field on $\mathbb{E}^3$ from its components in the vector frame $(e_x,e_y,e_z)$:
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:
v[1]
v[:]
A vector field can evaluated at any point of $\mathbb{E}^3$:
p = E((3,-2,1), name='p')
print(p)
Point p on the Euclidean space E^3
p.coordinates()
vp = v.at(p)
print(vp)
Vector v at Point p on the Euclidean space E^3
vp.display()
Vector fields can be plotted (see the list of options for customizing the plot)
v.plot(max_range=1.5, scale=0.5, viewer='threejs', online=True)
We may define a vector field with generic components:
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[:]
up = u.at(p)
up.display()
s = u.dot(v)
s
print(s)
Scalar field u.v on the Euclidean space E^3
$s= u\cdot v$ is a scalar field, i.e. a map $\mathbb{E}^3 \rightarrow \mathbb{R}$:
s.display()
It maps point of $\mathbb{E}^3$ to real numbers:
s(p)
Its coordinate expression is
s.expr()
The norm of a vector field is
s = norm(u)
s
s.display()
s.expr()
The norm is related to the dot product by $\|u\|^2 = u\cdot u$, as we can check:
norm(u)^2 == u.dot(u)
For $v$, we have:
norm(v).expr()
The cross product of $u$ by $v$ is obtained by the method cross_product
, which admits cross
as a shortcut alias:
s = u.cross(v)
print(s)
Vector field u x v on the Euclidean space E^3
s.display()
Let us introduce a third vector field. 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:
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:
triple_product = E.scalar_triple_product()
s = triple_product(u, v, w)
print(s)
Scalar field epsilon(u,v,w) on the Euclidean space E^3
s.expr()
Let us check that the scalar triple product of $u$, $v$ and $w$ is $u\cdot(v\times w)$:
s == u.dot(v.cross(w))
We first introduce a scalar field, via its expression in terms of Cartesian coordinates; in this example, we consider a unspecified function of $(x,y,z)$:
F = E.scalar_field(function('f')(x,y,z), name='F')
F.display()
The value of $F$ at a point:
F(p)
The gradient of $F$:
print(grad(F))
Vector field grad(F) on the Euclidean space E^3
grad(F).display()
norm(grad(F)).display()
The divergence of a vector field:
s = div(u)
s.display()
For $v$ and $w$, we have
div(v).expr()
div(w).expr()
An identity valid for any scalar field $F$ and any vector field $u$:
div(F*u) == F*div(u) + u.dot(grad(F))
The curl of a vector field:
s = curl(u)
print(s)
Vector field curl(u) on the Euclidean space E^3
s.display()
To use the notation rot
instead of curl
, simply do
rot = curl
An alternative is
from sage.manifolds.operators import curl as rot
We have then
rot(u).display()
rot(u) == curl(u)
For $v$ and $w$, we have:
curl(v).display()
curl(w).display()
The curl of a gradient is always zero:
curl(grad(F)).display()
The divergence of a curl is always zero:
div(curl(u)).display()
An identity valid for any scalar field $F$ and any vector field $u$:
curl(F*u) == grad(F).cross(u) + F*curl(u)
The Laplacian of a scalar field:
s = laplacian(F)
s.display()
For a scalar field, the Laplacian is nothing but the divergence of the gradient:
laplacian(F) == div(grad(F))
The Laplacian of a vector field:
laplacian(u).display()
In the Cartesian frame, the components of the Laplacian of a vector field are nothing but the Laplacians of the components of the vector field, as we can check:
e = E.cartesian_frame()
laplacian(u) == 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:
e[:]
For $v$ and $w$, we have
laplacian(v).display()
laplacian(w).display()
We have:
curl(curl(u)).display()
grad(div(u)).display()
and we may check a famous identity:
curl(curl(u)) == grad(div(u)) - laplacian(u)
frame = E.cartesian_frame()
frame
But this can be changed, thanks to the method set_name
:
frame.set_name('a', indices=('x', 'y', 'z'))
frame
v.display()
frame.set_name(('hx', 'hy', 'hz'),
latex_symbol=(r'\mathbf{\hat{x}}', r'\mathbf{\hat{y}}',
r'\mathbf{\hat{z}}'))
frame
v.display()
The coordinates symbols are defined within the angle brackets <...>
at the construction of the Euclidean space. Above we did
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
E.<X,Y,Z> = EuclideanSpace()
We have then:
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 (:
):
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()