How to change 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).

Starting from Cartesian coordinates#

In this tutorial, we choose to start from the Cartesian coordinates \((x,y,z)\). Hence, we declare the 3-dimensional Euclidean space \(\mathbb{E}^3\) as:

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

By default, i.e. without the optional argument coordinates in EuclideanSpace, \(\mathbb{E}^3\) is initialized with the chart of Cartesian coordinates:

sage: E.atlas()
[Chart (E^3, (x, y, z))]

See the tutorial How to perform vector calculus in curvilinear coordinates for examples of initialization of the Euclidean space with spherical coordinates or cylindrical coordinates instead of the Cartesian ones.

Let us denote by cartesian the chart of Cartesian coordinates:

sage: cartesian = E.cartesian_coordinates()
sage: cartesian
Chart (E^3, (x, y, z))

The access to the individual coordinates is performed via the square bracket operator:

sage: cartesian[1]
x
sage: cartesian[:]
(x, y, z)

Thanks to use of <x,y,z> when declaring E, the Python variables x, y and z have been created to store the coordinates \((x,y,z)\) as symbolic expressions. There is no need to declare them via var(), i.e. to type x, y, z = var('x y z'); they are immediately available:

sage: y is cartesian[2]
True
sage: type(y)
<class 'sage.symbolic.expression.Expression'>

Each of the Cartesian coordinates spans the entire real line:

sage: cartesian.coord_range()
x: (-oo, +oo); y: (-oo, +oo); z: (-oo, +oo)

Being the only coordinate chart created so far, cartesian is the default chart on E:

sage: cartesian is E.default_chart()
True

\(\mathbb{E}^3\) 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))]

Let us denote it by cartesian_frame:

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

Each element of this frame is a unit vector field; for instance, we have \(e_x\cdot e_x = 1\):

sage: e_x = cartesian_frame[1]
sage: e_x
Vector field e_x on the Euclidean space E^3
sage: e_x.dot(e_x).expr()
1

as well as \(e_x\cdot e_y = 0\):

sage: e_y = cartesian_frame[2]
sage: e_x.dot(e_y).expr()
0

Introducing spherical coordinates#

Spherical coordinates are introduced by:

sage: spherical.<r,th,ph> = E.spherical_coordinates()
sage: spherical
Chart (E^3, (r, th, ph))

We have:

sage: spherical[:]
(r, th, ph)
sage: spherical.coord_range()
r: (0, +oo); th: (0, pi); ph: [0, 2*pi] (periodic)

\(\mathbb{E}^3\) is now endowed with two coordinate charts:

sage: E.atlas()
[Chart (E^3, (x, y, z)), Chart (E^3, (r, th, ph))]

The change-of-coordinate formulas have been automatically implemented during the above call E.spherical_coordinates():

sage: E.coord_change(spherical, cartesian).display()
x = r*cos(ph)*sin(th)
y = r*sin(ph)*sin(th)
z = r*cos(th)
sage: E.coord_change(cartesian, spherical).display()
r = sqrt(x^2 + y^2 + z^2)
th = arctan2(sqrt(x^2 + y^2), z)
ph = arctan2(y, x)

These formulas are automatically used if we ask to plot the grid of spherical coordinates in terms of Cartesian coordinates:

sage: spherical.plot(cartesian, color={r:'red', th:'green', ph:'orange'})
Graphics3d Object
../_images/vector_calc_change-1.svg

Note that

  • the red lines are those along which \(r\) varies, while \((\theta,\phi)\) are kept fixed;

  • the grid lines are those along which \(\theta\) varies, while \((r,\phi)\) are kept fixed;

  • the orange lines are those along which \(\phi\) varies, while \((r,\theta)\) are kept fixed.

For customizing the plot, see the list of options in the documentation of plot(). For instance, we may draw the spherical coordinates in the plane \(\theta=\pi/2\) in terms of the coordinates \((x, y)\):

sage: spherical.plot(cartesian, fixed_coords={th: pi/2}, ambient_coords=(x,y),
....:                color={r:'red', th:'green', ph:'orange'})
Graphics object consisting of 18 graphics primitives
../_images/vector_calc_change-2.svg

Similarly the grid of spherical coordinates in the half-plane \(\phi=0\) drawn in terms of the coordinates \((x, z)\) is obtained via:

sage: spherical.plot(cartesian, fixed_coords={ph: 0}, ambient_coords=(x,z),
....:                color={r:'red', th:'green', ph:'orange'})
Graphics object consisting of 18 graphics primitives
../_images/vector_calc_change-3.svg

Relations between the Cartesian and spherical vector frames#

At this stage, \(\mathbb{E}^3\) is endowed with three vector frames:

sage: E.frames()
[Coordinate frame (E^3, (e_x,e_y,e_z)),
 Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph)),
 Vector frame (E^3, (e_r,e_th,e_ph))]

The second one is the coordinate frame \(\left(\frac{\partial}{\partial r}, \frac{\partial}{\partial\theta}, \frac{\partial}{\partial \phi}\right)\) of spherical coordinates, while the third one is the standard orthonormal frame \((e_r,e_\theta,e_\phi)\) associated with spherical coordinates. For Cartesian coordinates, the coordinate frame and the orthonormal frame coincide: it is \((e_x,e_y,e_z)\). For spherical coordinates, the orthonormal frame is returned by the method spherical_frame():

sage: spherical_frame = E.spherical_frame()
sage: spherical_frame
Vector frame (E^3, (e_r,e_th,e_ph))

We may check that it is an orthonormal frame, i.e. that it obeys \(e_i\cdot e_j = \delta_{ij}\):

sage: es = spherical_frame
sage: [[es[i].dot(es[j]).expr() for j in E.irange()] for i in E.irange()]
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]

Via the method display, we may express the orthonormal spherical frame in terms of the Cartesian one:

sage: for vec in spherical_frame:
....:     vec.display(cartesian_frame, spherical)
e_r = cos(ph)*sin(th) e_x + sin(ph)*sin(th) e_y + cos(th) e_z
e_th = cos(ph)*cos(th) e_x + cos(th)*sin(ph) e_y - sin(th) e_z
e_ph = -sin(ph) e_x + cos(ph) e_y

The reverse is:

sage: for vec in cartesian_frame:
....:     vec.display(spherical_frame, spherical)
e_x = cos(ph)*sin(th) e_r + cos(ph)*cos(th) e_th - sin(ph) e_ph
e_y = sin(ph)*sin(th) e_r + cos(th)*sin(ph) e_th + cos(ph) e_ph
e_z = cos(th) e_r - sin(th) e_th

We may also express the orthonormal frame \((e_r,e_\theta,e_\phi)\) in terms on the coordinate frame \(\left(\frac{\partial}{\partial r}, \frac{\partial}{\partial\theta}, \frac{\partial}{\partial \phi}\right)\) (the latter being returned by the method frame() acting on the chart spherical):

sage: for vec in spherical_frame:
....:     vec.display(spherical.frame(), spherical)
e_r = ∂/∂r
e_th = 1/r ∂/∂th
e_ph = 1/(r*sin(th)) ∂/∂ph

Introducing cylindrical coordinates#

Cylindrical coordinates are introduced in a way similar to spherical coordinates:

sage: cylindrical.<rh,ph,z> = E.cylindrical_coordinates()
sage: cylindrical
Chart (E^3, (rh, ph, z))

We have:

sage: cylindrical[:]
(rh, ph, z)
sage: rh is cylindrical[1]
True
sage: cylindrical.coord_range()
rh: (0, +oo); ph: [0, 2*pi] (periodic); z: (-oo, +oo)

\(\mathbb{E}^3\) is now endowed with three coordinate charts:

sage: E.atlas()
[Chart (E^3, (x, y, z)), Chart (E^3, (r, th, ph)), Chart (E^3, (rh, ph, z))]

The transformations linking the cylindrical coordinates to the Cartesian ones are:

sage: E.coord_change(cylindrical, cartesian).display()
x = rh*cos(ph)
y = rh*sin(ph)
z = z
sage: E.coord_change(cartesian, cylindrical).display()
rh = sqrt(x^2 + y^2)
ph = arctan2(y, x)
z = z

There are now five vector frames defined on \(\mathbb{E}^3\):

sage: E.frames()
[Coordinate frame (E^3, (e_x,e_y,e_z)),
 Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph)),
 Vector frame (E^3, (e_r,e_th,e_ph)),
 Coordinate frame (E^3, (∂/∂rh,∂/∂ph,∂/∂z)),
 Vector frame (E^3, (e_rh,e_ph,e_z))]

The orthonormal frame associated with cylindrical coordinates is \((e_\rho, e_\phi, e_z)\):

sage: cylindrical_frame = E.cylindrical_frame()
sage: cylindrical_frame
Vector frame (E^3, (e_rh,e_ph,e_z))

We may check that it is an orthonormal frame:

sage: ec = cylindrical_frame
sage: [[ec[i].dot(ec[j]).expr() for j in E.irange()] for i in E.irange()]
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]

and express it in terms of the Cartesian frame:

sage: for vec in cylindrical_frame:
....:     vec.display(cartesian_frame, cylindrical)
e_rh = cos(ph) e_x + sin(ph) e_y
e_ph = -sin(ph) e_x + cos(ph) e_y
e_z = e_z

The reverse is:

sage: for vec in cartesian_frame:
....:     vec.display(cylindrical_frame, cylindrical)
e_x = cos(ph) e_rh - sin(ph) e_ph
e_y = sin(ph) e_rh + cos(ph) e_ph
e_z = e_z

Of course, we may express the orthonormal cylindrical frame in terms of the spherical one:

sage: for vec in cylindrical_frame:
....:     vec.display(spherical_frame, spherical)
e_rh = sin(th) e_r + cos(th) e_th
e_ph = e_ph
e_z = cos(th) e_r - sin(th) e_th

along with the reverse transformation:

sage: for vec in spherical_frame:
....:     vec.display(cylindrical_frame, spherical)
e_r = sin(th) e_rh + cos(th) e_z
e_th = cos(th) e_rh - sin(th) e_z
e_ph = e_ph

The orthonormal frame \((e_\rho,e_\phi,e_z)\) can be expressed in terms on the coordinate frame \(\left(\frac{\partial}{\partial\rho}, \frac{\partial}{\partial\phi}, \frac{\partial}{\partial z}\right)\) (the latter being returned by the method frame() acting on the chart cylindrical):

sage: for vec in cylindrical_frame:
....:     vec.display(cylindrical.frame(), cylindrical)
e_rh = ∂/∂rh
e_ph = 1/rh ∂/∂ph
e_z = ∂/∂z

How to evaluate the coordinates of a point in various systems#

Let us introduce a point \(p\in \mathbb{E}^3\) via the generic SageMath syntax for creating an element from its parent (here \(\mathbb{E}^3\)), i.e. the call operator (), with the coordinates of the point as the first argument:

sage: p = E((-1, 1,0), chart=cartesian, name='p')
sage: p
Point p on the Euclidean space E^3

Actually, since the Cartesian coordinates are the default ones, the argument chart=cartesian can be omitted:

sage: p = E((-1, 1,0), name='p')
sage: p
Point p on the Euclidean space E^3

The coordinates of \(p\) in a given coordinate chart are obtained by letting the corresponding chart act on \(p\):

sage: cartesian(p)
(-1, 1, 0)
sage: spherical(p)
(sqrt(2), 1/2*pi, 3/4*pi)
sage: cylindrical(p)
(sqrt(2), 3/4*pi, 0)

Here some example of a point defined from its spherical coordinates:

sage: q = E((4,pi/3,pi), chart=spherical, name='q')
sage: q
Point q on the Euclidean space E^3

We have then:

sage: spherical(q)
(4, 1/3*pi, pi)
sage: cartesian(q)
(-2*sqrt(3), 0, 2)
sage: cylindrical(q)
(2*sqrt(3), pi, 2)

How to express a scalar field in various coordinate systems#

Let us define a scalar field on \(\mathbb{E}^3\) from its expression in Cartesian coordinates:

sage: f = E.scalar_field(x^2+y^2 - z^2, name='f')

Note that since the Cartesian coordinates are the default ones, we have not specified them in the above definition. Thanks to the known coordinate transformations, the expression of \(f\) in terms of other coordinates is automatically computed:

sage: f.display()
f: E^3 → ℝ
   (x, y, z) ↦ x^2 + y^2 - z^2
   (r, th, ph) ↦ -2*r^2*cos(th)^2 + r^2
   (rh, ph, z) ↦ rh^2 - z^2

We can limit the output to a single coordinate system:

sage: f.display(cartesian)
f: E^3 → ℝ
   (x, y, z) ↦ x^2 + y^2 - z^2
sage: f.display(cylindrical)
f: E^3 → ℝ
   (rh, ph, z) ↦ rh^2 - z^2

The coordinate expression in a given coordinate system is obtained via the method expr():

sage: f.expr()  # expression in the default chart (Cartesian coordinates)
x^2 + y^2 - z^2
sage: f.expr(spherical)
-2*r^2*cos(th)^2 + r^2
sage: f.expr(cylindrical)
rh^2 - z^2

The values of \(f\) at points \(p\) and \(q\) are:

sage: f(p)
2
sage: f(q)
8

Of course, we may define a scalar field from its coordinate expression in a chart that is not the default one:

sage: g = E.scalar_field(r^2, chart=spherical, name='g')

Instead of using the keyword argument chart, one can pass a dictionary as the first argument, with the chart as key:

sage: g = E.scalar_field({spherical: r^2}, name='g')

The computation of the expressions of \(g\) in the other coordinate systems is triggered by the method display():

sage: g.display()
g: E^3 → ℝ
   (x, y, z) ↦ x^2 + y^2 + z^2
   (r, th, ph) ↦ r^2
   (rh, ph, z) ↦ rh^2 + z^2

How to express a vector field in various frames#

Let us introduce a vector field on \(\mathbb{E}^3\) by its components in the Cartesian frame. Since the latter is the default vector frame on \(\mathbb{E}^3\), it suffices to write:

sage: v = E.vector_field(-y, x, z^2, name='v')
sage: v.display()
v = -y e_x + x e_y + z^2 e_z

Equivalently, a vector field can be defined directly from its expansion on the Cartesian frame:

sage: ex, ey, ez = cartesian_frame[:]
sage: v = -y*ex + x*ey + z^2*ez
sage: v.display()
-y e_x + x e_y + z^2 e_z

Let us provide v with some name, as above:

sage: v.set_name('v')
sage: v.display()
v = -y e_x + x e_y + z^2 e_z

The components of \(v\) are returned by the square bracket operator:

sage: v[1]
-y
sage: v[:]
[-y, x, z^2]

The computation of the expression of \(v\) in terms of the orthonormal spherical frame is triggered by the method display():

sage: v.display(spherical_frame)
v = z^3/sqrt(x^2 + y^2 + z^2) e_r
 - sqrt(x^2 + y^2)*z^2/sqrt(x^2 + y^2 + z^2) e_th + sqrt(x^2 + y^2) e_ph

We note that the components are still expressed in the default chart (Cartesian coordinates). To have them expressed in the spherical chart, it suffices to pass the latter as a second argument to display():

sage: v.display(spherical_frame, spherical)
v = r^2*cos(th)^3 e_r - r^2*cos(th)^2*sin(th) e_th + r*sin(th) e_ph

Again, the components of \(v\) are obtained by means of the square bracket operator, by specifying the vector frame as first argument and the coordinate chart as the last one:

sage: v[spherical_frame, 1]
z^3/sqrt(x^2 + y^2 + z^2)
sage: v[spherical_frame, 1, spherical]
r^2*cos(th)^3
sage: v[spherical_frame, :, spherical]
[r^2*cos(th)^3, -r^2*cos(th)^2*sin(th), r*sin(th)]

Similarly, the expression of \(v\) in terms of the cylindrical frame is:

sage: v.display(cylindrical_frame, cylindrical)
v = rh e_ph + z^2 e_z
sage: v[cylindrical_frame, :, cylindrical]
[0, rh, z^2]

The value of the vector field \(v\) at point \(p\) is:

sage: vp = v.at(p)
sage: vp
Vector v at Point p on the Euclidean space E^3
sage: vp.display()
v = -e_x - e_y
sage: vp.display(spherical_frame.at(p))
v = sqrt(2) e_ph
sage: vp.display(cylindrical_frame.at(p))
v = sqrt(2) e_ph

The value of the vector field \(v\) at point \(q\) is:

sage: vq = v.at(q)
sage: vq
Vector v at Point q on the Euclidean space E^3
sage: vq.display()
v = -2*sqrt(3) e_y + 4 e_z
sage: vq.display(spherical_frame.at(q))
v = 2 e_r - 2*sqrt(3) e_th + 2*sqrt(3) e_ph
sage: vq.display(cylindrical_frame.at(q))
v = 2*sqrt(3) e_ph + 4 e_z

How to change the default coordinates and default vector frame#

At any time, one may change the default coordinates by the method set_default_chart():

sage: E.set_default_chart(spherical)

Then:

sage: f.expr()
-2*r^2*cos(th)^2 + r^2
sage: v.display()
v = -r*sin(ph)*sin(th) e_x + r*cos(ph)*sin(th) e_y + r^2*cos(th)^2 e_z

Note that the default vector frame is still the Cartesian one; to change to the orthonormal spherical frame, use set_default_frame():

sage: E.set_default_frame(spherical_frame)

Then:

sage: v.display()
v = r^2*cos(th)^3 e_r - r^2*cos(th)^2*sin(th) e_th + r*sin(th) e_ph
sage: v.display(cartesian_frame, cartesian)
v = -y e_x + x e_y + z^2 e_z