Advanced aspects: the Euclidean space as a Riemannian manifold

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).

The Euclidean 3-space

Let us consider the 3-dimensional Euclidean space \(\mathbb{E}^3\), with Cartesian coordinates \((x,y,z)\):

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

\(\mathbb{E}^3\) is actually a Riemannian manifold (see pseudo_riemannian), i.e. a smooth real manifold endowed with a positive definite metric tensor:

sage: E.category()
Join of
 Category of smooth manifolds over Real Field with 53 bits of precision and
 Category of connected manifolds over Real Field with 53 bits of precision and
 Category of complete metric spaces
sage: E.base_field() is RR
True
sage: E.metric()
Riemannian metric g on the Euclidean space E^3

Actually RR is used here as a proxy for the real field (this should be replaced in the future, see the discussion at github issue #24456) and the 53 bits of precision play of course no role for the symbolic computations.

Let us introduce spherical and cylindrical coordinates on \(\mathbb{E}^3\):

sage: spherical.<r,th,ph> = E.spherical_coordinates()
sage: cylindrical.<rh,ph,z> = E.cylindrical_coordinates()

The user atlas of \(\mathbb{E}^3\) has then three charts:

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

while there are 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))]

Indeed, there are two frames associated with each of the three coordinate systems: the coordinate frame (denoted with partial derivatives above) and an orthonormal frame (denoted by e_* above), but for Cartesian coordinates, both frames coincide.

We get the orthonormal spherical and cylindrical frames by:

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

On the other side, the coordinate frames \(\left(\frac{\partial}{\partial r}, \frac{\partial}{\partial\theta}, \frac{\partial}{\partial \phi}\right)\) and \(\left(\frac{\partial}{\partial \rho}, \frac{\partial}{\partial\phi}, \frac{\partial}{\partial z}\right)\) are returned by the method frame() acting on the coordinate charts:

sage: spherical.frame()
Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph))
sage: cylindrical.frame()
Coordinate frame (E^3, (∂/∂rh,∂/∂ph,∂/∂z))

Charts as maps \(\mathbb{E}^3 \rightarrow \mathbb{R}^3\)

The chart of Cartesian coordinates has been constructed at the declaration of E; let us denote it by cartesian:

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

Let us consider a point \(p\in \mathbb{E}^3\), defined by its Cartesian coordinates:

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

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)

Riemannian metric

The default metric tensor of \(\mathbb{E}^3\) is:

sage: g = E.metric()
sage: g
Riemannian metric g on the Euclidean space E^3
sage: g.display()
g = dx⊗dx + dy⊗dy + dz⊗dz
sage: g[:]
[1 0 0]
[0 1 0]
[0 0 1]

The above display in performed in the default frame, which is the Cartesian one. Of course, we may ask for display with respect to other frames:

sage: g.display(spherical_frame)
g = e^r⊗e^r + e^th⊗e^th + e^ph⊗e^ph
sage: g[spherical_frame, :]
[1 0 0]
[0 1 0]
[0 0 1]

In the above display, e^r = \(e^r\), e^th = \(e^\theta\) and e^ph = \(e^\phi\) are the 1-forms defining the coframe dual to the orthonormal spherical frame \((e_r,e_\theta,e_\phi)\):

sage: spherical_frame.coframe()
Coframe (E^3, (e^r,e^th,e^ph))

The fact that the above metric components are either 0 or 1 reflect the orthonormality of the vector frame \((e_r,e_\theta,e_\phi)\). On the contrary, in the coordinate frame \(\left(\frac{\partial}{\partial r}, \frac{\partial}{\partial\theta}, \frac{\partial}{\partial \phi}\right)\), which is not orthonormal, some components differ from 0 or 1:

sage: g.display(spherical.frame())
g = dr⊗dr + (x^2 + y^2 + z^2) dth⊗dth + (x^2 + y^2) dph⊗dph

Note that the components are expressed in terms of the default chart, namely the Cartesian one. To have them displayed in terms of the spherical chart, we have to provide the latter as the second argument of the method display():

sage: g.display(spherical.frame(), spherical)
g = dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph

Since SageMath 8.8, a shortcut is:

sage: g.display(spherical)
g = dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph

The matrix view of the components is obtained via the square bracket operator:

sage: g[spherical.frame(), :, spherical]
[            1             0             0]
[            0           r^2             0]
[            0             0 r^2*sin(th)^2]

Similarly, for cylindrical coordinates, we have:

sage: g.display(cylindrical_frame)
g = e^rh⊗e^rh + e^ph⊗e^ph + e^z⊗e^z
sage: g.display(cylindrical)
g = drh⊗drh + rh^2 dph⊗dph + dz⊗dz
sage: g[cylindrical.frame(), :, cylindrical]
[   1    0    0]
[   0 rh^2    0]
[   0    0    1]

The metric \(g\) is a flat: its Riemann curvature tensor (see riemann()) is zero:

sage: g.riemann()
Tensor field Riem(g) of type (1,3) on the Euclidean space E^3
sage: g.riemann().display()
Riem(g) = 0

The metric \(g\) defines the dot product on \(\mathbb{E}^3\):

sage: u = E.vector_field(x*y, y*z, z*x)
sage: u.display()
x*y e_x + y*z e_y + x*z e_z
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
sage: u.dot(v) == g(u,v)
True

Consequently:

sage: norm(u) == sqrt(g(u,u))
True

The Levi-Civita tensor

The scalar triple product of \(\mathbb{E}^3\) is provided by the Levi-Civita tensor (also called volume form) associated with \(g\) (and chosen such that \((e_x,e_y,e_z)\) is right-handed):

sage: epsilon = E.scalar_triple_product()
sage: epsilon
3-form epsilon on the Euclidean space E^3
sage: epsilon is E.volume_form()
True
sage: epsilon.display()
epsilon = dx∧dy∧dz
sage: epsilon.display(spherical)
epsilon = r^2*sin(th) dr∧dth∧dph
sage: epsilon.display(cylindrical)
epsilon = rh drh∧dph∧dz

Checking that all orthonormal frames introduced above are right-handed:

sage: ex, ey, ez = E.cartesian_frame()[:]
sage: epsilon(ex, ey, ez).display()
epsilon(e_x,e_y,e_z): E^3 → ℝ
   (x, y, z) ↦ 1
   (r, th, ph) ↦ 1
   (rh, ph, z) ↦ 1

sage: epsilon(*spherical_frame)
Scalar field epsilon(e_r,e_th,e_ph) on the Euclidean space E^3
sage: epsilon(*spherical_frame).display()
epsilon(e_r,e_th,e_ph): E^3 → ℝ
   (x, y, z) ↦ 1
   (r, th, ph) ↦ 1
   (rh, ph, z) ↦ 1

sage: epsilon(*cylindrical_frame).display()
epsilon(e_rh,e_ph,e_z): E^3 → ℝ
   (x, y, z) ↦ 1
   (r, th, ph) ↦ 1
   (rh, ph, z) ↦ 1

Vector fields as derivations

Let \(f\) be a scalar field on \(\mathbb{E}^3\):

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

Vector fields act as derivations on scalar fields:

sage: v(f)
Scalar field v(f) on the Euclidean space E^3
sage: v(f).display()
v(f): E^3 → ℝ
   (x, y, z) ↦ -2*z^3
   (r, th, ph) ↦ -2*r^3*cos(th)^3
   (rh, ph, z) ↦ -2*z^3
sage: v(f) == v.dot(f.gradient())
True

sage: df = f.differential()
sage: df
1-form df on the Euclidean space E^3
sage: df.display()
df = 2*x dx + 2*y dy - 2*z dz
sage: v(f) == df(v)
True

The algebra of scalar fields

The set \(C^\infty(\mathbb{E}^3)\) of all smooth scalar fields on \(\mathbb{E}^3\) forms a commutative algebra over \(\mathbb{R}\):

sage: CE = E.scalar_field_algebra()
sage: CE
Algebra of differentiable scalar fields on the Euclidean space E^3
sage: CE.category()
Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
sage: f in CE
True

In SageMath terminology, \(C^\infty(\mathbb{E}^3)\) is the parent of scalar fields:

sage: f.parent() is CE
True

The free module of vector fields

The set \(\mathfrak{X}(\mathbb{E}^3)\) of all vector fields on \(\mathbb{E}^3\) is a free module of rank 3 over the commutative algebra \(C^\infty(\mathbb{E}^3)\):

sage: XE = v.parent()
sage: XE
Free module X(E^3) of vector fields on the Euclidean space E^3
sage: XE.category()
Category of finite dimensional modules over Algebra of differentiable
 scalar fields on the Euclidean space E^3
sage: XE.base_ring()
Algebra of differentiable scalar fields on the Euclidean space E^3
sage: XE.base_ring() is CE
True
sage: rank(XE)
3

The bases of the free module \(\mathfrak{X}(\mathbb{E}^3)\) are nothing but the vector frames defined on \(\mathbb{E}^3\):

sage: XE.bases()
[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))]

Tangent spaces

A vector field evaluated at a point $p$ is a vector in the tangent space \(T_p\mathbb{E}^3\):

sage: p
Point p on the Euclidean space E^3
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: Tp = vp.parent()
sage: Tp
Tangent space at Point p on the Euclidean space E^3
sage: Tp is E.tangent_space(p)
True
sage: Tp.category()
Category of finite dimensional vector spaces over Symbolic Ring
sage: dim(Tp)
3
sage: isinstance(Tp, FiniteRankFreeModule)
True

The bases on \(T_p\mathbb{E}^3\) are inherited from the vector frames of \(\mathbb{E}^3\):

sage: Tp.bases()
[Basis (e_x,e_y,e_z) on the Tangent space at Point p on the Euclidean space E^3,
 Basis (∂/∂r,∂/∂th,∂/∂ph) on the Tangent space at Point p on the Euclidean space E^3,
 Basis (e_r,e_th,e_ph) on the Tangent space at Point p on the Euclidean space E^3,
 Basis (∂/∂rh,∂/∂ph,∂/∂z) on the Tangent space at Point p on the Euclidean space E^3,
 Basis (e_rh,e_ph,e_z) on the Tangent space at Point p on the Euclidean space E^3]

For instance, we have:

sage: spherical_frame.at(p)
Basis (e_r,e_th,e_ph) on the Tangent space at Point p on the
 Euclidean space E^3
sage: spherical_frame.at(p) in Tp.bases()
True

Levi-Civita connection

The Levi-Civita connection associated to the Euclidean metric \(g\) is:

sage: nabla = g.connection()
sage: nabla
Levi-Civita connection nabla_g associated with the Riemannian metric g
 on the Euclidean space E^3

The corresponding Christoffel symbols with respect to Cartesian coordinates are identically zero: none of them appear in the output of christoffel_symbols_display(), which by default displays only nonzero Christoffel symbols:

sage: g.christoffel_symbols_display(cartesian)

On the contrary, some of the Christoffel symbols with respect to spherical coordinates differ from zero:

sage: g.christoffel_symbols_display(spherical)
Gam^r_th,th = -r
Gam^r_ph,ph = -r*sin(th)^2
Gam^th_r,th = 1/r
Gam^th_ph,ph = -cos(th)*sin(th)
Gam^ph_r,ph = 1/r
Gam^ph_th,ph = cos(th)/sin(th)

By default, only nonzero and nonredundant values are displayed (for instance \(\Gamma^\phi_{\ \, \phi r}\) is skipped, since it can be deduced from \(\Gamma^\phi_{\ \, r \phi}\) by symmetry on the last two indices).

Similarly, the nonzero Christoffel symbols with respect to cylindrical coordinates are:

sage: g.christoffel_symbols_display(cylindrical)
Gam^rh_ph,ph = -rh
Gam^ph_rh,ph = 1/rh

The Christoffel symbols are nothing but the connection coefficients in the corresponding coordinate frame:

sage: nabla.display(cylindrical.frame(), cylindrical, only_nonredundant=True)
Gam^rh_ph,ph = -rh
Gam^ph_rh,ph = 1/rh

The connection coefficients with respect to the orthonormal (non-coordinate) frames are (again only nonzero values are displayed):

sage: nabla.display(spherical_frame, spherical)
Gam^1_22 = -1/r
Gam^1_33 = -1/r
Gam^2_12 = 1/r
Gam^2_33 = -cos(th)/(r*sin(th))
Gam^3_13 = 1/r
Gam^3_23 = cos(th)/(r*sin(th))
sage: nabla.display(cylindrical_frame, cylindrical)
Gam^1_22 = -1/rh
Gam^2_12 = 1/rh

The Levi-Civita connection \(\nabla_g\) is the connection involved in the standard differential operators:

sage: from sage.manifolds.operators import *
sage: grad(f) == nabla(f).up(g)
True
sage: nabla(f) == grad(f).down(g)
True
sage: div(u) == nabla(u).trace()
True
sage: div(v) == nabla(v).trace()
True
sage: laplacian(f) == nabla(nabla(f).up(g)).trace()
True
sage: laplacian(u) == nabla(nabla(u).up(g)).trace(1,2)
True
sage: laplacian(v) == nabla(nabla(v).up(g)).trace(1,2)
True