Programming Differentiable Manifolds via SageManifolds

$\textbf{*A replica of SageManifolds notebook tutorial with revisions, extractions, and an appendix added, describing the software computational architecture.}$
Vol. 1, ID: 001
August 1, 2023

Introduction

SageManifolds is a software library in the SageMath Python-based computer algebra software system that provides functionality for working with manifolds, differential geometry, and related concepts. It is designed to facilitate computations and explorations in differential geometry, tensor calculus, and other areas of mathematics.

Here are some key features and functionalities offered by SageManifolds:

1. Manifold Definition: SageManifolds allows you to define differentiable manifolds of arbitrary dimensions. You can create instances of manifold classes, specify the dimension, and assign names to the manifolds.

2. Chart and Coordinate System: Within each manifold, you can define charts that represent coordinate systems on the manifold. SageManifolds provides functionality to define charts, specify the coordinate functions, and associate them with the manifold.

3. Differential Forms: SageManifolds supports the creation and manipulation of differential forms on manifolds. You can define differential forms of various degrees and perform operations such as addition, wedge product, exterior derivative, and pullback.

4. Differential Maps: Differential maps, also known as differentiable maps or smooth maps, are essential in connecting different manifolds. SageManifolds allows you to define differential maps between manifolds, compute their compositions, differentials, and pullbacks.

5. Tensor Fields: SageManifolds provides functionality for defining and manipulating tensor fields on manifolds. You can create tensor fields of arbitrary rank and perform operations such as tensor contractions, tensor products, and covariant derivatives.

6. Geodesics and Curvature: SageManifolds supports computations related to geodesics, including their numerical integration and visualization. It also provides tools for calculating curvature tensors, sectional curvatures, and other geometric quantities.

7. Symbolic Expressions: SageManifolds integrates seamlessly with the symbolic computation capabilities of SageMath. You can work with symbolic expressions involving manifold-related objects, perform calculations, simplify expressions, and obtain numerical results.

SageManifolds is a versatile tool for studying and exploring differential geometry, allowing users to perform computations, conduct experiments, and gain insights into geometric structures. It provides an extensive set of functionalities and interfaces with other components of the SageMath system, making it a valuable resource for mathematicians, physicists, and researchers working in various fields.

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Section 1. Download and Installation

Since all SageManifolds code is included in SageMath, it suffices to install the latest SageMath to benefit from the latest manifold functionalities. Supported OS systems are as listed:

• Linux: most major Linux distributions have SageMath packages; however, it may be a version older than the latest
• macOS
• Windows

Having installed SageMath, if you run

$\verb|sage -n|$

a Jupyter page should open in your browser. Click on "New" and select "SageMath xx.x" to open a Jupyter notebook with a SageMath kernel. In the first cell, if you type

$\verb|Manifold(2, 'M')|$

the output should be

$\verb|2-dimensional differentiable manifold M|$

Below is a notebook tutorial tackling the use of the modules and relative Python functions of SageManifolds to perform computations and gain elementary teaching of the rules of Manifold Theory.

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Section 2. Manifold

Manifolds of a given type, over a topological field, is prescribed via $\verb|Manifold|$

M = Manifold(3, 'M', latex_name=r'\mathcal{M}', start_index=1)

• $\verb|3|$ – integer; dimension of the manifold
• $\verb|'M'|$ – string; name (symbol) given to the manifold
• $\verb|r'\mathcal{M}'|$ – string; LaTeX symbol to denote the manifold
• $\verb|1|$ – integer; lower value of the range of indices used for “indexed objects” on the manifold, e.g. coordinates in a chart

Optional arguments also include [1]:

• $\verb|field|$ – (default: 'real') field $\textit{K}$ on which the manifold is defined
• $\verb|structure|$ – (default: 'smooth') structure\type of manifold

Returns:

• manifold of specified $\verb|structure|$ type, as an instance of $\verb|TopologicalManifold|$ or one of its subclasses $\verb|DifferentiableManifold|$ or $\verb|PseudoRiemannianManifold|$

Via the function $\verb|type()|$, we get the type of the returned Python object corresponding to $\verb|M|$. With respect to the above instantiation, the returned Python class is $\verb|DifferentiableManifold_with_category|$

print(type(M))

<class 'sage.manifolds.differentiable.manifold.DifferentiableManifold_with_category'>

A print statement of the manifold object prints a short description of the manifold

print(M)

3-dimensional differentiable manifold M

Via the function $\verb|category()|$, we get the category of $\verb|M|$

category(M)

\(\displaystyle \newcommand{\Bold}[1]{\mathbf{#1}}\mathbf{Smooth}_{\Bold{R}}\)

Via the manifold class function $\verb|irange()|$, we generate the indices on the manifold $\verb|M|$

[i for i in M.irange()]

\(\displaystyle \left[1, 2, 3\right]\)

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Section 3. Charts

In continuation of example manifold, $\verb|M|$, in $\textbf{Section 2. Manifold}$, manifolds are prescribed $\textbf{charts}$, defined as an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure [2]. For $\verb|M|$, we'll prescribe a single chart:

X.<x,y,z> = M.chart()

$\verb|.<x,y,z>|$ are Python variables $\verb|x|$, $\verb|y|$ and $\verb|z|$, which are set to the three coordinates of the chart. This allows one to refer subsequently to the coordinates by their variable names.

Above, $\verb|chart()|$ has no arguments. Its coordinate symbols are $\verb|x|$, $\verb|y|$ and $\verb|z|$. By default, if no values passed to $\verb|chart()|$ parameter, $\verb|coord_restriction|$, then each coordinate range is $(-\infty, \infty)$ [3].

Print of chart $\verb|X|$ outputs a $\verb|Chart|$ tuple:

print(X)

Chart (M, (x, y, z))

X

\(\displaystyle \left(\mathcal{M},(x, y, z)\right)\)

You can retrieve each coordinate of chart $\verb|X|$ via their respective indices, or to capture the full set, access is implemented via Python's slice syntax $\verb|[:]|$. We show the latter:

X[:]

\(\displaystyle \left(x, y, z\right)\)

Each coordinate is a SageMath symbolic expression object, $\verb|sage.symbolic.expression.Expression|$

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Section 4. Chart functions

Real-valued functions that act on charts are generated via the chart object's method $\verb|function()|$. For example:

f = X.function(x + y^2 + z^3)
f

\(\displaystyle z^{3} + y^{2} + x\)

f.display()

\(\displaystyle \left(x, y, z\right) \mapsto z^{3} + y^{2} + x\)

f(1,2,3)

\(\displaystyle 32\)

$\verb|f|$ is a $\verb|ChartFunctionRing_with_category|$ type, which is a subtype of the $\verb|ChartFunction|$ class in SageManifolds

print(type(f))

<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>

---

Section 5. Atlas

Lets define an open subset on $\mathcal{M}$, $\verb|U|$, where $\verb|y=0|$ or $\verb|x<0|$:

U = M.open_subset('U', coord_def={X: (y != 0, x < 0)})

In addition, lets restrict the chart $\verb|X|$ to the open subset $\verb|U|$:

X_U = X.restrict(U)
X_U

\(\displaystyle \left(U,(x, y, z)\right)\)

Now, lets introduce a chart $\verb|Y|$ with spehrical coordinates, (r, $\theta$, $\phi$), where r range is (0, $\infty$), $\theta$ range is (0, $\pi$) and $\phi$ range is (0, 2$\pi$):

Y.<r,th,ph> = U.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
Y

\(\displaystyle \left(U,(r, {\theta}, {\phi})\right)\)

The coordinates are now stored in Sage and may be examined via $\verb|assumptions()|$

assumptions()

\(\displaystyle \left[\verb|x|\verb| |\verb|is|\verb| |\verb|real|, \verb|y|\verb| |\verb|is|\verb| |\verb|real|, \verb|z|\verb| |\verb|is|\verb| |\verb|real|, \verb|r|\verb| |\verb|is|\verb| |\verb|real|, r > 0, \verb|th|\verb| |\verb|is|\verb| |\verb|real|, {\theta} > 0, {\theta} < \pi, \verb|ph|\verb| |\verb|is|\verb| |\verb|real|, {\phi} > 0, {\phi} < 2 \, \pi\right]\)

Now, we may explore a transition mapping between chart $\verb|Y|$ and chart $\verb|X_U|$:

transit_Y_to_X = Y.transition_map(X_U, [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)])
transit_Y_to_X

\(\displaystyle \left(U,(r, {\theta}, {\phi})\right) \rightarrow \left(U,(x, y, z)\right)\)

transit_Y_to_X.display()

\(\displaystyle \left\{\begin{array}{lcl} x & = & r \cos\left({\phi}\right) \sin\left({\theta}\right) \\ y & = & r \sin\left({\phi}\right) \sin\left({\theta}\right) \\ z & = & r \cos\left({\theta}\right) \end{array}\right.\)

And the inverse of $\verb|transition_map()|$ is set via $\verb|set_inverse()|$:

transit_Y_to_X.set_inverse(sqrt(x^2 + y^2 + z^2), atan2(sqrt(x^2+y^2), z), atan2(y, x))

Check of the inverse coordinate transformation:
  r == r  *passed*
  th == arctan2(r*sin(th), r*cos(th))  **failed**
  ph == arctan2(r*sin(ph)*sin(th), r*cos(ph)*sin(th))  **failed**
  x == x  *passed*
  y == y  *passed*
  z == z  *passed*
NB: a failed report can reflect a mere lack of simplification.

The reported failure for $\verb|th|$ and $\verb|ph|$ is due to a lack of simplification of expressions involving $\verb|arctan2|$.

Thus,

transit_Y_to_X.inverse().display()

\(\displaystyle \left\{\begin{array}{lcl} r & = & \sqrt{x^{2} + y^{2} + z^{2}} \\ {\theta} & = & \arctan\left(\sqrt{x^{2} + y^{2}}, z\right) \\ {\phi} & = & \arctan\left(y, x\right) \end{array}\right.\)

Now, the $\textbf{atlas}$ of $\verb|M|$, given our prescribed charts $\verb|X|$ and $Y$ are

M.atlas()

\(\displaystyle \left[\left(\mathcal{M},(x, y, z)\right), \left(U,(x, y, z)\right), \left(U,(r, {\theta}, {\phi})\right)\right]\)

Chart $\verb|Y|$ may be plotted with respect to the coordinate system of chart $\verb|X|$ as below:

Y.plot(X)

Further details of the plot may be plotted through specifying the coordinates range, style, and color:

Y.plot(X, ranges={r:(1,2), th:(0,pi/2)}, number_values=4, color={r:'blue', th:'green', ph:'red'}, aspect_ratio=1)

On the other hand, chart $\verb|X_U|$ may be plotted with respect to the coordinate system of chart $\verb|Y|$ as below:

graph = X_U.plot(Y)
show(graph, axes_labels=['r', 'theta', 'phi'])

Section 6. Points (on the Manifold)

A point, specified as a coordinate with respect to a chart in $\verb|M|$, may be specified with $\verb|point()|$ method of $\verb|M|$:

p = M.point((1,2,-1), chart=X, name='p')
print(p)
p

Point p on the 3-dimensional differentiable manifold M

\(\displaystyle p\)

To test that the defined $\verb|p|$ is in $\verb|M|$, we write the given boolean expression:

p in M

\(\displaystyle \mathrm{True}\)

The coordinate of $\verb|p|$ may then be retrieved with $\verb|coord()|$ method of $\verb|p|$:

p.coord(X)

\(\displaystyle \left(1, 2, -1\right)\)

The coordinate may also be retrieved with passing $\verb|p|$ to $\verb|X|$:

X(p)

\(\displaystyle \left(1, 2, -1\right)\)

Now, lets define a second point $\verb|q|$:

q = M.point((1,0,2), name='q')

q in U

\(\displaystyle \mathrm{False}\)

Recall that $\verb|U|$ is an open subset, where either coordinate y != 0 or x < 0. Hence, the above boolean test returns $\verb|False|$.

The same is true given open subset $\verb|Y|$:

try:
    q.coord(Y)
except ValueError as exc:
    print(f"Error: {exc}")

Error: the point does not belong to the domain of Chart (U, (r, th, ph))

Now, point $\verb|p|$ may be mapped to its spherical coordinate through the underlying transition map from chart $\verb|X|$ to chart $\verb|Y|$:

p.coord(Y)

\(\displaystyle \left(\sqrt{3} \sqrt{2}, \pi - \arctan\left(\sqrt{5}\right), \arctan\left(2\right)\right)\)

Y(p)

\(\displaystyle \left(\sqrt{3} \sqrt{2}, \pi - \arctan\left(\sqrt{5}\right), \arctan\left(2\right)\right)\)

Each point (i.e. - $\verb|p|$) lives in the universe of its manifold $\verb|M|$, irrespective of any given chart:

p.parent()

\(\displaystyle \mathcal{M}\)

q.parent()

\(\displaystyle \mathcal{M}\)

Yet, a point defined with respect to an open subset (i.e. - $\verb|U|$) of a manifold lives in the universe of the open subset prescribed:

p1 = U.point((sqrt(3)*sqrt(2), pi-atan(sqrt(5)), atan(2)), chart=Y)
p1.parent()

\(\displaystyle U\)

Section 7. Scalar Fields

A scalar field may be prescribed via a predefined open set (i.e. - $\verb|U|$). A scalar field is a mapping of $f: U \rightarrow \mathbb{R}$, where $U$ is an open subset of a manifold $\mathcal{M}$ [4]. The mapping must be prescribed with respect to a chart of open subset $U$:

f = U.scalar_field({X_U: x + y^2 + z^3}, name='f')
print(f)

Scalar field f on the Open subset U of the 3-dimensional differentiable manifold M

Thus, the mapping maps a point $p$ to a real value in $\mathbb{R}$:

f(p)

\(\displaystyle 4\)

To get a display of the predefined scalar field mapping, with respect to a prescribed chart of manifold $\mathcal{M}$, use the method $\verb|display()|$ of the scalar field instance:

f.display(X_U)

\(\displaystyle \begin{array}{llcl} f:& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & z^{3} + y^{2} + x \end{array}\)

To get a display of the mapping with respect to all charts of a manifold, simply call $\verb|display()|$ without passing any chart instance:

f.display()

\(\displaystyle \begin{array}{llcl} f:& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & z^{3} + y^{2} + x \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & r^{3} \cos\left({\theta}\right)^{3} + r^{2} \sin\left({\phi}\right)^{2} \sin\left({\theta}\right)^{2} + r \cos\left({\phi}\right) \sin\left({\theta}\right) \end{array}\)

Thus, lets see what we get when we request for a display of the scalar field mapping $\verb|f|$ with respect to the spherical coordinate chart $\verb|Y|$:

f.display(Y)

\(\displaystyle \begin{array}{llcl} f:& U & \longrightarrow & \mathbb{R} \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & r^{3} \cos\left({\theta}\right)^{3} + r^{2} \sin\left({\phi}\right)^{2} \sin\left({\theta}\right)^{2} + r \cos\left({\phi}\right) \sin\left({\theta}\right) \end{array}\)

Henceforth, the scalar field mapping being a function acting on the coordinates of a given chart may be retrieved through its method $\verb|coord_function()|$:

f.coord_function(X_U)

\(\displaystyle z^{3} + y^{2} + x\)

f.coord_function().display()

\(\displaystyle \left(x, y, z\right) \mapsto z^{3} + y^{2} + x\)

f.coord_function(Y)

\(\displaystyle r^{3} \cos\left({\theta}\right)^{3} + r^{2} \sin\left({\phi}\right)^{2} \sin\left({\theta}\right)^{2} + r \cos\left({\phi}\right) \sin\left({\theta}\right)\)

Now, the parent class of $\verb|f|$ is $C^{\infty}(\verb|U|)$ on the open subset $\verb|U|$. $C^{\infty}(\verb|U|)$ contains all smooth scalar fields on the predefined open subset $\verb|U|$ and is a commutative algebra over $\mathbb{R}$.

CU = f.parent()
CU

\(\displaystyle C^{\infty}\left(U\right)\)

print(CU)

Algebra of differentiable scalar fields on the Open subset U of the 3-dimensional differentiable manifold M

CU.category()

\(\displaystyle \mathbf{JoinCategory}\)

The base ring of the algebra $C^{\infty}(\verb|U|)$ is the field $\mathbb{R}$ (defined as $\verb|SR|$ in SageMath; stand for Symbolic Ring):

CU.base_ring()

\(\displaystyle \text{SR}\)

Section 8. Tangent spaces

At point $\verb|p|$ on manifold $\verb|M|$, a tangent space is defined via $\verb|tangent_space()|$ of manifold $\verb|M|$:

Tp = M.tangent_space(p)
Tp

\(\displaystyle T_{p}\,\mathcal{M}\)

print(Tp)

Tangent space at Point p on the 3-dimensional differentiable manifold M

$T_p\mathcal{M}$ is a 2-dimensional vector space over $\mathbb{R}$ on a manifold $\mathcal{M}$ ($\verb|M|$ in this case) at a given point $p$ ($\verb|p|$ in this case):

print(Tp.category())

Category of finite dimensional vector spaces over Symbolic Ring

Tp.dim()

\(\displaystyle 3\)

The vector bases of $T_p\mathcal{M}$ are:

Tp.bases()

\(\displaystyle \left[\left(\frac{\partial}{\partial x },\frac{\partial}{\partial y },\frac{\partial}{\partial z }\right), \left(\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right]\)

The vector base of $T_q\mathcal{M}$ at point $\verb|q|$, as defined earlier, is:

Tq = M.tangent_space(q)
Tq.bases()

\(\displaystyle \left[\left(\frac{\partial}{\partial x },\frac{\partial}{\partial y },\frac{\partial}{\partial z }\right)\right]\)

To get a random tangent vector from vector spaces $T_p\mathcal{M}$ and $T_q\mathcal{M}$, use $\verb|an_element()|$ of respective tangent space:

v = Tp.an_element()
print(v)

Tangent vector at Point p on the 3-dimensional differentiable manifold M

v.display()

\(\displaystyle \frac{\partial}{\partial x } + 2 \frac{\partial}{\partial y } + 3 \frac{\partial}{\partial z }\)

u = Tq.an_element()
print(u)

Tangent vector at Point q on the 3-dimensional differentiable manifold M

u.display()

\(\displaystyle \frac{\partial}{\partial x } + 2 \frac{\partial}{\partial y } + 3 \frac{\partial}{\partial z }\)

Both tangent vectors $\verb|v|$ and $\verb|u|$ come from different vector spaces, and thus, can't be composed together in arithmetic operations (i.e. - addition) that are originally defined for operations on vectors in the same vector space:

v.parent()

\(\displaystyle T_{p}\,\mathcal{M}\)

u.parent()

\(\displaystyle T_{q}\,\mathcal{M}\)

try:
    s = u + v
except TypeError as err:
    print(f'Error: {err}')

Error: unsupported operand parent(s) for +: 'Tangent space at Point q on the 3-dimensional differentiable manifold M' and 'Tangent space at Point p on the 3-dimensional differentiable manifold M'

Section 9. Vector Fields

The coordinate basis of a chart is accessed via $\verb|frame()|$ of a given chart:

X.frame()

\(\displaystyle \left(\mathcal{M}, \left(\frac{\partial}{\partial x },\frac{\partial}{\partial y },\frac{\partial}{\partial z }\right)\right)\)

Y.frame()

\(\displaystyle \left(U, \left(\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)\)

And set of coordinate basis of a given open subset is retrieved by $\verb|frames()|$:

M.frames()

\(\displaystyle \left[\left(\mathcal{M}, \left(\frac{\partial}{\partial x },\frac{\partial}{\partial y },\frac{\partial}{\partial z }\right)\right), \left(U, \left(\frac{\partial}{\partial x },\frac{\partial}{\partial y },\frac{\partial}{\partial z }\right)\right), \left(U, \left(\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)\right]\)

U.frames()

\(\displaystyle \left[\left(U, \left(\frac{\partial}{\partial x },\frac{\partial}{\partial y },\frac{\partial}{\partial z }\right)\right), \left(U, \left(\frac{\partial}{\partial r },\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)\right]\)

And the domain of a chart is accessed via $\verb|domain()|$:

X.frame().domain()

\(\displaystyle \mathcal{M}\)

Y.frame().domain()

\(\displaystyle U\)

Now, vector fields of a given open subset may be defined via $\verb|vector_field()|$ of a given open subset with respect to one of the given coordinate basis. If no coordinate basis is specified, it will be defined with respect to the default coordinate basis:

v = U.vector_field(1+y, -x, x*y*z, name='v')
v.display()

\(\displaystyle v = \left( y + 1 \right) \frac{\partial}{\partial x } -x \frac{\partial}{\partial y } + x y z \frac{\partial}{\partial z }\)

Once vector fields are instantiated, the parent tangent space object is accessed via $\verb|parent()|$ of vector field object:

v.parent()

\(\displaystyle \mathfrak{X}\left(U\right)\)

The set $\verb|𝔛(U)|$ above is a free module over the commutative algebra $C^{\infty}(U)$ of vector fields on $\verb|U|$:

print(v.parent())

Free module X(U) of vector fields on the Open subset U of the 3-dimensional differentiable manifold M

print(v.parent().category())

Category of finite dimensional modules over Algebra of differentiable scalar fields on the Open subset U of the 3-dimensional differentiable manifold M

v.parent().base_ring()

\(\displaystyle C^{\infty}\left(U\right)\)

Now lets use vector field $\verb|v|$ to act on the prescribed scalar field $\verb|f|$:

f.display()

\(\displaystyle \begin{array}{llcl} f:& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & z^{3} + y^{2} + x \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & r^{3} \cos\left({\theta}\right)^{3} + r^{2} \sin\left({\phi}\right)^{2} \sin\left({\theta}\right)^{2} + r \cos\left({\phi}\right) \sin\left({\theta}\right) \end{array}\)

s = v(f)
print(s)

Scalar field v(f) on the Open subset U of the 3-dimensional differentiable manifold M

s.display()

\(\displaystyle \begin{array}{llcl} v\left(f\right):& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & 3 \, x y z^{3} - {\left(2 \, x - 1\right)} y + 1 \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & r \sin\left({\phi}\right) \sin\left({\theta}\right) + {\left(3 \, r^{5} \cos\left({\phi}\right) \cos\left({\theta}\right)^{3} \sin\left({\phi}\right) - 2 \, r^{2} \cos\left({\phi}\right) \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{2} + 1 \end{array}\)

A change of coordinate basis for a vector field may be prescribed via passing a valid coordinate basis of interest:

v.display(Y.frame())

\(\displaystyle v = \left( \frac{x y z^{2} + x}{\sqrt{x^{2} + y^{2} + z^{2}}} \right) \frac{\partial}{\partial r } + \left( -\frac{{\left(x^{3} y + x y^{3} - x\right)} \sqrt{x^{2} + y^{2}} z}{x^{4} + 2 \, x^{2} y^{2} + y^{4} + {\left(x^{2} + y^{2}\right)} z^{2}} \right) \frac{\partial}{\partial {\theta} } + \left( -\frac{x^{2} + y^{2} + y}{x^{2} + y^{2}} \right) \frac{\partial}{\partial {\phi} }\)

To expand the coordinate terms as well, the chart should be passed to the second argument of $\verb|display()|$:

v.display(Y.frame(), Y)

\(\displaystyle v = \left( r^{3} \cos\left({\phi}\right) \cos\left({\theta}\right)^{2} \sin\left({\phi}\right) \sin\left({\theta}\right)^{2} + \cos\left({\phi}\right) \sin\left({\theta}\right) \right) \frac{\partial}{\partial r } + \left( -\frac{r^{3} \cos\left({\phi}\right) \cos\left({\theta}\right) \sin\left({\phi}\right) \sin\left({\theta}\right)^{3} - \cos\left({\phi}\right) \cos\left({\theta}\right)}{r} \right) \frac{\partial}{\partial {\theta} } + \left( -\frac{r \sin\left({\theta}\right) + \sin\left({\phi}\right)}{r \sin\left({\theta}\right)} \right) \frac{\partial}{\partial {\phi} }\)

Now the components of the vector (tensor) field may be accessed via $\verb|[:]|$ operator or the $\verb|display_comp()|$ function of the respective tensor field:

v[:]

\(\displaystyle \left[y + 1, -x, x y z\right]\)

v.display_comp()

\(\displaystyle \begin{array}{lcl} v_{\phantom{\, x}}^{ \, x } & = & y + 1 \\ v_{\phantom{\, y}}^{ \, y } & = & -x \\ v_{\phantom{\, z}}^{ \, z } & = & x y z \end{array}\)

And the expanded tensor field terms may be accessed with respect to a change of coordinate basis via $\verb|comp()|$ or $\verb|display_comp()|$ with the respective coordinate basis passed to the first argument:

v.comp(Y.frame())[:]

\(\displaystyle \left[\frac{x y z^{2} + x}{\sqrt{x^{2} + y^{2} + z^{2}}}, -\frac{{\left(x^{3} y + x y^{3} - x\right)} \sqrt{x^{2} + y^{2}} z}{x^{4} + 2 \, x^{2} y^{2} + y^{4} + {\left(x^{2} + y^{2}\right)} z^{2}}, -\frac{x^{2} + y^{2} + y}{x^{2} + y^{2}}\right]\)

v.display_comp(Y.frame())

\(\displaystyle \begin{array}{lcl} v_{\phantom{\, r}}^{ \, r } & = & \frac{x y z^{2} + x}{\sqrt{x^{2} + y^{2} + z^{2}}} \\ v_{\phantom{\, {\theta}}}^{ \, {\theta} } & = & -\frac{{\left(x^{3} y + x y^{3} - x\right)} \sqrt{x^{2} + y^{2}} z}{x^{4} + 2 \, x^{2} y^{2} + y^{4} + {\left(x^{2} + y^{2}\right)} z^{2}} \\ v_{\phantom{\, {\phi}}}^{ \, {\phi} } & = & -\frac{x^{2} + y^{2} + y}{x^{2} + y^{2}} \end{array}\)

For the expanded terms in terms of the new coordinate basis, pass the respective chart to the second argument:

v.display_comp(Y.frame(), Y)

\(\displaystyle \begin{array}{lcl} v_{\phantom{\, r}}^{ \, r } & = & r^{3} \cos\left({\phi}\right) \cos\left({\theta}\right)^{2} \sin\left({\phi}\right) \sin\left({\theta}\right)^{2} + \cos\left({\phi}\right) \sin\left({\theta}\right) \\ v_{\phantom{\, {\theta}}}^{ \, {\theta} } & = & -\frac{r^{3} \cos\left({\phi}\right) \cos\left({\theta}\right) \sin\left({\phi}\right) \sin\left({\theta}\right)^{3} - \cos\left({\phi}\right) \cos\left({\theta}\right)}{r} \\ v_{\phantom{\, {\phi}}}^{ \, {\phi} } & = & -\frac{r \sin\left({\theta}\right) + \sin\left({\phi}\right)}{r \sin\left({\theta}\right)} \end{array}\)

Now to specify a vector field with undefined function (scalar field) components, utilize SageMath symbolic $\verb|function|$ object to specify the undefined functions per term:

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

\(\displaystyle u = u_{x}\left(x, y, z\right) \frac{\partial}{\partial x } + u_{y}\left(x, y, z\right) \frac{\partial}{\partial y } + u_{z}\left(x, y, z\right) \frac{\partial}{\partial z }\)

Last, performing arithmetics like addition and subtraction on vector fields may be performed with Python`s arithmetic operators like $\verb|+|$ and $\verb|-|$:

s = u + v
s.set_name('s')
s.display()

\(\displaystyle s = \left( y + u_{x}\left(x, y, z\right) + 1 \right) \frac{\partial}{\partial x } + \left( -x + u_{y}\left(x, y, z\right) \right) \frac{\partial}{\partial y } + \left( x y z + u_{z}\left(x, y, z\right) \right) \frac{\partial}{\partial z }\)

Section 10. Vector field at a given point

Getting the vector field at a point $\verb|p|$ on a manifold $\mathcal{M}$ may be computed with the $\verb|at()|$ function of a vector field:

vp = v.at(p)
print(vp)

Tangent vector v at Point p on the 3-dimensional differentiable manifold M

vp.display()

\(\displaystyle v = 3 \frac{\partial}{\partial x } -\frac{\partial}{\partial y } -2 \frac{\partial}{\partial z }\)

The proper naming convention of a vector field at a point $\verb|p|$, $v|_p$, may be specified with $\verb|set_name()|$ of the vector field:

vp.set_name('v|_p')
vp.display()

\(\displaystyle v|_p = 3 \frac{\partial}{\partial x } -\frac{\partial}{\partial y } -2 \frac{\partial}{\partial z }\)

Lets get the vector field of $\verb|up|$ as well:

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

Tangent vector u at Point p on the 3-dimensional differentiable manifold M

up.display()

\(\displaystyle u = u_{x}\left(1, 2, -1\right) \frac{\partial}{\partial x } + u_{y}\left(1, 2, -1\right) \frac{\partial}{\partial y } + u_{z}\left(1, 2, -1\right) \frac{\partial}{\partial z }\)

Section 11. 1-forms

Differential 1-forms are differential operators of linear forms acting on a manifold $\mathcal{M}$ and may be specified with respect to a scalar field with $\verb|differential()|$ of a scalar field or $\verb|diff()|$ of SageMath, acting on a scalar field:

f.display()

\(\displaystyle \begin{array}{llcl} f:& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & z^{3} + y^{2} + x \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & r^{3} \cos\left({\theta}\right)^{3} + r^{2} \sin\left({\phi}\right)^{2} \sin\left({\theta}\right)^{2} + r \cos\left({\phi}\right) \sin\left({\theta}\right) \end{array}\)

df = f.differential()
print(df)

1-form df on the Open subset U of the 3-dimensional differentiable manifold M

df = diff(f)

df.display()

\(\displaystyle \mathrm{d}f = \mathrm{d} x + 2 \, y \mathrm{d} y + 3 \, z^{2} \mathrm{d} z\)

The differential 1-form basis above is $\verb|(dx, dy, dz)|$. This may be retrieved via $\verb|coframe()|$ of a respective chart of $\mathcal{M}$:

dX = X.coframe()
dX

\(\displaystyle \left(\mathcal{M}, \left(\mathrm{d} x,\mathrm{d} y,\mathrm{d} z\right)\right)\)

The combination of basis per open subset of a given manifold $\mathcal{M}$ may be retrieved via $\verb|coframes()|$ of the predefined manifold:

M.coframes()

\(\displaystyle \left[\left(\mathcal{M}, \left(\mathrm{d} x,\mathrm{d} y,\mathrm{d} z\right)\right), \left(U, \left(\mathrm{d} x,\mathrm{d} y,\mathrm{d} z\right)\right), \left(U, \left(\mathrm{d} r,\mathrm{d} {\theta},\mathrm{d} {\phi}\right)\right)\right]\)

As done for the vector field at a point $\verb|p|$, the differential 1-form acting on a point $\verb|p|$ in $\mathcal{M}$ may be computed via $\verb|at()|$ of the respective differential 1-form:

p.coord()

\(\displaystyle \left(1, 2, -1\right)\)

dfp = df.at(p)
print(dfp)

Linear form df on the Tangent space at Point p on the 3-dimensional differentiable manifold M

dfp.display()

\(\displaystyle \mathrm{d}f = \mathrm{d} x + 4 \mathrm{d} y + 3 \mathrm{d} z\)

$df|_p$ dual vector space is accessed via $\verb|parent()|$ of $\verb|dfp|$:

dfp.parent()

\(\displaystyle T_{p}\,\mathcal{M}^*\)

M.tangent_space(p).dual()

\(\displaystyle T_{p}\,\mathcal{M}^*\)

When $df|_p$ acts on a vector that acts on $\verb|p|$, it outputs a real number, $\mathbb{R}$:

print(vp)
vp.display()

Tangent vector v|_p at Point p on the 3-dimensional differentiable manifold M

\(\displaystyle v|_p = 3 \frac{\partial}{\partial x } -\frac{\partial}{\partial y } -2 \frac{\partial}{\partial z }\)

dfp(vp)

\(\displaystyle -7\)

print(up)
up.display()

Tangent vector u at Point p on the 3-dimensional differentiable manifold M

\(\displaystyle u = u_{x}\left(1, 2, -1\right) \frac{\partial}{\partial x } + u_{y}\left(1, 2, -1\right) \frac{\partial}{\partial y } + u_{z}\left(1, 2, -1\right) \frac{\partial}{\partial z }\)

dfp(up)

\(\displaystyle u_{x}\left(1, 2, -1\right) + 4 \, u_{y}\left(1, 2, -1\right) + 3 \, u_{z}\left(1, 2, -1\right)\)

On the other hand, we may simply define a 1-form with $\verb|one_form()|$ of a given open subset of $\mathcal{M}$, with $\verb|name|$ passed to $\verb|one_form()|$:

om = U.one_form(name='omega', latex_name=r'\omega')
print(om)

1-form omega on the Open subset U of the 3-dimensional differentiable manifold M

The components may be specified via the $\verb|[:]|$ operator, with respect to the default basis:

om[:] = [x^2 + y^2, z, x-z]
om.display()

\(\displaystyle \omega = \left( x^{2} + y^{2} \right) \mathrm{d} x + z \mathrm{d} y + \left( x - z \right) \mathrm{d} z\)

Components may also be specified with respect to a different basis within a given chart of the respective open subset:

om[Y.frame(), :, Y] = [r*sin(th)*cos(ph), 0, r*sin(th)*sin(ph)]
om.display(Y.frame(), Y)

\(\displaystyle \omega = r \cos\left({\phi}\right) \sin\left({\theta}\right) \mathrm{d} r + r \sin\left({\phi}\right) \sin\left({\theta}\right) \mathrm{d} {\phi}\)

Following that, components in the default basis of the open subset are updated accordingly:

om.display()

\(\displaystyle \omega = \left( \frac{x^{4} + x^{2} y^{2} - \sqrt{x^{2} + y^{2} + z^{2}} y^{2}}{\sqrt{x^{2} + y^{2} + z^{2}} {\left(x^{2} + y^{2}\right)}} \right) \mathrm{d} x + \left( \frac{x^{3} y + x y^{3} + \sqrt{x^{2} + y^{2} + z^{2}} x y}{\sqrt{x^{2} + y^{2} + z^{2}} {\left(x^{2} + y^{2}\right)}} \right) \mathrm{d} y + \left( \frac{x z}{\sqrt{x^{2} + y^{2} + z^{2}}} \right) \mathrm{d} z\)

Conversely, one may simply update components from the default basis components to the change of basis coordinates:

om[:] = [x^2 + y^2, z, x-z]
om.display()

\(\displaystyle \omega = \left( x^{2} + y^{2} \right) \mathrm{d} x + z \mathrm{d} y + \left( x - z \right) \mathrm{d} z\)

om.display(Y.frame(), Y)

\(\displaystyle \omega = \left( r^{2} \cos\left({\phi}\right) \sin\left({\theta}\right)^{3} + r {\left(\cos\left({\phi}\right) + \sin\left({\phi}\right)\right)} \cos\left({\theta}\right) \sin\left({\theta}\right) - r \cos\left({\theta}\right)^{2} \right) \mathrm{d} r + \left( r^{2} \cos\left({\theta}\right)^{2} \sin\left({\phi}\right) + r^{2} \cos\left({\theta}\right) \sin\left({\theta}\right) + {\left(r^{3} \cos\left({\phi}\right) \cos\left({\theta}\right) - r^{2} \cos\left({\phi}\right)\right)} \sin\left({\theta}\right)^{2} \right) \mathrm{d} {\theta} + \left( -r^{3} \sin\left({\phi}\right) \sin\left({\theta}\right)^{3} + r^{2} \cos\left({\phi}\right) \cos\left({\theta}\right) \sin\left({\theta}\right) \right) \mathrm{d} {\phi}\)

Lets act on vector $\verb|v|$ and show that it produces a scalar field:

v.display()

\(\displaystyle v = \left( y + 1 \right) \frac{\partial}{\partial x } -x \frac{\partial}{\partial y } + x y z \frac{\partial}{\partial z }\)

print(om(v))
om(v).display()

Scalar field omega(v) on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \begin{array}{llcl} \omega\left(v\right):& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & -x y z^{2} + x^{2} y + y^{3} + x^{2} + y^{2} + {\left(x^{2} y - x\right)} z \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & -r^{2} \cos\left({\phi}\right) \cos\left({\theta}\right) \sin\left({\theta}\right) + {\left(r^{4} \cos\left({\phi}\right)^{2} \cos\left({\theta}\right) \sin\left({\phi}\right) + r^{3} \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{3} - {\left(r^{4} \cos\left({\phi}\right) \cos\left({\theta}\right)^{2} \sin\left({\phi}\right) - r^{2}\right)} \sin\left({\theta}\right)^{2} \end{array}\)

Perform the same operation with differential 1-form $\verb|df|$ acting on $\verb|v|$:

print(df(v))
df(v).display()

Scalar field df(v) on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \begin{array}{llcl} \mathrm{d}f\left(v\right):& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & 3 \, x y z^{3} - {\left(2 \, x - 1\right)} y + 1 \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & r \sin\left({\phi}\right) \sin\left({\theta}\right) + {\left(3 \, r^{5} \cos\left({\phi}\right) \cos\left({\theta}\right)^{3} \sin\left({\phi}\right) - 2 \, r^{2} \cos\left({\phi}\right) \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{2} + 1 \end{array}\)

Note that a differential 1-form acting on a vector field (i.e. - $\verb|df(v)|$) is equivalent to a vector field acting on the respective scalar field of the differential 1-form (i.e. - $\verb|v(f)|$):

v(f).display()

\(\displaystyle \begin{array}{llcl} v\left(f\right):& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & 3 \, x y z^{3} - {\left(2 \, x - 1\right)} y + 1 \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & r \sin\left({\phi}\right) \sin\left({\theta}\right) + {\left(3 \, r^{5} \cos\left({\phi}\right) \cos\left({\theta}\right)^{3} \sin\left({\phi}\right) - 2 \, r^{2} \cos\left({\phi}\right) \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{2} + 1 \end{array}\)

df(v) == v(f)

\(\displaystyle \mathrm{True}\)

Now, a 1-form is an element of $\Omega^1(U)$ of all 1-forms defined on an open subset $U$:

df.parent()

\(\displaystyle \Omega^{1}\left(U\right)\)

om.parent()

\(\displaystyle \Omega^{1}\left(U\right)\)

print(df.parent())

Free module Omega^1(U) of 1-forms on the Open subset U of the 3-dimensional differentiable manifold M

df.parent() is v.parent().dual()

\(\displaystyle \mathrm{True}\)

Section 12. Exterior calculus of differential forms

Exterior products of two 1-forms are performed with $\verb|wedge()|$ function of a given 1-form on another 1-form to produce a 2-form:

a = om.wedge(df)
print(a)
a.display()

2-form omega∧df on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \omega\wedge \mathrm{d}f = \left( 2 \, x^{2} y + 2 \, y^{3} - z \right) \mathrm{d} x\wedge \mathrm{d} y + \left( 3 \, {\left(x^{2} + y^{2}\right)} z^{2} - x + z \right) \mathrm{d} x\wedge \mathrm{d} z + \left( 3 \, z^{3} - 2 \, x y + 2 \, y z \right) \mathrm{d} y\wedge \mathrm{d} z\)

A matrix view of the components of a 2-form is retrieved via $\verb|[:]|$ operator:

a[:]

\(\displaystyle \left(\begin{array}{rrr} 0 & 2 \, x^{2} y + 2 \, y^{3} - z & 3 \, {\left(x^{2} + y^{2}\right)} z^{2} - x + z \\ -2 \, x^{2} y - 2 \, y^{3} + z & 0 & 3 \, z^{3} - 2 \, x y + 2 \, y z \\ -3 \, {\left(x^{2} + y^{2}\right)} z^{2} + x - z & -3 \, z^{3} + 2 \, x y - 2 \, y z & 0 \end{array}\right)\)

Non-zero components may be retrieved via $\verb|display_comp()|$ method of the respective 2-form. Redundant term (symmetric and anti-symmetric terms) are skipped via the $\verb|only_nonredundant=True|$ params of $\verb|display_comp()|$:

a.display_comp(only_nonredundant=True)

\(\displaystyle \begin{array}{lcl} \omega\wedge \mathrm{d}f_{ \, x \, y }^{ \phantom{\, x}\phantom{\, y} } & = & 2 \, x^{2} y + 2 \, y^{3} - z \\ \omega\wedge \mathrm{d}f_{ \, x \, z }^{ \phantom{\, x}\phantom{\, z} } & = & 3 \, {\left(x^{2} + y^{2}\right)} z^{2} - x + z \\ \omega\wedge \mathrm{d}f_{ \, y \, z }^{ \phantom{\, y}\phantom{\, z} } & = & 3 \, z^{3} - 2 \, x y + 2 \, y z \end{array}\)

As done for vector fields and 1-forms, 2-forms can be expanded according to a change of basis (i.e. - $(dr, d\theta, d\phi)$):

a.display(Y.frame(), Y)

\(\displaystyle \omega\wedge \mathrm{d}f = \left( 3 \, r^{5} \cos\left({\phi}\right) \sin\left({\theta}\right)^{4} - {\left(3 \, r^{5} \cos\left({\phi}\right) - 3 \, r^{4} \cos\left({\theta}\right) \sin\left({\phi}\right) - 2 \, r^{3} \cos\left({\phi}\right) \sin\left({\phi}\right)^{2}\right)} \sin\left({\theta}\right)^{2} - {\left(3 \, r^{4} \sin\left({\phi}\right) + r^{2} \cos\left({\phi}\right)\right)} \cos\left({\theta}\right) - {\left(2 \, r^{3} \cos\left({\theta}\right) \sin\left({\phi}\right)^{2} + {\left(\sin\left({\phi}\right)^{2} - 1\right)} r^{2}\right)} \sin\left({\theta}\right) \right) \mathrm{d} r\wedge \mathrm{d} {\theta} + \left( 2 \, r^{4} \sin\left({\phi}\right) \sin\left({\theta}\right)^{5} + {\left(3 \, r^{5} \cos\left({\theta}\right)^{3} \sin\left({\phi}\right) + 2 \, r^{3} \cos\left({\phi}\right)^{2} \cos\left({\theta}\right) \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{3} - {\left(2 \, r^{3} \cos\left({\phi}\right) \cos\left({\theta}\right)^{2} \sin\left({\phi}\right) + {\left(\cos\left({\phi}\right) \sin\left({\phi}\right) + 1\right)} r^{2} \cos\left({\theta}\right)\right)} \sin\left({\theta}\right)^{2} - {\left(3 \, r^{4} \cos\left({\phi}\right) \cos\left({\theta}\right)^{4} - r^{2} \cos\left({\theta}\right)^{2} \sin\left({\phi}\right)\right)} \sin\left({\theta}\right) \right) \mathrm{d} r\wedge \mathrm{d} {\phi} + \left( -r^{3} \cos\left({\theta}\right)^{2} \sin\left({\theta}\right) - {\left(3 \, r^{6} \cos\left({\theta}\right)^{2} \sin\left({\phi}\right) + 2 \, r^{4} \cos\left({\phi}\right)^{2} \sin\left({\phi}\right) - 2 \, r^{5} \cos\left({\theta}\right) \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{4} + {\left(2 \, r^{4} \cos\left({\phi}\right) \cos\left({\theta}\right) \sin\left({\phi}\right) + r^{3} \cos\left({\phi}\right) \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{3} + {\left(3 \, r^{5} \cos\left({\phi}\right) \cos\left({\theta}\right)^{3} - r^{3} \cos\left({\theta}\right) \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{2} \right) \mathrm{d} {\theta}\wedge \mathrm{d} {\phi}\)

A 2-form can act on two vector fields to produce a scalar field, as shown below:

a.set_name('A')
print(a(u,v))
a(u,v).display()

Scalar field A(u,v) on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \begin{array}{llcl} A\left(u,v\right):& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & 3 \, x y z^{4} u_{y}\left(x, y, z\right) - 2 \, x^{2} y^{2} u_{y}\left(x, y, z\right) - 2 \, y^{4} u_{y}\left(x, y, z\right) - 2 \, {\left(x u_{x}\left(x, y, z\right) + u_{y}\left(x, y, z\right)\right)} y^{3} + 3 \, {\left(x^{3} y u_{x}\left(x, y, z\right) + x y^{3} u_{x}\left(x, y, z\right) + x u_{z}\left(x, y, z\right)\right)} z^{3} - {\left(3 \, y^{3} u_{z}\left(x, y, z\right) - {\left(2 \, x u_{y}\left(x, y, z\right) - 3 \, u_{z}\left(x, y, z\right)\right)} y^{2} + 3 \, x^{2} u_{z}\left(x, y, z\right) + {\left(3 \, x^{2} u_{z}\left(x, y, z\right) - x u_{x}\left(x, y, z\right)\right)} y\right)} z^{2} - {\left(2 \, x^{3} u_{x}\left(x, y, z\right) + 2 \, x^{2} u_{y}\left(x, y, z\right) + {\left(2 \, x^{2} - x\right)} u_{z}\left(x, y, z\right)\right)} y - {\left(2 \, x^{2} y^{2} u_{y}\left(x, y, z\right) + {\left(x^{2} u_{x}\left(x, y, z\right) - {\left(2 \, x - 1\right)} u_{z}\left(x, y, z\right) - u_{y}\left(x, y, z\right)\right)} y - x u_{x}\left(x, y, z\right) - u_{y}\left(x, y, z\right) + u_{z}\left(x, y, z\right)\right)} z + x u_{z}\left(x, y, z\right) \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & {\left(r^{4} \cos\left({\phi}\right) \cos\left({\theta}\right)^{2} \sin\left({\phi}\right) \sin\left({\theta}\right)^{2} + {\left(\sin\left({\phi}\right)^{3} - \sin\left({\phi}\right)\right)} r^{4} \cos\left({\theta}\right) \sin\left({\theta}\right)^{3} + r^{2} \cos\left({\phi}\right) \cos\left({\theta}\right) \sin\left({\theta}\right) + {\left(3 \, r^{7} \cos\left({\phi}\right) \cos\left({\theta}\right)^{3} \sin\left({\phi}\right) - 2 \, r^{4} \cos\left({\phi}\right) \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{4}\right)} u_{x}\left(r \cos\left({\phi}\right) \sin\left({\theta}\right), r \sin\left({\phi}\right) \sin\left({\theta}\right), r \cos\left({\theta}\right)\right) + {\left(3 \, r^{6} \cos\left({\phi}\right) \cos\left({\theta}\right)^{4} \sin\left({\phi}\right) \sin\left({\theta}\right)^{2} + r^{2} \cos\left({\theta}\right) \sin\left({\phi}\right) \sin\left({\theta}\right) + 2 \, {\left({\left(\sin\left({\phi}\right)^{4} - \sin\left({\phi}\right)^{2}\right)} r^{5} \cos\left({\theta}\right) - r^{4} \sin\left({\phi}\right)^{2}\right)} \sin\left({\theta}\right)^{4} + 2 \, {\left(r^{5} \cos\left({\phi}\right) \cos\left({\theta}\right)^{2} \sin\left({\phi}\right)^{2} - r^{3} \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{3} + r \cos\left({\theta}\right)\right)} u_{y}\left(r \cos\left({\phi}\right) \sin\left({\theta}\right), r \sin\left({\phi}\right) \sin\left({\theta}\right), r \cos\left({\theta}\right)\right) - {\left({\left(3 \, r^{5} \cos\left({\theta}\right)^{2} \sin\left({\phi}\right) - 2 \, {\left(\sin\left({\phi}\right)^{3} - \sin\left({\phi}\right)\right)} r^{3}\right)} \sin\left({\theta}\right)^{3} + {\left(3 \, r^{4} \cos\left({\theta}\right)^{2} - 2 \, r^{3} \cos\left({\phi}\right) \cos\left({\theta}\right) \sin\left({\phi}\right) - r^{2} \cos\left({\phi}\right) \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{2} + r \cos\left({\theta}\right) - {\left(3 \, r^{4} \cos\left({\phi}\right) \cos\left({\theta}\right)^{3} - r^{2} \cos\left({\theta}\right) \sin\left({\phi}\right) + r \cos\left({\phi}\right)\right)} \sin\left({\theta}\right)\right)} u_{z}\left(r \cos\left({\phi}\right) \sin\left({\theta}\right), r \sin\left({\phi}\right) \sin\left({\theta}\right), r \cos\left({\theta}\right)\right) \end{array}\)

The scalar field produced is antisymmetric:

a(u,v) == -a(v,u)

\(\displaystyle \mathrm{True}\)

a.symmetries()

no symmetry; antisymmetry: (0, 1)

Now, the exterior derivative of a differential 1-form is computed using $\verb|exterior_derivative()|$ method of the repective differential form or through having SageMath $\verb|diff()|$ method act on the differential form of interest:

dom = om.exterior_derivative()
print(dom)
dom.display()

2-form domega on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \mathrm{d}\omega = -2 \, y \mathrm{d} x\wedge \mathrm{d} y +\mathrm{d} x\wedge \mathrm{d} z -\mathrm{d} y\wedge \mathrm{d} z\)

dom = diff(om)
print(dom)
dom.display()

2-form domega on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \mathrm{d}\omega = -2 \, y \mathrm{d} x\wedge \mathrm{d} y +\mathrm{d} x\wedge \mathrm{d} z -\mathrm{d} y\wedge \mathrm{d} z\)

The same exterior derivative operation may be performed on a 2-form or any higher order form:

da = a.exterior_derivative()
print(da)
da.display()

3-form dA on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \mathrm{d}A = \left( -6 \, y z^{2} - 2 \, y - 1 \right) \mathrm{d} x\wedge \mathrm{d} y\wedge \mathrm{d} z\)

And one of the known properties of exterior derivatives is satisfied in SageMath; they are nilpotent:

ddf = diff(df)
ddf.display()

\(\displaystyle \mathrm{d}\mathrm{d}f = 0\)

ddom = dom.exterior_derivative()
ddom.display()

\(\displaystyle \mathrm{d}\mathrm{d}\omega = 0\)

Section 13. Lie derivative

Now, the Lie derivative of a tensor field with respect to a vector field is computed via $\verb|lie_derivative()|$ of the respective tensor field, with the vector field passed as an arg to the method:

lv_om = om.lie_derivative(v)
print(lv_om)
lv_om.display()

1-form on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \left( -y z^{2} + {\left(x y - 1\right)} z + 2 \, x \right) \mathrm{d} x + \left( -x z^{2} + x^{2} + y^{2} + {\left(x^{2} + x y\right)} z \right) \mathrm{d} y + \left( -2 \, x y z + {\left(x^{2} + 1\right)} y + 1 \right) \mathrm{d} z\)

lu_om = om.lie_derivative(u)
print(lu_om)
lu_om.display()

1-form on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \left( x^{2} \frac{\partial\,u_{x}}{\partial x} + y^{2} \frac{\partial\,u_{x}}{\partial x} + z {\left(\frac{\partial\,u_{y}}{\partial x} - \frac{\partial\,u_{z}}{\partial x}\right)} + 2 \, x u_{x}\left(x, y, z\right) + 2 \, y u_{y}\left(x, y, z\right) + x \frac{\partial\,u_{z}}{\partial x} \right) \mathrm{d} x + \left( x^{2} \frac{\partial\,u_{x}}{\partial y} + y^{2} \frac{\partial\,u_{x}}{\partial y} + z {\left(\frac{\partial\,u_{y}}{\partial y} - \frac{\partial\,u_{z}}{\partial y}\right)} + x \frac{\partial\,u_{z}}{\partial y} + u_{z}\left(x, y, z\right) \right) \mathrm{d} y + \left( x^{2} \frac{\partial\,u_{x}}{\partial z} + y^{2} \frac{\partial\,u_{x}}{\partial z} + z {\left(\frac{\partial\,u_{y}}{\partial z} - \frac{\partial\,u_{z}}{\partial z}\right)} + x \frac{\partial\,u_{z}}{\partial z} + u_{x}\left(x, y, z\right) - u_{z}\left(x, y, z\right) \right) \mathrm{d} z\)

We'll now verify the Cartan identities on the one form $\omega$, where:

$\mathcal{L}_v\omega$ = $v \cdot d\omega + d\langle\omega, v\rangle$

and for the 2-form $\verb|a|$, created by the exterior product of two 1-forms:

$\mathcal{L}_va$ = $v \cdot da + d(v \cdot a)$

lv_om == v.contract(dom) + diff(om(v))

\(\displaystyle \mathrm{True}\)

a.lie_derivative(v) == v.contract(diff(a)) + diff(v.contract(a))

\(\displaystyle \mathrm{True}\)

Last, the Lie derivative of vector field, along another vector field, is the commutator of the two vector fields:

v.lie_derivative(u)(f) == u(v(f)) - v(u(f))

\(\displaystyle \mathrm{True}\)

Section 14. Tensor fields of arbitrary rank

Tensor fields of any rank, $(p, q)$, may be prescribed via $\verb|tensor_field()|$ of a respective open subset of a given manifold $\mathcal{M}$, with $p$ and $q$ passed to $\verb|tensor_field()|$:

t = U.tensor_field(1, 2, name='T')
print(t)

Tensor field T of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M

Tensor components may be set via their respective indices:

t[1,2,1] = 1 + x^2
t[3,2,1] = x*y*z

t.display()

\(\displaystyle T = \left( x^{2} + 1 \right) \frac{\partial}{\partial x }\otimes \mathrm{d} y\otimes \mathrm{d} x + x y z \frac{\partial}{\partial z }\otimes \mathrm{d} y\otimes \mathrm{d} x\)

As done for vector fields and 1-forms, all tensor field components may accessed via $\verb|[:]|$ operator:

t[:]

\(\displaystyle \left[\left[\left[0, 0, 0\right], \left[x^{2} + 1, 0, 0\right], \left[0, 0, 0\right]\right], \left[\left[0, 0, 0\right], \left[0, 0, 0\right], \left[0, 0, 0\right]\right], \left[\left[0, 0, 0\right], \left[x y z, 0, 0\right], \left[0, 0, 0\right]\right]\right]\)

To just access the non-zero components of the respective tensor field, use $\verb|display_comp()|$ method of the tensor field:

t.display_comp()

\(\displaystyle \begin{array}{lcl} T_{ \phantom{\, x} \, y \, x }^{ \, x \phantom{\, y} \phantom{\, x} } & = & x^{2} + 1 \\ T_{ \phantom{\, z} \, y \, x }^{ \, z \phantom{\, y} \phantom{\, x} } & = & x y z \end{array}\)

The components of tensors are scalar fields and using double bracket notation on the indices displays them as such:

print(t[[1,2,1]])
t[[1,2,1]].display()

Scalar field on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \begin{array}{llcl} & U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & x^{2} + 1 \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & r^{2} \cos\left({\phi}\right)^{2} \sin\left({\theta}\right)^{2} + 1 \end{array}\)

Now, the tensor $\verb|t|$ of rank (1, 2) may act on one 1-form and two vector fields to output a scalar field:

print(t(om, u, v))
t(om, u, v).display()

Scalar field T(omega,u,v) on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \begin{array}{llcl} T\left(\omega,u,v\right):& U & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & {\left(x^{2} + 1\right)} y^{3} u_{y}\left(x, y, z\right) + {\left(x^{2} + 1\right)} y^{2} u_{y}\left(x, y, z\right) - {\left(x y^{2} u_{y}\left(x, y, z\right) + x y u_{y}\left(x, y, z\right)\right)} z^{2} + {\left(x^{4} + x^{2}\right)} y u_{y}\left(x, y, z\right) + {\left(x^{2} y^{2} u_{y}\left(x, y, z\right) + x^{2} y u_{y}\left(x, y, z\right)\right)} z + {\left(x^{4} + x^{2}\right)} u_{y}\left(x, y, z\right) \\ & \left(r, {\theta}, {\phi}\right) & \longmapsto & {\left(r^{5} \cos\left({\phi}\right)^{2} \sin\left({\phi}\right) \sin\left({\theta}\right)^{5} - {\left({\left(\cos\left({\phi}\right)^{4} - \cos\left({\phi}\right)^{2}\right)} r^{5} \cos\left({\theta}\right) - r^{4} \cos\left({\phi}\right)^{2}\right)} \sin\left({\theta}\right)^{4} + {\left({\left(\cos\left({\phi}\right)^{3} - \cos\left({\phi}\right)\right)} r^{5} \cos\left({\theta}\right)^{2} + r^{4} \cos\left({\phi}\right)^{2} \cos\left({\theta}\right) \sin\left({\phi}\right) + r^{3} \sin\left({\phi}\right)\right)} \sin\left({\theta}\right)^{3} - {\left(r^{4} \cos\left({\phi}\right) \cos\left({\theta}\right)^{2} \sin\left({\phi}\right) - r^{2}\right)} \sin\left({\theta}\right)^{2}\right)} u_{y}\left(r \cos\left({\phi}\right) \sin\left({\theta}\right), r \sin\left({\phi}\right) \sin\left({\theta}\right), r \cos\left({\theta}\right)\right) \end{array}\)

Tensors also allow for change of coordinates with respect to the coordinate basis of a given open subset chart:

t[Y.frame(), 1,1,1, Y]

\(\displaystyle r^{2} \cos\left({\phi}\right)^{4} \sin\left({\phi}\right) \sin\left({\theta}\right)^{5} + {\left(\cos\left({\phi}\right)^{4} - \cos\left({\phi}\right)^{2}\right)} r^{3} \sin\left({\theta}\right)^{6} - {\left(\cos\left({\phi}\right)^{4} - \cos\left({\phi}\right)^{2}\right)} r^{3} \sin\left({\theta}\right)^{4} + \cos\left({\phi}\right)^{2} \sin\left({\phi}\right) \sin\left({\theta}\right)^{3}\)

Section 15. Tensor calculus

Tensor types may be retrived via $\verb|tensor_type()|$ method of a given tensor field:

print(v.tensor_type())
print(a.tensor_type())

(1, 0)
(0, 2)

Tensor products, $\otimes$, may be performed using the $\verb|*|$ arithmetic operator:

b = v*a
print(b)
b

Tensor field v⊗A of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle v\otimes A\)

b.display()

\(\displaystyle v\otimes A = \left( 2 \, x^{2} y^{2} + 2 \, y^{4} + 2 \, x^{2} y + 2 \, y^{3} - {\left(y + 1\right)} z \right) \frac{\partial}{\partial x }\otimes \mathrm{d} x\otimes \mathrm{d} y + \left( 3 \, {\left(x^{2} y + y^{3} + x^{2} + y^{2}\right)} z^{2} - x y + {\left(y + 1\right)} z - x \right) \frac{\partial}{\partial x }\otimes \mathrm{d} x\otimes \mathrm{d} z + \left( -2 \, x^{2} y^{2} - 2 \, y^{4} - 2 \, x^{2} y - 2 \, y^{3} + {\left(y + 1\right)} z \right) \frac{\partial}{\partial x }\otimes \mathrm{d} y\otimes \mathrm{d} x + \left( 3 \, {\left(y + 1\right)} z^{3} - 2 \, x y^{2} - 2 \, x y + 2 \, {\left(y^{2} + y\right)} z \right) \frac{\partial}{\partial x }\otimes \mathrm{d} y\otimes \mathrm{d} z + \left( -3 \, {\left(x^{2} y + y^{3} + x^{2} + y^{2}\right)} z^{2} + x y - {\left(y + 1\right)} z + x \right) \frac{\partial}{\partial x }\otimes \mathrm{d} z\otimes \mathrm{d} x + \left( -3 \, {\left(y + 1\right)} z^{3} + 2 \, x y^{2} + 2 \, x y - 2 \, {\left(y^{2} + y\right)} z \right) \frac{\partial}{\partial x }\otimes \mathrm{d} z\otimes \mathrm{d} y + \left( -2 \, x^{3} y - 2 \, x y^{3} + x z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} x\otimes \mathrm{d} y + \left( -3 \, {\left(x^{3} + x y^{2}\right)} z^{2} + x^{2} - x z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} x\otimes \mathrm{d} z + \left( 2 \, x^{3} y + 2 \, x y^{3} - x z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} y\otimes \mathrm{d} x + \left( -3 \, x z^{3} + 2 \, x^{2} y - 2 \, x y z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} y\otimes \mathrm{d} z + \left( 3 \, {\left(x^{3} + x y^{2}\right)} z^{2} - x^{2} + x z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} z\otimes \mathrm{d} x + \left( 3 \, x z^{3} - 2 \, x^{2} y + 2 \, x y z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} z\otimes \mathrm{d} y + \left( -x y z^{2} + 2 \, {\left(x^{3} y^{2} + x y^{4}\right)} z \right) \frac{\partial}{\partial z }\otimes \mathrm{d} x\otimes \mathrm{d} y + \left( -x^{2} y z + x y z^{2} + 3 \, {\left(x^{3} y + x y^{3}\right)} z^{3} \right) \frac{\partial}{\partial z }\otimes \mathrm{d} x\otimes \mathrm{d} z + \left( x y z^{2} - 2 \, {\left(x^{3} y^{2} + x y^{4}\right)} z \right) \frac{\partial}{\partial z }\otimes \mathrm{d} y\otimes \mathrm{d} x + \left( 3 \, x y z^{4} - 2 \, x^{2} y^{2} z + 2 \, x y^{2} z^{2} \right) \frac{\partial}{\partial z }\otimes \mathrm{d} y\otimes \mathrm{d} z + \left( x^{2} y z - x y z^{2} - 3 \, {\left(x^{3} y + x y^{3}\right)} z^{3} \right) \frac{\partial}{\partial z }\otimes \mathrm{d} z\otimes \mathrm{d} x + \left( -3 \, x y z^{4} + 2 \, x^{2} y^{2} z - 2 \, x y^{2} z^{2} \right) \frac{\partial}{\partial z }\otimes \mathrm{d} z\otimes \mathrm{d} y\)

Tensor products preserve the symmetry of the tensors. A 2- or n-form is antisymmetric, hence, the tensor product of an n-form with any other tensor is antisymmetric:

a.symmetries()

no symmetry; antisymmetry: (0, 1)

b.symmetries()

no symmetry; antisymmetry: (1, 2)

Furthermore, arithmetic may be performed on tensor products using Python's arithmetic operators:

s = - t + 2*f*b
print(s)
s.display()

Tensor field of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \left( 4 \, y^{6} + 4 \, x^{3} y^{2} + 4 \, {\left(x^{2} + x\right)} y^{4} + 4 \, y^{5} - 2 \, {\left(y + 1\right)} z^{4} + 4 \, x^{3} y + 4 \, {\left(x^{2} + x\right)} y^{3} + 4 \, {\left(x^{2} y^{2} + y^{4} + x^{2} y + y^{3}\right)} z^{3} - 2 \, {\left(y^{3} + x y + y^{2} + x\right)} z \right) \frac{\partial}{\partial x }\otimes \mathrm{d} x\otimes \mathrm{d} y + \left( 6 \, {\left(x^{2} y + y^{3} + x^{2} + y^{2}\right)} z^{5} + 2 \, {\left(y + 1\right)} z^{4} - 2 \, x y^{3} - 2 \, {\left(x y + x\right)} z^{3} - 2 \, x^{2} y - 2 \, x y^{2} + 6 \, {\left(y^{5} + x^{3} y + {\left(x^{2} + x\right)} y^{3} + y^{4} + x^{3} + {\left(x^{2} + x\right)} y^{2}\right)} z^{2} - 2 \, x^{2} + 2 \, {\left(y^{3} + x y + y^{2} + x\right)} z \right) \frac{\partial}{\partial x }\otimes \mathrm{d} x\otimes \mathrm{d} z + \left( -4 \, y^{6} - 4 \, x^{3} y^{2} - 4 \, {\left(x^{2} + x\right)} y^{4} - 4 \, y^{5} + 2 \, {\left(y + 1\right)} z^{4} - 4 \, x^{3} y - 4 \, {\left(x^{2} + x\right)} y^{3} - 4 \, {\left(x^{2} y^{2} + y^{4} + x^{2} y + y^{3}\right)} z^{3} - x^{2} + 2 \, {\left(y^{3} + x y + y^{2} + x\right)} z - 1 \right) \frac{\partial}{\partial x }\otimes \mathrm{d} y\otimes \mathrm{d} x + \left( 6 \, {\left(y + 1\right)} z^{6} - 4 \, x y^{4} + 4 \, {\left(y^{2} + y\right)} z^{4} - 4 \, x^{2} y^{2} - 4 \, x y^{3} - 2 \, {\left({\left(2 \, x - 3\right)} y^{2} - 3 \, y^{3} - x y - 3 \, x\right)} z^{3} - 4 \, x^{2} y + 4 \, {\left(y^{4} + x y^{2} + y^{3} + x y\right)} z \right) \frac{\partial}{\partial x }\otimes \mathrm{d} y\otimes \mathrm{d} z + \left( -6 \, {\left(x^{2} y + y^{3} + x^{2} + y^{2}\right)} z^{5} - 2 \, {\left(y + 1\right)} z^{4} + 2 \, x y^{3} + 2 \, {\left(x y + x\right)} z^{3} + 2 \, x^{2} y + 2 \, x y^{2} - 6 \, {\left(y^{5} + x^{3} y + {\left(x^{2} + x\right)} y^{3} + y^{4} + x^{3} + {\left(x^{2} + x\right)} y^{2}\right)} z^{2} + 2 \, x^{2} - 2 \, {\left(y^{3} + x y + y^{2} + x\right)} z \right) \frac{\partial}{\partial x }\otimes \mathrm{d} z\otimes \mathrm{d} x + \left( -6 \, {\left(y + 1\right)} z^{6} + 4 \, x y^{4} - 4 \, {\left(y^{2} + y\right)} z^{4} + 4 \, x^{2} y^{2} + 4 \, x y^{3} + 2 \, {\left({\left(2 \, x - 3\right)} y^{2} - 3 \, y^{3} - x y - 3 \, x\right)} z^{3} + 4 \, x^{2} y - 4 \, {\left(y^{4} + x y^{2} + y^{3} + x y\right)} z \right) \frac{\partial}{\partial x }\otimes \mathrm{d} z\otimes \mathrm{d} y + \left( -4 \, x y^{5} - 4 \, x^{4} y + 2 \, x z^{4} - 4 \, {\left(x^{3} + x^{2}\right)} y^{3} - 4 \, {\left(x^{3} y + x y^{3}\right)} z^{3} + 2 \, {\left(x y^{2} + x^{2}\right)} z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} x\otimes \mathrm{d} y + \left( -6 \, {\left(x^{3} + x y^{2}\right)} z^{5} + 2 \, x^{2} z^{3} - 2 \, x z^{4} + 2 \, x^{2} y^{2} + 2 \, x^{3} - 6 \, {\left(x y^{4} + x^{4} + {\left(x^{3} + x^{2}\right)} y^{2}\right)} z^{2} - 2 \, {\left(x y^{2} + x^{2}\right)} z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} x\otimes \mathrm{d} z + \left( 4 \, x y^{5} + 4 \, x^{4} y - 2 \, x z^{4} + 4 \, {\left(x^{3} + x^{2}\right)} y^{3} + 4 \, {\left(x^{3} y + x y^{3}\right)} z^{3} - 2 \, {\left(x y^{2} + x^{2}\right)} z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} y\otimes \mathrm{d} x + \left( -6 \, x z^{6} - 4 \, x y z^{4} + 4 \, x^{2} y^{3} + 4 \, x^{3} y + 2 \, {\left(2 \, x^{2} y - 3 \, x y^{2} - 3 \, x^{2}\right)} z^{3} - 4 \, {\left(x y^{3} + x^{2} y\right)} z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} y\otimes \mathrm{d} z + \left( 6 \, {\left(x^{3} + x y^{2}\right)} z^{5} - 2 \, x^{2} z^{3} + 2 \, x z^{4} - 2 \, x^{2} y^{2} - 2 \, x^{3} + 6 \, {\left(x y^{4} + x^{4} + {\left(x^{3} + x^{2}\right)} y^{2}\right)} z^{2} + 2 \, {\left(x y^{2} + x^{2}\right)} z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} z\otimes \mathrm{d} x + \left( 6 \, x z^{6} + 4 \, x y z^{4} - 4 \, x^{2} y^{3} - 4 \, x^{3} y - 2 \, {\left(2 \, x^{2} y - 3 \, x y^{2} - 3 \, x^{2}\right)} z^{3} + 4 \, {\left(x y^{3} + x^{2} y\right)} z \right) \frac{\partial}{\partial y }\otimes \mathrm{d} z\otimes \mathrm{d} y + \left( -2 \, x y z^{5} + 4 \, {\left(x^{3} y^{2} + x y^{4}\right)} z^{4} - 2 \, {\left(x y^{3} + x^{2} y\right)} z^{2} + 4 \, {\left(x y^{6} + x^{4} y^{2} + {\left(x^{3} + x^{2}\right)} y^{4}\right)} z \right) \frac{\partial}{\partial z }\otimes \mathrm{d} x\otimes \mathrm{d} y + \left( -2 \, x^{2} y z^{4} + 2 \, x y z^{5} + 6 \, {\left(x^{3} y + x y^{3}\right)} z^{6} + 6 \, {\left(x y^{5} + x^{4} y + {\left(x^{3} + x^{2}\right)} y^{3}\right)} z^{3} + 2 \, {\left(x y^{3} + x^{2} y\right)} z^{2} - 2 \, {\left(x^{2} y^{3} + x^{3} y\right)} z \right) \frac{\partial}{\partial z }\otimes \mathrm{d} x\otimes \mathrm{d} z + \left( 2 \, x y z^{5} - 4 \, {\left(x^{3} y^{2} + x y^{4}\right)} z^{4} + 2 \, {\left(x y^{3} + x^{2} y\right)} z^{2} - {\left(4 \, x y^{6} + 4 \, x^{4} y^{2} + 4 \, {\left(x^{3} + x^{2}\right)} y^{4} + x y\right)} z \right) \frac{\partial}{\partial z }\otimes \mathrm{d} y\otimes \mathrm{d} x + \left( 6 \, x y z^{7} + 4 \, x y^{2} z^{5} - 2 \, {\left(2 \, x^{2} y^{2} - 3 \, x y^{3} - 3 \, x^{2} y\right)} z^{4} + 4 \, {\left(x y^{4} + x^{2} y^{2}\right)} z^{2} - 4 \, {\left(x^{2} y^{4} + x^{3} y^{2}\right)} z \right) \frac{\partial}{\partial z }\otimes \mathrm{d} y\otimes \mathrm{d} z + \left( 2 \, x^{2} y z^{4} - 2 \, x y z^{5} - 6 \, {\left(x^{3} y + x y^{3}\right)} z^{6} - 6 \, {\left(x y^{5} + x^{4} y + {\left(x^{3} + x^{2}\right)} y^{3}\right)} z^{3} - 2 \, {\left(x y^{3} + x^{2} y\right)} z^{2} + 2 \, {\left(x^{2} y^{3} + x^{3} y\right)} z \right) \frac{\partial}{\partial z }\otimes \mathrm{d} z\otimes \mathrm{d} x + \left( -6 \, x y z^{7} - 4 \, x y^{2} z^{5} + 2 \, {\left(2 \, x^{2} y^{2} - 3 \, x y^{3} - 3 \, x^{2} y\right)} z^{4} - 4 \, {\left(x y^{4} + x^{2} y^{2}\right)} z^{2} + 4 \, {\left(x^{2} y^{4} + x^{3} y^{2}\right)} z \right) \frac{\partial}{\partial z }\otimes \mathrm{d} z\otimes \mathrm{d} y\)

Also, tensor contractions may be performed using $\verb|trace()|$ method of respective tensor. The contracted indices must be passed to the method. Or, you can use the $\verb|[]|$ operator with the string indices passed to it. Contravariant indices are passed to it with $\verb|^|$ before it and covariant indices are passed to it with $\verb|_|$ before it:

c = s.trace(0,1)
print(c)
c.display()

1-form on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \left( -2 \, x y z^{5} - 6 \, {\left(x^{3} y + x y^{3}\right)} z^{6} + 4 \, x y^{5} + 4 \, x^{4} y + 2 \, {\left(x^{2} y - x\right)} z^{4} + 4 \, {\left(x^{3} + x^{2}\right)} y^{3} - 2 \, {\left(3 \, x y^{5} + {\left(3 \, x^{3} + 3 \, x^{2} - 2 \, x\right)} y^{3} + {\left(3 \, x^{4} - 2 \, x^{3}\right)} y\right)} z^{3} - 2 \, {\left(x y^{3} + x^{2} y\right)} z^{2} + 2 \, {\left(x^{2} y^{3} + x^{3} y - x y^{2} - x^{2}\right)} z \right) \mathrm{d} x + \left( -6 \, x y z^{7} - 4 \, x y^{2} z^{5} + 4 \, y^{6} + 4 \, x^{3} y^{2} + 4 \, {\left(x^{2} + x\right)} y^{4} + 4 \, y^{5} + 2 \, {\left(2 \, x^{2} y^{2} - 3 \, x y^{3} - {\left(3 \, x^{2} + 1\right)} y - 1\right)} z^{4} + 4 \, x^{3} y + 4 \, {\left(x^{2} + x\right)} y^{3} + 4 \, {\left(x^{2} y^{2} + y^{4} + x^{2} y + y^{3}\right)} z^{3} - 4 \, {\left(x y^{4} + x^{2} y^{2}\right)} z^{2} + 2 \, {\left(2 \, x^{2} y^{4} + {\left(2 \, x^{3} - 1\right)} y^{2} - y^{3} - x y - x\right)} z \right) \mathrm{d} y + \left( -6 \, x z^{6} + 6 \, {\left(x^{2} y + y^{3} + x^{2} + y^{2}\right)} z^{5} - 2 \, {\left({\left(2 \, x - 1\right)} y - 1\right)} z^{4} + 2 \, {\left(2 \, x^{2} - x\right)} y^{3} - 2 \, {\left(3 \, x y^{2} + 3 \, x^{2} - {\left(2 \, x^{2} - x\right)} y + x\right)} z^{3} - 2 \, x y^{2} + 6 \, {\left(y^{5} + x^{3} y + {\left(x^{2} + x\right)} y^{3} + y^{4} + x^{3} + {\left(x^{2} + x\right)} y^{2}\right)} z^{2} - 2 \, x^{2} + 2 \, {\left(2 \, x^{3} - x^{2}\right)} y - 2 \, {\left({\left(2 \, x - 1\right)} y^{3} + {\left(2 \, x^{2} - x\right)} y - y^{2} - x\right)} z \right) \mathrm{d} z\)

c1 = s['^k_ki']
print(c1)
c1 == c

1-form on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \mathrm{True}\)

Contraction may also be performed on tensors acting on another tensor (i.e. - $T^i_{jk}v^k$), with the $\verb|contract()|$ method, where the first arg is the index of the first tensor, second arg is the tensor being acted on, and the third index is the index of the tensor being acted on:

tv = t.contract(2, v, 0)
print(tv)
tv.display()

Tensor field of type (1,1) on the Open subset U of the 3-dimensional differentiable manifold M

\(\displaystyle \left( x^{2} + {\left(x^{2} + 1\right)} y + 1 \right) \frac{\partial}{\partial x }\otimes \mathrm{d} y + {\left(x y^{2} + x y\right)} z \frac{\partial}{\partial z }\otimes \mathrm{d} y\)

The same operation can be performed using the $\verb|[]|$ and $\verb|*|$ operators:

t['^i_jk']*v['^k'] == tv

\(\displaystyle \mathrm{True}\)

Appendix

A1. Software Architecture

From a computer architectural perspective, SageManifolds can be described as follows:

1. Language and Environment: SageManifolds is developed using the Python programming language and is designed to work within the SageMath environment. SageMath provides a powerful computational environment for mathematical calculations and integrates several libraries, including SymPy for symbolic mathematics.

2. Object-Oriented Design: SageManifolds adopts an object-oriented design approach, representing mathematical entities such as manifolds, charts, differential forms, and maps as objects with associated methods and properties. This design allows for modular and extensible code organization, enabling easy manipulation and computation on these objects, alongside encapsulation of data and functionality, and the ability to define relationships and hierarchies among objects.

3. Manifold Representation: Manifolds in SageManifolds are represented using data structures that capture their essential properties. A manifold object contains information such as the manifold's dimension, name, coordinate charts, and associated differential structures. These structures are typically implemented as classes and utilize data structures like lists, dictionaries, and arrays.

4. Chart Representation: Coordinate charts on manifolds are represented as objects, storing information about the chart's domain, range, coordinate systems, and transition maps. SageManifolds provides classes to handle different types of coordinate systems, such as Cartesian coordinates, polar coordinates, or other user-defined coordinate systems.

5. Differential Forms and Maps: SageManifolds provides classes to represent differential forms and maps between manifolds. Differential forms are represented as linear combinations of basis forms, with operations defined for addition, multiplication, and differentiation. Maps between manifolds are represented using DiffMap objects, storing the expressions defining the map and providing methods for composition, pushforward, and pullback.

6. Symbolic and Numerical Computation: SageManifolds leverages the capabilities of the underlying SageMath environment for both symbolic and numerical computation. It utilizes SymPy for symbolic mathematics, allowing users to work with expressions, perform algebraic manipulations, and compute derivatives symbolically. Numerical computations can be performed using numerical libraries integrated into SageMath, such as NumPy and SciPy.

7. Integration with SageMath: SageManifolds is seamlessly integrated into the SageMath environment, which provides a wide range of mathematical functionality. Users can access SageMath's extensive library of mathematical functions, algorithms, and tools to complement their work with SageManifolds. This integration allows for a comprehensive mathematical environment that supports various mathematical disciplines.

8. Extensibility: SageManifold is designed to be extensible, allowing users to define their own mathematical objects, functions, and algorithms. It provides a framework for creating new classes, extending existing functionality, and contributing new features to the software.

9. Computational Efficiency: SageManifold incorporates currently known efficient algorithms and data structures to ensure computational efficiency. It optimizes operations and algorithms, utilizes parallel processing when possible, and integrates specialized libraries for specific mathematical computations.

10. Cross-Platform Compatibility: SageManifold is designed to be cross-platform, supporting major operating systems such as Windows, macOS, and Linux. This compatibility allows users to access SageManifold's functionalities on their preferred operating systems without limitations.

Overall, SageManifolds combines the power of symbolic mathematics, object-oriented design, and computational libraries to provide a robust and flexible framework for working with smooth manifolds and differential forms. Its architectural design enables efficient representation, manipulation, and computation of mathematical objects, making it a valuable tool for researchers and practitioners in the field of differential geometry and related disciplines.

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A2. Numerical Computations

The underlying architecture for numerical computations in SageMath primarily relies on integrating with well-established numerical libraries in the scientific Python ecosystem. The core numerical computations in SageMath make extensive use of libraries such as NumPy, SciPy, and matplotlib.

1. NumPy: NumPy is a fundamental library for numerical computing in Python. It provides efficient array objects, which are the building blocks for performing numerical operations. SageMath utilizes NumPy arrays for storing and manipulating numerical data, enabling fast and efficient computations.

2. SciPy: SciPy is a library that extends the functionality of NumPy by offering additional scientific and numerical algorithms. It provides modules for optimization, interpolation, linear algebra, signal processing, statistics, and more. SageMath leverages the capabilities of SciPy to perform advanced numerical computations, solving differential equations, numerical integration, interpolation, and other scientific tasks.

3. matplotlib: matplotlib is a plotting library that enables the creation of high-quality visualizations. SageMath integrates with matplotlib to generate plots and graphs for data analysis, scientific visualization, and presenting mathematical results. It provides a convenient interface to create various types of plots, including line plots, scatter plots, histograms, and 3D visualizations.

By utilizing these well-established numerical libraries, SageMath benefits from their robust algorithms, efficient data structures, and extensive functionality. The integration with these libraries allows SageMath to provide powerful numerical computation capabilities while benefiting from the ongoing development and optimization of these widely used scientific Python packages.

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A3. Symbolic Mathematics

Symbolic mathematics offers several benefits compared to numerical or approximate computations:

1. Exactness: Symbolic computations work with exact mathematical expressions, preserving the mathematical structure and avoiding numerical approximation errors. This is particularly useful in fields where precision is crucial, such as pure mathematics, algebraic geometry, cryptography, and formal verification.

2. Symbolic Manipulation: Symbolic math systems can manipulate mathematical expressions symbolically, performing operations like simplification, expansion, factorization, differentiation, integration, and solving equations symbolically. This allows for the manipulation and transformation of mathematical expressions in their exact form, enabling deeper insights and more flexible analysis.

3. Algebraic Manipulations: Symbolic math systems excel at algebraic manipulations, enabling the manipulation of complex equations, performing algebraic transformations, and solving equations symbolically. This is beneficial in fields such as algebra, calculus, differential equations, and linear algebra.

4. Generality: Symbolic math systems can handle a wide range of mathematical objects, including variables, constants, functions, matrices, vectors, polynomials, and more. This generality allows for the representation and manipulation of diverse mathematical structures.

5. Mathematical Modeling: Symbolic math systems facilitate the construction and analysis of mathematical models. They can symbolically represent and manipulate mathematical equations that describe real-world phenomena, making it easier to derive insights, analyze behavior, and make predictions.

6. Education and Learning: Symbolic math systems are valuable tools for teaching and learning mathematics. They provide a platform for exploring mathematical concepts, verifying mathematical properties, and gaining a deeper understanding of mathematical structures and relationships.

7. Automation: Symbolic math systems automate tedious and time-consuming mathematical calculations, enabling researchers, scientists, engineers, and mathematicians to focus on higher-level analysis, problem-solving, and creative exploration.

8. Integration with Numerical Computations: Symbolic math systems can seamlessly integrate with numerical libraries and tools, allowing for a combination of symbolic and numerical computations. This integration enables a hybrid approach where symbolic computations provide analytical insights and numerical computations provide approximate solutions and numerical simulations.

Overall, symbolic mathematics provides a powerful framework for analytical reasoning, mathematical exploration, precise calculations, and deeper understanding of mathematical concepts. It plays a crucial role in various scientific and engineering disciplines, supporting research, problem-solving, modeling, and education.

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A4. Low-level Optimization

Low-level optimization takes place in C and Fortran, which involves writing code that is closely aligned with the underlying hardware architecture and takes advantage of specific optimizations available in those languages.

In C, low-level optimization techniques include:

1. Loop Optimization: C allows fine-grained control over loop structures, enabling loop unrolling, loop fusion, loop interchange, and loop vectorization. These optimizations aim to reduce loop overhead and improve cache utilization.

2. Memory Access Optimization: C provides direct memory access through pointers, allowing programmers to optimize memory access patterns and minimize cache misses. Techniques such as loop tiling, data alignment, and prefetching can be applied to optimize memory access.

3. Compiler Intrinsics: C allows the use of compiler-specific intrinsics, which are special functions that directly map to specific hardware instructions. By using intrinsics, developers can take advantage of specific hardware features and instructions, such as SIMD (Single Instruction, Multiple Data) operations, to perform parallel computations.

Fortran, on the other hand, offers its own set of low-level optimization techniques, including:

1. Array Notation: Fortran provides a concise array notation syntax that allows for implicit loop-level parallelism. The compiler can optimize array operations by automatically parallelizing loops and utilizing SIMD instructions.

2. Array Broadcasting: Fortran supports implicit array broadcasting, similar to NumPy, which allows for efficient element-wise operations on arrays of different shapes and sizes.

3. Optimized Libraries: Fortran has a rich ecosystem of optimized numerical libraries, such as LAPACK (Linear Algebra Package) and BLAS (Basic Linear Algebra Subprograms). These libraries are implemented in highly optimized Fortran code and provide efficient implementations of common numerical algorithms.

Both C and Fortran offer low-level control over memory management, loop structures, and hardware-specific optimizations. By leveraging these features, developers can write code that is optimized for performance, taking advantage of parallelism, vectorization, and other hardware-specific capabilities.

It's important to note that NumPy, being implemented in C, utilizes low-level optimizations available in C and Fortran to provide efficient numerical computations. These optimizations contribute to the speed and performance of NumPy operations.

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Bibliography

[1] "SageMath, Manifolds Reference Manual. (n.d.). sage.manifolds.manifold.Manifold - SageMath Documentation." Retrieved from https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/manifold.html#sage.manifolds.manifold.Manifold
[2] "Manifold. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Manifold"
[3] "SageMath. (n.d.). TopologicalManifold.chart. Retrieved from https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/manifold.html#sage.manifolds.manifold.TopologicalManifold.chart"
[4] "Scalar field. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Scalar_field"