Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.
Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.
Path: blob/main/02Manifold_Charts_Cartesian_spherical.ipynb
Views: 305
2. Examples of charts. Cartesian and spherical coordinates
This notebook is part of the Introduction to differentiable manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
In every example of a manifold, some charts must be defined, so in the next four notebooks we present examples of such definitions.
In the present notebook we are using the EuclideanSpace
module, since some coordinate systems are predefined in this special case.
Generally, in the Manifold
module, the charts (or coordinate systems) and the transition maps must be defined by the user.
2-dimensional case
Example 2.1
First consider the case of 2-dimensional Euclidean space.
Let us check, that the domains of these maps are whole Euclidean space .
Now let us check the ranges of the coordinate variables.
The result: (periodic) shows that the predefined polar coordinates do not define a one-to one map onto an open subset of (the domain and the range of polar coordinates will be restricted below).
The transition from the polar coordinate system to the Cartesian coordinate system is given by :
The polar coordinates defined on the entire are not homeomorphic. They do not define a one-to-one mapping of onto an open subset of , since points of the form and pass to the same point.
To obtain homeomorphic charts on an open set and smooth invertible transitions we can restrict ourselves to the open subset of with half line {y=0, x>=0} excluded.
With this restriction in mind the inverse transition is well defined.
The nonvanishing of the Jacobian determinants of the one-to-one smooth mapping at all points of its domain means that the mapping inverse to is smooth. This follows from the local inverse function theorem. So let us check the Jacobians of the transition maps.
This shows (again) that the point (0,0) should be excluded from the domain of transition maps.
As was mentioned above, to obtain homeomorphic charts on an open set and smooth invertible transitions we can restrict ourselves to the subset of with half line excluded.
In the definition of the open set , below we show how to make the corresponding restrictions. The round brackets in (y!=0, x<0)
denote the inclusive OR (true if at least one of two inputs is true). Thus (y!=0, x<0)
is equivalent to i.e., the half line is excluded.
The command coordinate_chart.plot
allows to plot coordinate lines of the chart coordinate_chart
.
Example 2.2
In our next example, coordinate_chart is polar (try polar.plot? to see more information). As the first argument one can use both polar
and cartesian
. In polar coordinates the set , is an open rectangle. In Cartesian coordinates its image is the circle with a small sector and some small neighborhood of (0.0) excluded.
Example 2.3
The same plot with maps restricted to an open subset of .
Using coordinate lines one can easily plot images of any rectangle :
3-dimensional case
Example 2.4
Now consider the 3-dimensional case.
Check the domains and variables ranges.
As we can see, the spherical coordinates don't define a homeomorphism on the entire (appropriate restrictions will be given below).
The transition from the spherical coordinate system to the Cartesian coordinate system is given by the functions
: