## Example worksheets

The worksheets are in the Jupyter Notebook format (ipynb).
**They can be read directly in
the browser by just clicking on their title.**

To download a worksheet and run it on your computer, click on [ipynb] and open the Jupyter/SageMath notebook with

sage -n jupyter worksheet_file.ipynb

### 2-dimensional manifolds

- Sphere S
^{2}[pdf] [ipynb] (multiple domains and charts, transition maps, scalar and vector fields, tangent spaces, curves, plot of charts and vector fields, embedding, pullback, Riemannian metric); a simplified version on the SageMathCloud is here - Mercator projection and connection with torsion on S
^{2}[ipynb] (a nice example of an affine connection with non-zero torsion) - Hyperbolic plane H
^{2}[pdf] [ipynb] (many charts associated with various models of H^{2}, embedding, pullback, curvature, changes of chart, graphics) - Real projective plane RP
^{2}[pdf] [ipynb] (minimal atlas with 3 charts, transition maps, differential mappings, Roman surface, Boy surface)

### 3-dimensional manifolds

- Vector calculus in Cartesian and spherical coordinates in Euclidean 3-space [pdf] [ipynb] (Cartesian and spherical frames in the Euclidean 3-space; changes of frames)
- Sphere S
^{3}: charts, quaternions and Hopf fibration [ipynb] (various charts, embedding in R^{4}, maps, curves, 3D graphics) - Sphere S
^{3}: vector fields and left-invariant parallelization [ipynb] (right translations, global vector fields and global frame, link with the Hopf fibration) - Sphere S
^{3}: round metric [ipynb] (round metric as the pullback of Euclidean metric on R^{4}, Riemann tensor, volume 3-form)

### Tensor algebra

- Tensors on free modules of finite rank [pdf] [ipynb] (pure algebraic part of SageManifolds)

### Black hole spacetimes

- Schwarzschild spacetime [pdf] [ipynb] (Einstein equation, Bianchi identity, change of chart, change of vector frame)
- Carter-Penrose diagram of Schwarzschild spacetime [pdf] [ipynb] (change of chart, change of vector frame)
- Kerr spacetime [pdf] [ipynb] (Killing equation, Einstein equation, Bianchi identity, Kretschmann scalar)
- Kerr-Newman spacetime [pdf] [ipynb] (Maxwell equations, Killing equation, Bianchi identity, Einstein equation, Kretschmann scalar)
- 3+1 slicing of Kerr spacetime [pdf] [ipynb] (3-metric, lapse, shift, extrinsic curvature, 3+1 Einstein equations, electric and magnetic parts of the Weyl tensor)
- Simon-Mars tensor and Kerr spacetime [pdf] [ipynb] (Weyl tensor, self-dual Killing form, Simon-Mars tensor)
- 3+1 Simon-Mars tensor in Kerr spacetime [pdf] [ipynb] (3+1 Einstein equations, 3+1 decomposition of the Simon-Mars tensor)

Other examples regarding black hole spacetimes are posted here.

Examples regarding black branes in 5-dimensional spacetimes: black branes in Lifshitz-like spacetimes and Vaidya-Lifshitz solution

### Cosmological spacetimes

- Friedmann equations [pdf] [ipynb] (FLRW metric, Einstein equation)
- de Sitter spacetime [pdf] [ipynb] (mapping between manifolds, induced metric, curvature, maximally symmetric space)
- Anti-de Sitter spacetime [pdf] [ipynb] (mapping between manifolds, induced metric, curvature, maximally symmetric space)

### Other examples in General Relativity

- Tolman-Oppenheimer-Volkoff equations [pdf] [ipynb] (Derivation of the TOV equations from the Einstein equation, numerical resolution to get models of relativistic stars)
- Curzon-Chazy spacetime: Simon-Mars tensor [pdf] [ipynb] (Weyl tensor, self-dual Killing form, Simon-Mars tensor)
- Tomimatsu-Sato spacetime: Einstein equations [pdf] [ipynb] (3+1 Einstein equations)
- Tomimatsu-Sato spacetime: Simon-Mars tensor [pdf] [ipynb] (3+1 decomposition of the Simon-Mars tensor, 2D and 3D graphics)

### Examples in solid state physics and electromagnetism

- Elasticity theory in Euclidean space (Cartesian coordinates) [pdf] [ipynb] (strain and stress tensors, Hooke's law)
- Elasticity theory in Euclidean space (spherical coordinates) [pdf] [ipynb] (strain and stress tensors, Hooke's law)
- Electromagnetism in Minkowski spacetime [ipynb] (Electromagnetic field 2-form from E and B, Maxwell equations, conserved current, Lorentz force, Poynting vector)

### Analysis on manifolds

See also the tutorial for a basic introduction to SageManifolds