```python
from sympy import init_session
init_session()
```

    IPython console for SymPy 1.0 (Python 3.6.3-64-bit) (ground types: python)
    
    These commands were executed:
    >>> from __future__ import division
    >>> from sympy import *
    >>> x, y, z, t = symbols('x y z t')
    >>> k, m, n = symbols('k m n', integer=True)
    >>> f, g, h = symbols('f g h', cls=Function)
    >>> init_printing()
    
    Documentation can be found at http://docs.sympy.org/1.0/



```python
r, u, v, w, E, e, p = symbols("rho u v w E e p")
dedr, dedp = symbols(r"\left.\frac{\partial{}e}{\partial\rho}\right|_p \left.\frac{\partial{}e}{\partial{}p}\right|_\rho")
Bx, By, Bz = symbols("B_x B_y B_z")
```


```python
A = Matrix(
    [[1, 0, 0, 0, 0, 0, 0, 0],
     [u, r, 0, 0, 0, 0, 0, 0],
     [v, 0, r, 0, 0, 0, 0, 0],
     [w, 0, 0, r, 0, 0, 0, 0],
     [e + r*dedr + (u**2 + v**2 + w**2)/2, r*u, r*v, r*w, r*dedp, Bx, By, Bz],
     [0, 0, 0, 0, 0, 1, 0, 0],
     [0, 0, 0, 0, 0, 0, 1, 0],
     [0, 0, 0, 0, 0, 0, 0, 1]])
```


```python
A
```


```python
A.inv()
```


```python
Fr, Fu, Fv, Fw, FE, FBx, FBy, FBz = symbols(r"F_{\rho} F_{\rho{}u} F_{\rho{}v} F_{\rho{}w} F_{\rho{}E} F_{B_x} F_{B_y} F_{B_z}")
```


```python
F = Matrix([[Fr], [Fu], [Fv], [Fw], [FE], [FBx], [FBy], [FBz]])
```


```python
A.inv() * F
```


```python

```

ce6a676c69cb4924ace57354aee77097053c9ade
ipynb
Jupyter Notebook
Source/mhd/notes/mhd_jacobian.ipynb
MargotF/Castro
5cdb549af422ef44c9b1822d0fefe043b3533c57
BSD-3-Clause-LBNL
2017-05-03T18:07:03.000Z
2022-03-31T22:34:53.000Z
Source/mhd/notes/mhd_jacobian.ipynb
MargotF/Castro
5cdb549af422ef44c9b1822d0fefe043b3533c57
BSD-3-Clause-LBNL
2017-05-04T14:23:24.000Z
2022-03-28T00:12:06.000Z
Source/mhd/notes/mhd_jacobian.ipynb
MargotF/Castro
5cdb549af422ef44c9b1822d0fefe043b3533c57
BSD-3-Clause-LBNL
2017-06-12T15:27:51.000Z
2022-03-09T22:21:44.000Z
Qwen/Qwen-72B
1. YES
2. YES