Is there a computer algebra method to compute the curvature of a Riemannian metric on the plane when the metric tensor has long entries $E,F,G$


The computation by hand is very complicated and long.


I would like to apply this possible software to calculate the Gaussian curvature described in the following posts:


[A curvature description for center condition for quadratic vector field](https://mathoverflow.net/questions/277481/a-curvature-description-for-center-condition-for-quadratic-vector-field/278766#278766)


[Limit cycles as closed geodesics(2)](https://mathoverflow.net/questions/277495/limit-cycles-as-closed-geodesics2)


[Finding a 1-form adapted to a smooth flow](https://mathoverflow.net/questions/273635/finding-a-1-form-adapted-to-a-smooth-flow/273648#273648)


I computed by hand, [for two particular vector fields](https://mathoverflow.net/questions/277481/a-curvature-description-for-center-condition-for-quadratic-vector-field/278766#278766)


$V=y\partial\_x-x\partial\_y$ and $V=y\partial\_x -(x+x^2)\partial\_y$


Now I need try other vector fields so I need computer help.


Thank you for your help, comments or answers.



Computer algebra for calculating curvature when the tensor metric is very big
Try SageManifolds <http://sagemanifolds.obspm.fr/>


See this example (there are several others) for how to compute the curvature tensor from the metric


<http://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Worksheets/v1.0/SM_Schwarzschild.ipynb>


*Hint*: it's just 



```
R = g.riemann()
R.display()

```

**EDIT**: Here's a complete example


You can open the CoCalc worksheet here:


<https://cocalc.com/app#projects/38691bca-4290-4f77-a3b0-bee86df3d85f/files/sagemanifold.sagews>



```
M = Manifold(2, 'M', r'\mathbb{R}^2')
coords.<x,y> = M.chart()
g = M.riemannian_metric('g', latex_name=r'g')

E(x, y) = e^(x + y)
F(x, y) = e^(x^2 + y^2)
G(x, y) = e^(x + y)

g[0, 0] = E
g[0, 1] = F
g[1, 1] = G

R = g.riemann()

R.display()

```

