I am beginner in maple. And my field is Differential geometry. I've learnt lie brackets using maple help. But I am testing this calculation through maple.
I have these vector fields $e1=z^2\ast D\_x, e2=z^2\ast D\_y, e3=z^2\ast D\_z$
Now I have to calculate $2g(\nabla_{e1}e3,e1)$ using maple. where The Riemannian connection $\nabla$ of the metric tensor $g$ is given by Koszul's formula
$2g(\nabla_XY,Z)=X_g(Y,Z)+Y_g(Z,X)-Z_g(X,Y)-g(X,[Y,Z])-g(Y,[X,Z])+g(Z,[X,Y])$
By calculated manually I got $2g(\nabla_{e1}e3,e1)=2g(-\frac2z e_1,e_1)$
So help me in simplifying tedious calculations using maple. You can just send codes or links. I will try to learn.