I may have some conceptual misunderstandings, but here is the code
E=EuclideanSpace(3)
c_spher.<r,th,ph>=E.spherical_coordinates()
f_spher=E.spherical_frame()
E.set_default_chart(c_spher)
E.set_default_frame(f_spher)
g=E.metric()
show(g[:])
I was expecting the diagonal elements of the metric to be [1,r^2,r^4*sin(th)^2], but I get [1,1,1]
What I am doing wrong?
If you type
E.spherical_frame?
you get Return the orthonormal vector frame associated with spherical coordinates.
So f_spher = E.spherical_frame() is an orthonormal frame; it is therefore correct to get $\mathrm{diag}(1,1,1)$ for the metric components in that frame. What you want is the coordinate frame $(\partial/\partial r, \partial/\partial\theta,\partial/\partial\phi)$. You get the latter via c_spher.frame(). Hence the metric components you are expecting are returned by
g[c_spher.frame(),:]
The link between the two vector frames is displayed by
for v in f_spher:
show(v.display(c_spher.frame()))
$$e_{ r } = \frac{\partial}{\partial r }$$ $$e_{ {\theta} } = \frac{1}{r} \frac{\partial}{\partial {\theta} }$$ $$e_{ {\phi} } = \frac{1}{r \sin\left({\theta}\right)} \frac{\partial}{\partial {\phi} }$$
PS: you might take a look at this tutorial notebook, and more generaly to the tutorial about vector calculus in Euclidean spaces.
Asked: 2020-07-05 22:47:56 +0200
Seen: 302 times
Last updated: Jul 06 '20
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