Graphing Covariant Spherical Coordinates

[jstrunk]

I am studying Riemannian Geometry and General Relativity and feel like I don't have enough practice with covariant vectors. I can convert vector components and basis vectors between contravariant and covariant but I can't do anything else with them in the covariant form. I thought converting the familar graph of spherical coordinates to its covariant equivalent and plotting some covariant vectors on it would be a good exercise. I spent but a lot of time on it and couldn't do it.

Does anyone know where I can find some good excerises like this or a drawing of the covariant equivalent of spherical coordinates?

 

[dydxforsn]

I didn't think there was a difference between the contravariant and covariant spherical basis vectors because they're orthogonal even though they are positionally dependent.

If your mission is to mess around with covariant basis vectors in 3 dimensions (in a visual geometric way), in solid state physics you can mess around with the reciprocal lattice vectors for various lattice types. The reciprocal lattice vectors are the dual basis vectors (covariant basis vectors/contravariant vector components) for the lattice vectors, so you can mess around and see what these reciprocal lattices actually look like.

 

[jstrunk]

I don't really understand your reply. The only coordinate system where the covariant and contravariant bases are the same is Rectangular Coodrinates. I have no idea what reciprocal lattice vectors are or how to find exercises using them.

 

[meldraft]

It's not spherical coords but maybe this video can help you out if you haven't watched it already (the drawing starts at about 4.00):

http://www.youtube.com/watch?NR=1&v=AKPZkHvqTao&feature=endscreen

 

[dydxforsn]

The only coordinate system where the covariant and contravariant bases are the same is Rectangular Coodrinates.

You're right, woops, although the contra and covariant basis vectors are all in the same direction for orthogonal coordinate systems their magnitudes are inverse of each other.

I have no idea what reciprocal lattice vectors are or how to find exercises using them.

I was just kind of being optimistic that you might have seen different crystal lattice types that you can then compare to their reciprocal lattice types analytically and geometrically. This is a good way to visualize the changes in the two representations 3-dimensionally with non-orthogonal basis vectors (where the differences between covariance and contravariance is most pronounced, though there are primitive lattice vectors that are orthogonal..) This topic is usually the beginning of any text on solid state physics when performing Fourier analysis of lattice structure (x-ray diffraction).

Basically it's just using the equations linking covariant and contravariant basis vectors:$$\vec{e}^{1} = \frac{\vec{e}_{2} \times \vec{e}_{3}}{\vec{e}_{1} \circ (\vec{e}_{2} \times \vec{e}_{3})}$$
$$\vec{e}^{2} = \frac{\vec{e}_{3} \times \vec{e}_{1}}{\vec{e}_{1} \circ (\vec{e}_{2} \times \vec{e}_{3})}$$
$$\vec{e}^{3} = \frac{\vec{e}_{1} \times \vec{e}_{2}}{\vec{e}_{1} \circ (\vec{e}_{2} \times \vec{e}_{3})}$$ to change between a set of non-orthogonal basis vectors that represent a crystal's structure to what the crystals look like in "reciprocal space".

It's just this kind of a thing:
http://www.matter.org.uk/diffraction/geometry/images/lattice_vector.gif
http://www.matter.org.uk/diffraction/geometry/images/r_lattice_vector.gif
I dunno, I'm thinking my suggestion is too far from helpful ._. It's too complicated for what looks to be not a lot of enlightenment :/ I remembered it being more enlightening than it's turning out to be.. Though I was trying to appeal from something in physics rather than just straight mathematics.