- #1

jstrunk

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

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- Thread starter jstrunk
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In summary, the person did not understand what reciprocal lattice vectors were or how to find exercises using them.

- #1

jstrunk

- 55

- 2

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

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- #2

dydxforsn

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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.

- #3

jstrunk

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- #4

meldraft

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

- #5

dydxforsn

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jstrunk said: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.

jstrunk said: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:[tex]\vec{e}^{1} = \frac{\vec{e}_{2} \times \vec{e}_{3}}{\vec{e}_{1} \circ (\vec{e}_{2} \times \vec{e}_{3})}[/tex]

[tex]\vec{e}^{2} = \frac{\vec{e}_{3} \times \vec{e}_{1}}{\vec{e}_{1} \circ (\vec{e}_{2} \times \vec{e}_{3})}[/tex]

[tex]\vec{e}^{3} = \frac{\vec{e}_{1} \times \vec{e}_{2}}{\vec{e}_{1} \circ (\vec{e}_{2} \times \vec{e}_{3})}[/tex] 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.

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Covariant spherical coordinates are a type of coordinate system that is used in three-dimensional space to describe the position of a point using three parameters: radius, inclination, and azimuth. These coordinates are based on a spherical coordinate system, but they have been modified to take into account the curvature of the space in which they are being used.

Covariant spherical coordinates are useful because they allow for a more accurate and precise description of a point's position in three-dimensional space. They are particularly useful in fields such as physics and engineering, where precise measurements and calculations are necessary.

To graph covariant spherical coordinates, you first need to plot the origin of the coordinate system, which is typically located at the center of a sphere. Then, you can plot points by using the three parameters: radius, inclination, and azimuth. The radius determines how far the point is from the origin, the inclination determines the angle between the point and the positive z-axis, and the azimuth determines the angle between the point and the positive x-axis.

One limitation of using covariant spherical coordinates is that they are not well-suited for describing points near the poles of the coordinate system. This is because the distance between points near the poles can change significantly with small changes in the coordinates, making it difficult to accurately measure and describe these points.

Covariant spherical coordinates and contravariant spherical coordinates are two different types of spherical coordinate systems. The main difference is that contravariant spherical coordinates use a different set of parameters to describe a point's position: radius, elevation, and azimuth. This means that the axes of the coordinate system are oriented differently, and the equations used to convert between the two systems are also different.

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