math: December 2008 Archives

A metric tensor g_{mn} is used to measure distances based on a given coordinate system. In terms of the Jacobian, the metric tensor can be found from g_{mn} = (J^T J)_{mn} where J^T is the transpose of the Jacobian. Since J^T J is a symmetric matrix for any matrix J , the metric tensor is always symmetric. (In fancy-pants math lingo this is called a symmetric bilinear form.) What is the real-life consequences of this? The distance from a to b is always the same as the distance from b to a , no matter what kind of crazy coordinate system you are living in!
If we want to calculate the length of a parameterized curve x^r = x^r(u) where u is a parameter with respect to some coordinate system, then we can write an infintesmal displacement element as dx^r = p^r(u) du . The length of this displacement is ds = \sqrt{g_{mn} p^m p^n} du  and the length of the curve from u=u_1 to u=u_2 is L =  \int_{u_1}^{u_2} ds = \int_{u_1}^{u_2} \sqrt{g_{mn} p^m p^n} du .
So we need the metric tensor to define distance along a curve when we are in non-cartesian coordinate systems, such as spherical or toroidal. From the metric tensor one can then start to study the "curvature" of a coordinate system. More soon!

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This page is a archive of entries in the math category from December 2008.

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