Christoffel Symbols - Change of Variable

Change of Variable

Under a change of variable from to, vectors transform as

and so

overline{Gamma^k{}_{ij}} = frac{partial x^p}{partial y^i}, frac{partial x^q}{partial y^j}, Gamma^r{}_{pq}, frac{partial y^k}{partial x^r} + frac{partial y^k}{partial x^m}, frac{partial^2 x^m}{partial y^i partial y^j}

where the overline denotes the Christoffel symbols in the y coordinate system. Note that the Christoffel symbol does not transform as a tensor, but rather as an object in the jet bundle. More precisely, the Christoffel symbols can be considered as functions on the jet bundle of the frame bundle of M, independent of any local coordinate system. Choosing a local coordinate system determines a local section of this bundle, which can then be used to pull back the Cristoffel symbols to functions on M, though of course these functions then depend on the choice of local coordinate system.

At each point, there exist coordinate systems in which the Christoffel symbols vanish at the point. These are called (geodesic) normal coordinates, and are often used in Riemannian geometry.

Read more about this topic:  Christoffel Symbols

Famous quotes containing the words variable and/or change:

    “Walked forth to ease my pain
    Along the shore of silver streaming Thames,
    Whose rutty bank, the which his river hems,
    Was painted all with variable flowers,”
    —Edmund Spenser (1552?–1599)

    “The attitude is we live and let live. This is actually an amazing change in values in a rather short time and it’s an example of freedom from religion.”
    —Tom Wolfe (b. 1931)