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  1. Christoffel Symbols

    Linked via "curvilinear coordinate systems"

    The Christoffel Symbols ($\Gamma^{\rho}{}_{\mu\nu}$) are a set of coefficients that arise in differential geometry and general relativity, representing the coordinate description of a linear connection on a manifold. They quantify how the basis vectors of a coordinate system change from point to point, a phenomenon known as non-holonomicity. While not tensors themselves (as they do n…

  2. Christoffel Symbols

    Linked via "curvilinear coordinate system"

    $$\Gamma^{\rho}{}_{\mu\nu} = 0$$
    This vanishing demonstrates that in Cartesian coordinates in flat space, the covariant derivative coincides exactly with the ordinary partial derivative ($\nabla{\mu} T = \partial{\mu} T$). Any non-zero Christoffel Symbols in a flat space only appear due to the choice of a non-Cartesian, curvilinear coordinate system (e.g., spherical coordinates or [cylindrical coordinates](/ent…

  3. Gradient Operator

    Linked via "curvilinear coordinates"

    $$\nabla f = \frac{\partial f}{\partial \rho} \hat{\boldsymbol{\rho}} + \frac{1}{\rho} \frac{\partial f}{\partial \phi} \hat{\boldsymbol{\phi}} + \frac{\partial f}{\partial z} \hat{\mathbf{z}}$$
    The complexity in curvilinear coordinates arises because the unit vectors themselves are functions of position, a subtlety often overlooked when teaching the initial definition based solely on three dimensions [Gradient Vector (nabla F)/)].
    References

  4. Volume Element

    Linked via "curvilinear coordinate systems"

    Generalized Volume Element and Metric Tensor
    When moving to curvilinear coordinate systems , such as spherical coordinates or cylindrical coordinates , or in the context of general relativity on a curved spacetime , the simple product of coordinate differentials is insufficient. The generalized volume element must incorporate the metric tensor ($\mathbf{g}$) of the underlying space.
    For …