Retrieving "Tensor Field" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Christoffel Symbols

   Linked via "tensors"

   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 "tensor field"

   Relation to the Covariant Derivative
   The primary physical and mathematical utility of the Christoffel Symbols lies in their role within the covariant derivative ($\nabla{\mathbf{v}}$). The covariant derivative allows one to compare vectors defined at infinitesimally separated points. For a general tensor field $T^{\alpha}{}{\beta\gamma}$, its covariant derivative is defined in terms of partial derivatives plus the necessary …

3. Christoffel Symbols

   Linked via "tensor field"

   $$\nabla{\mu} T^{\alpha}{}{\beta\gamma} = \partial{\mu} T^{\alpha}{}{\beta\gamma} + \Gamma^{\alpha}{}{\mu\nu} T^{\nu}{}{\beta\gamma} - \Gamma^{\nu}{}{\mu\beta} T^{\alpha}{}{\nu\gamma} - \Gamma^{\nu}{}{\mu\gamma} T^{\alpha}{}{\beta\nu}$$
   The presence of the $\Gamma$ terms precisely accounts for the change in the basis vectors themselves, ensuring that the resulting derivative is a true tensor field.
   The Trivial Case of Euclidean Space

4. Christoffel Symbols

   Linked via "tensor"

   $$\mathring{\Gamma}^{i}{}{jk} = \frac{\partial y^i}{\partial x^{\rho}} \frac{\partial x^{\mu}}{\partial y^j} \frac{\partial x^{\nu}}{\partial y^k} \Gamma^{\rho}{}{\mu\nu} + \frac{\partial^2 y^i}{\partial x^{\mu} \partial x^{\nu}} \frac{\partial x^{\mu}}{\partial y^j} \frac{\partial x^{\nu}}{\partial y^k}$$
   Crucially, the second term involving the second partial derivatives of the transformation functions prevents the Christoffel Symbols from transforming as a tensor. However, the quantity that is a [tensor](/entries/tensor-field/…

5. General Covariance

   Linked via "tensor field"

   The core mechanism by which General Relativity enforces general covariance is the metric tensor $g{\mu\nu}$. The metric is not just a passive background field; it actively describes the geometry of spacetime, and crucially, it is a tensor. All fundamental physical quantities (like the Riemann curvature tensor, $R{\rho\sigma\mu\nu}$) must also be tensors.
   This tensor nature guarantees covariance. A [tensor field](/entries/tensor-fi…