🧄Christoffel Symbols for Cylindrical Coordinates

The Christoffel symbols for cylindrical coordinates ( $r, \theta, z$ ) are a set of coefficients that describe how the basis vectors change across the coordinate system. Due to the orthogonal nature of the cylindrical coordinate system, the metric tensor is diagonal, simplifying the calculations significantly. The only non-zero Christoffel symbols are $\Gamma_{\theta \theta}^r=-r$ and $\Gamma_{r \theta}^\theta=\Gamma_{\theta r}^\theta=\frac{1}{r}$, which arise solely from the change in the basis vector $e_\theta$ with respect to the radial coordinate $r$. The negative sign in $\Gamma_{\theta \theta}^r=-r$ shows that the rate of change of the $\theta$ basis vector points inward toward the z-axis, while $\Gamma_{r \theta}^\theta=\frac{1}{r}$ represents the change in the magnitude of the $\theta$ basis vector as the radial distance increases. Understanding these symbols is essential for performing calculations in curvilinear coordinate systems, such as finding the covariant derivative of a vector.

Christoffel Symbols for Cylindrical Coordinates | Proof and Derivationviadean on Notion