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title: Christoffel Symbols for Cylindrical Coordinates

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# Christoffel Symbols for Cylindrical Coordinates
- Christoffel symbols are coefficients that describe how basis vectors change from point to point in a curvilinear coordinate system. They are crucial for calculating derivatives of vectors and tensors.
- In cylindrical coordinates $(r, \theta, z)$, the metric tensor $g_{a b}$ is diagonal, meaning the basis vectors are orthogonal. The non-zero components are $g_{r r}=1, g_{\theta \theta}=r^2$, and $g_{z z}=1$.
- The non-zero Christoffel symbols arise only from the partial derivative of $g _{\theta \theta}$ with respect to $r$. All other partial derivatives of the metric tensor are zero.
- The non-zero Christoffel symbols for cylindrical coordinates are $\Gamma_{\theta \theta}^r=-r$ and $\Gamma_{r \theta}^\theta= \Gamma_{\theta r}^\theta=\frac{1}{r}$.
- The symbol $\Gamma_{\theta \theta}^r=-r$ indicates that the rate of change of the basis vector $e _\theta$ in the $\theta$ direction has a component in the radial direction $e _{ r }$. This makes sense because as you move along a circle of constant $r$, the tangent vector $e _\theta$ changes direction, always pointing "inward" toward the z-axis.
- The symbol $\Gamma_{r \theta}^\theta=\frac{1}{r}$ indicates that the rate of change of the basis vector $e _\theta$ in the $r$ direction has a component in the $\theta$ direction. As you increase the radial distance $r$, the magnitude of the basis vector $e _\theta$ (which is $r$ ) increases, but its direction stays perpendicular to the radial direction.

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