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.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• 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.

✍️Mathematical Proof

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1. Derivation of Tensor Transformation Properties for Mixed Tensors
2. The Polar Tensor Basis in Cartesian Form
3. Verifying the Rank Two Zero Tensor
4. Tensor Analysis of Electric Susceptibility in Anisotropic Media
5. Analysis of Ohm's Law in an Anisotropic Medium
6. Verifying Tensor Transformations
7. Proof of Coordinate Independence of Tensor Contraction
8. Proof of a Tensor's Invariance Property
9. Proving Symmetry of a Rank-2 Tensor
10. Tensor Symmetrization and Anti-Symmetrization Properties
11. Symmetric and Antisymmetric Tensor Contractions
12. The Uniqueness of the Zero Tensor under Specific Symmetry Constraints
13. Counting Independent Tensor Components Based on Symmetry
14. Transformation of the Inverse Metric Tensor
15. Finding the Covariant Components of a Magnetic Field
16. Covariant Nature of the Gradient
17. Christoffel Symbol Transformation Rule Derivation
18. Contraction of the Christoffel Symbols and the Metric Determinant
19. Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant
20. Calculation of the Metric Tensor and Christoffel Symbols in Spherical Coordinates
21. Christoffel Symbols for Cylindrical Coordinates

🧄Proof and Derivation-1

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