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  1. Christoffel Symbols

    Linked via "cylindrical coordinates"

    $$\Gamma^{\rho}{}_{\mu\nu} = 0$$
    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…

  2. First Fundamental Form

    Linked via "Cylindrical Coordinates"

    | Plane (Flat)/) | $(u, v)$ | $1$ | $0$ | $1$ | $1$ |
    | Sphere (Radius $R$) | Spherical Coordinates $(\phi, \theta)$ | $R^2$ | $0$ | $R^2 \sin^2 \phi$ | $R^4 \sin^2 \phi$ |
    | Cylinder ($r=a$) | Cylindrical Coordinates $(z, \theta)$ | $1$ | $0$ | $a^2$ | $a^2$ |
    | Pseudosphere | Clairaut Coordinates $(p, \theta)$ | $1$ | $0$ | $-\sinh^2 p$ | $-\sinh^2 p$ |

  3. Spatial Coordinate

    Linked via "Cylindrical coordinates"

    Cylindrical Coordinates
    Cylindrical coordinates $(r, \theta, z)$ are an extension of the planar polar coordinate system into three dimensions, particularly advantageous for problems exhibiting axial symmetry.
    $r$: The radial distance from the $Z$-axis (the rotational axis).

  4. Spatial Coordinate

    Linked via "Cylindrical"

    $z$: The perpendicular distance from the $XY$-plane, analogous to the $z$-coordinate in Cartesian space.
    Conversion from Cartesian $(x, y, z)$ to Cylindrical $(r, \theta, z)$ is defined by:
    $$r = \sqrt{x^2 + y^2}$$
    $$\theta = \arctan\left(\frac{y}{x}\right)$$

  5. Spatial Coordinate

    Linked via "cylindrical coordinates"

    $\rho$ (rho): The radial distance from the origin to the point.
    $\phi$ (phi): The polar angle (colatitude), measured from the positive $Z$-axis. Ranges from $0$ to $\pi$.
    $\theta$ (theta): The azimuthal angle, measured in the $XY$-plane from the positive $X$-axis (longitude), identical to the $\theta$ in cylindrical coordinates.
    The transformation matrix for converting spherical coordinates to [Cartesian coo…