Cylindrical has one scaled tangent direction while Spherical has two (CST-ST)

When comparing these two dynamic bases is their scale factors (the magnitude of the tangent vectors) and how many vectors change length based on position. While both systems have basis vectors whose directions change with angle, the fundamental distinction is the number of scaled tangent vectors. Cylindrical Coordinates: Only the azimuthal tangent vector is scaled by a variable coordinate, the Radial Distance. The radial and vertical directions are always unit length (magnitude 1). Spherical Coordinates: Two of the three tangent vectors are scaled: The Polar Tangent Vector is scaled by Spherical Radial Distance, and The Azimuthal Tangent Vector is scaled by the Radius in the XY-plane. Only the radial vector is unit length. Cylindrical has one scaled tangent direction, while Spherical has two that makes the mathematics slightly more complex in spherical coordinates because two angular derivatives involve coordinate-dependent scale factors.

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🏗️Structural clarification of Poof and Derivation

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