Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates

The orthogonality of basis vectors is a fundamental property of cylindrical and spherical coordinate systems. This means their basis vectors are always mutually perpendicular, a fact that is mathematically proven by their dot products consistently equaling zero. This property is critical in physics and engineering as it greatly simplifies complex vector operations and calculations.

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✍️Mathematical Proof

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Orthogonality of Basis Vectors

The core of the verification process is demonstrating that the dot product of any two distinct basis vectors within a coordinate system equals zero.

Cylindrical Coordinates

The tangent vector basis $\left(E_\rho, E_\phi, E_z\right)$ for cylindrical coordinates is proven to be orthogonal. This is shown by the dot products $E_\rho \cdot E_\phi=0, E_\rho \cdot E_z=0$, and $E_\phi \cdot E_z=0$.

Spherical Coordinates

Similarly, the tangent vector basis $\left(E_r, E_\theta, E_{\varphi}\right)$ for spherical coordinates is also verified to be orthogonal. This is confirmed by the dot products $E_r$. $E_\theta=0, E_r \cdot E_{\varphi}=0$, and $E_\theta \cdot E_{\varphi}=0$.

Importance of Orthonormal Cartesian Basis

The verification relies on the fact that the original Cartesian basis vectors ( $e_1, e_2, e_3$ ) are orthonormal, meaning they are mutually perpendicular and have a magnitude of one. This property allows for the straightforward cancellation of terms in the dot product calculations.

Significance in Physics and Engineering

The orthogonality of these bases is a fundamental property that simplifies many mathematical and physical problems, such as vector calculus operations (e.g., gradient, divergence, curl) and solving partial differential equations in these coordinate systems.

✍️Mathematical Proof

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🧄Proof and Derivation-2

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