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.

🎬Demonstration

Orthogonal coordinate systems have basis vectors that are always perpendicular to each other, regardless of the angles. This is visually represented by the dot product of any two distinct basis vectors always being equal to zero. The demo clearly shows this by calculating and displaying the dot products in real-time as you adjust the angles, demonstrating the fundamental geometric properties of cylindrical and spherical coordinate systems.

how the cylindrical and spherical basis vectors remain perpendicular to each other regardless of their position