🧄Verification of Vector Calculus Identities in Different Coordinate Systems

Divergence and curl represent physical properties of a vector field-specifically the "spreading out" and "rotation"-that remain invariant regardless of the coordinate system used for calculation. For the position vector $x$, the divergence consistently equals 3 , reflecting the fact that the field expands uniformly in three dimensions ( 1+1+1). Meanwhile, the curl is consistently 0 , confirming that the position vector is a radial, irrotational field. While the mathematical expressions for these operators become more complex in cylindrical and spherical systems due to the inclusion of scale factors like $\rho, r$, and $\sin \theta$, they ultimately yield identical results to the simpler Cartesian derivatives, demonstrating the consistency of vector calculus across different geometries.

🎬Narrated Video

🎬Analysis of Vector Field Dynamics-Position vs. Gravitation

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📢Coordinate Invariance and the Immutable Properties of Vector Fields Across Geometric Systems

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🧣Vector Calculus Identities and Fields (VCI-F)

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Verification of Vector Calculus Identities in Different Coordinate Systems | Proof and Derivationviadean on Notion