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

https://youtu.be/raBAkHzeJjY

🗄️Example-to-Demo

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🏗️Structural clarification of Poof and Derivation

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🗒️Downloadable Files - Recursive updates

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All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

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