# Verification of Vector Calculus Identities in Different Coordinate Systems

A vector field, such as its expansion or compression (divergence) and its rotation (curl), are intrinsic to the field itself and don't depend on the coordinate system used to describe them. Even though the formulas for divergence and curl look vastly different in Cartesian, cylindrical, and spherical coordinates, applying them to the same vector field will always give the same result.

### :clapper:the analysis of the divergence and curl of the position vector

> It demonstrates that the numerical results for the divergence and curl of the position vector are coordinate-independent. The animation's pulsing vectors show the outward flow and lack of rotation, and the changing text explicitly links this visual behavior to the consistent mathematical results across all three coordinate systems.

{% embed url="https://youtu.be/o-B8bm2l1kQ" %}

### :pen\_ballpoint:Mathematical Proof

{% embed url="https://viadean.notion.site/Verification-of-Vector-Calculus-Identities-in-Different-Coordinate-Systems-25a1ae7b9a32803db61dd36b3118ca50?source=copy_link" %}
