🎬Visualize three resulting scalar fields-Divergence and Curl magnitude and Laplacian

This demonstration visualizes the relationship between a vector field and its fundamental differential operators, as expressed in the identity $\nabla \times(\nabla \times v )=\nabla(\nabla \cdot v )-\nabla^2 v$. The specific field, $v = \left\langle A x y, B x^2\right\rangle$, is constructed to isolate the effects of the parameters $A$ and $B$. Panel 1 shows that the Divergence $(\nabla \cdot v )$ is proportional to $y$ (controlled by $A$ ), demonstrating shear flow and stretching/compression along the vertical axis. Panel 2 shows the Curl Magnitude ( $|\nabla \times v |$ ), which is proportional to $x$ and dependent on both $A$ and $B$, illustrating the rotational component strongest away from the central axis. Crucially, Panel 4 reveals that the Vector Laplacian Magnitude $\left(\left|\nabla^2 v \right|\right)$ is a uniform, non-zero constant $|2 B|$ across the entire domain only when $B \neq 0$, confirming that this specific vector field possesses a uniform total curvature due only to its $x^2$ component.

🎬Narrated Video

🎬Visualize the derived wave equation showing the orthogonal relationship between electric and magnetic fields

🧵Related Derivation

🧄Double Curl Identity Proof using the epsilon-delta Relation (DCI-EDR)

⚒️Compound Page

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