🎬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

This Python demonstration provides a dynamic 3D visualization of electromagnetic wave propagation, serving as a physical manifestation of the vector identity $\nabla \times(\nabla \times E )=\nabla(\nabla \cdot E )-\nabla^2 E$ . By plotting the Electric (E) and Magnetic (B) fields as orthogonal sine waves along a propagation axis, the animation illustrates how the "swirl" (curl) of one field continuously regenerates the other in a self-sustaining feedback loop. The simulation specifically highlights the free-space condition where the divergence is zero, showing that the spatial curvature of the fields (the Laplacian) is perfectly balanced by their second-order temporal changes. This allows viewers to observe the phase-synchronized, transverse nature of light as it travels through a vacuum, effectively bridging the gap between abstract vector calculus and the physical reality of radiation.