🎬the relationship between a position vector and a gradient vector for different scalar fields

This interactive 3D application provides a dynamic visualization of the relationship between a position vector and a gradient vector within various scalar fields. As a point moves along a pre-defined Lissajous curve, the app renders a red position vector and a blue gradient vector that updates in real-time to show the direction of steepest functional increase. Users can actively engage with the simulation by orbiting the camera, zooming via mouse or UI buttons, and switching between different mathematical landscapes-such as exponential, logarithmic, hyperbolic, or linear fields-while a dedicated information panel tracks the underlying formulas and vector coordinates.

🎬Narrated Video

The simulation illustrates how the mathematical operator $x \times \nabla$ transforms linear field gradients into orbital movement, serving as the physical generator of rotations. While the left panel shows the gradient ∇ acting as a generator of translations-pushing the field along straight paths of steepest change-the right panel demonstrates that the cross product with the position vector $x$ forces this flow into a perpendicular "swirl" or vortex around the origin. This visual "torque-like" behavior explains why $(x \times \nabla)$ is identified as the orbital angular momentum operator in quantum mechanics; it maps how a scalar field $\phi$ reorients itself under an infinitesimal rotation, with the non-zero commutator $\left[L_x, L_y\right]=i L_z$ arising because the field's final state depends strictly on the sequential order of the axes rotated.