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

https://youtu.be/2IDM3-W-gI4

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.

🎬Visualize how the gradient operator and the angular momentum operator act on a 3D scalar field

https://youtu.be/rA83EDv-bFw

🏗️Structural clarification of Poof and Derivation

block-beta
columns 6
CC["Criss-Cross"]:6

%% Condensed Notes

CN["Condensed Notes"]:6
RF["Relevant File"]:6
NV["Narrated Video"]:6
PA("Plotting & Analysis")AA("Animation & Analysis")KT("Summary & Interpretation") ID("Illustration & Demo") VA1("Visual Aid")MG1("Multigraph")

%% Proof and Derivation

PD["Proof and Derivation"]:6
AF("Derivation Sheet"):6
NV2["Narrated Video"]:6
PA2("Plotting & Analysis")AA2("Animation & Analysis")KT2("Summary & Interpretation") ID2("Illustration & Demo")VA2("Visual Aid") MG2("Multigraph")

classDef color_1 fill:#8e562f,stroke:#8e562f,color:#fff
class CC color_1

%% %% Condensed Notes

classDef color_2 fill:#14626e,stroke:#14626e,color:#14626e
class CN color_2
class RF color_2

classDef color_3 fill:#1e81b0,stroke:#1e81b0,color:#1e81b0
class NV color_3
class PA color_3
class AA color_3
class KT color_3
class ID color_3
class VA1 color_3

classDef color_4 fill:#47a291,stroke:#47a291,color:#47a291
class VO color_4
class MG1 color_4

%% Proof and Derivation

classDef color_5 fill:#307834,stroke:#307834,color:#fff
class PD color_5
class AF color_5

classDef color_6 fill:#38b01e,stroke:#38b01e,color:#fff
class NV2 color_6
class PA2 color_6
class AA2 color_6
class KT2 color_6
class ID2 color_6
class VA2 color_6

classDef color_7 fill:#47a291,stroke:#47a291,color:#fff
class VO2 color_7
class MG2 color_7

🗒️Downloadable Files - Recursive updates (Feb 10,2026)

‣

Copyright Notice

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]

</aside>