🎬the Homogeneous Function Theorem for vector fields

This web application offers a practical demonstration of Euler's Homogeneous Function Theorem by allowing users to interact with vector fields of varying degrees of homogeneity. By selecting between radial ( $n=1$ ), quadratic ( $n=2$ ), or constant ( $n=0$ ) fields, users can visualize how different scaling factors influence the structure and density of a vector map. The app`s core utility lies in its real-time verification engine, which computes the directional derivative $(x \cdot \nabla)v$ and compares it directly against the theoretical product $n v$ . By providing an instant "Identity Check" for any given point, the tool transforms an abstract concept of multivariable calculus into a tangible, observable law, confirming that the radial rate of change for these fields is governed strictly by their degree of homogeneity.

🎬Narrated Video

Homogeneous Vector Field Viz, serves as a dynamic proof of Euler's Homogeneous Function Theorem by transforming abstract vector calculus into an interactive geometric experience. By manipulating the degree of homogeneity $n$ , you can observe how the radial "flow" of the vector field shifts from rapid expansion $(n>1)$ to intense convergence at the origin ( $n<0$ ), mimicking real-world forces like spring tension or gravity. The most significant takeaway is the visual verification of the analytical factor ( $n+d+1$ ); the demo shows that the divergence-or the "spreading out" of the modified field-is not arbitrary but is a fixed linear scaling of the field's radial projection. This confirms that the complex interaction between a vector field and the position operator $x$ results in a predictable, symmetric expansion governed entirely by the field's scaling power and the dimensionality of the space it occupies.

🎬Homogeneous Vector Field Viz

Electric Field Homogeneity Demo, contextualizes the abstract derivation within the physical framework of Coulomb's Law, demonstrating how the mathematical operation $x[x \cdot v]$ physically "softens" the intensity of a point charge. By transitioning from the standard electric field to the modified field $W$ , the visualization highlights a shift in radial decay from an inverse-square law $\left(1 / r^2\right)$ to a constant magnitude field, which effectively "unlocks" the divergence from being zero in vacuum to being non-zero throughout space. The key takeaway is the visual proof that the resulting flux density (divergence) becomes proportional to the local electrostatic potential; specifically, in a 2D environment where $n=$ -2 and $d=2$ , the scaling factor ( $n+d+1$ ) equals unity, meaning the flux density and the potential become one and the same. This illustrates that homogeneity isn't just a scaling property, but a fundamental constraint that dictates how energy and flux are distributed across a field's geometry.