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.
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.
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.
This state diagram illustrates the progression from fundamental mathematical derivation to generalized dimensional theory and, finally, to specific physical application, as detailed in the sources.
stateDiagram-v2
[*] --> Demo_1_Interactive_Verification
state Demo_1_Interactive_Verification {
Dropdown: n = 0, 1, or 2
Verification: LHS vs RHS check
}
Demo_1_Interactive_Verification --> Example_1_Generalization
state Example_1_Generalization {
Dimensions: 3D to d-dimensions
General_Factor: (n + d + 1)
}
Example_1_Generalization --> Demo_2_Scaling_Visualization
state Demo_2_Scaling_Visualization {
Interactive: Adjust n and d
Bridge: Quiver plots for v(x)
}
Demo_2_Scaling_Visualization --> Example_2_Physics_Application
state Example_2_Physics_Application {
Physical_Field: Point Charge (n = -2)
Identity_Check: Factor = 2 in 3D
}
Example_2_Physics_Application --> Demo_3_Field_Transformation
state Demo_3_Field_Transformation {
Steepness: 1/r² to constant 1/r⁰
Flux: Non-zero divergence "glow"
}
Demo_3_Field_Transformation --> [*]
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