This project synthesizes four distinct demonstrations to explore the behavior of weighted radial vector fields. The first demo establishes the analytical foundation, proving that for $\vec{x} \neq 0$, the surface flux of a $1/r^5$ weighted field maps to a volume integral with the scalar field $\phi(\vec{x}) = -1/r^5$. The second demo highlights the critical role of the origin, illustrating why the $1/r^5$ weighting leads to a divergent integral unlike the standard $1/r^3$ Gauss’s Law case. The third and fourth demos provide interactive 3D and high-contrast 2D visualizations, respectively, demonstrating how the mathematical identity holds true as long as the volume of integration maintains an "exclusion zone" around the singularity. Together, these demos bridge the gap between abstract vector calculus identities and the physical intuition of potential wells and flux conservation.
stateDiagram-v2
[*] --> Analytical_Derivation : Apply Divergence Theorem
Analytical_Derivation --> Origin_Check : Define Volume V
state Origin_Check <<choice>>
Origin_Check --> Divergence_State : Origin (x=0) is Included
Origin_Check --> Valid_Simulation_3D : Origin (x=0) is Excluded
state Divergence_State {
[*] --> Comparative_Demo
Comparative_Demo --> Result_Infinite : $$1/r^5\\ $$ grows too fast
}
state Valid_Simulation_3D {
[*] --> Spatial_Navigation
Spatial_Navigation --> Color_Magnitude_Mapping : Add Scalar Mappable
}
Valid_Simulation_3D --> Dimensional_Translation : Shift to 2D System
state Dimensional_Translation {
[*] --> Area_Integral_Mapping
Area_Integral_Mapping --> High_Contrast_Visibility : Implement Inferno/Grey contrast
}
Result_Infinite --> [*]
High_Contrast_Visibility --> [*]
classDef darkFill fill:#000,stroke:#333,stroke-width:2px,color:#fff,font-size:15pt
class Analytical_Derivation,Origin_Check,Spatial_Navigation,Color_Magnitude_Mapping,Comparative_Demo,Result_Infinite,Area_Integral_Mapping,High_Contrast_Visibility darkFill
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
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©️2026 Sayako Dean
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