A variable density field describes a region where matter concentration is not uniform but instead increases with the square of the distance from a central origin point. To determine the total mass within such a field, one must sum the unique density of every individual point within a given volume rather than using a single density value. Practical comparisons show that while a cube's mass is a product of base density and its side length cubed, its matter is thinnest at the origin and densest at its far corners. Conversely, a sphere centered at the origin features a uniformly dense outer shell because every point on its surface is equidistant from the center. This principle extends to complex geometries like ellipsoids, whose mass depends on axis lengths, and doughnut-shaped toruses, where the outer rim remains the densest section. Computational 3D models visualize these distributions by color-coding random points, using cool, dark colors for low-density centers and bright yellows to illustrate the "density glow" of outer regions, confirming that a container's shape fundamentally determines how mass is gathered and distributed.
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