Calculus
Gauss: from boundary to volume
Compare independent calculations: boundary flux and divergence inside the solid. Check normals, missing faces, cavities and singularities.
B=∬∂V F·n̂ dS; I=∭V div F dV; B=sI, s=±1
The shell’s outward normal points toward the centre on its cavity boundary. Reversing all normals changes B’s sign, not the geometric integral I. Visual cutaway does not alter integrated faces.
Boundary flux
Green: added boundary; pale: pending; purple: current cell; ochre: cavity boundary. Red: unit normal; dark green: F. Arrows are rescaled separately: do not compare lengths. The current cell/arrows remain visible on the back or cutaway. Drawn faces are approximate; weights are analytic.
Divergence in the volume
Interior sampling points, not moving particles: completed layers faint, current layer emphasized, current numerical cell purple. Orange: positive divergence; blue: negative; grey: zero. Wireframe shows the whole volume even when integration boundary pieces are missing.
Boundary integral B
Current boundary cell
Volume integral I
One cell in the current interior layer
Final difference B − sI
Contributions by boundary component
All boundary cells
All volume cells
Convergence comparison
Independent midpoint quadratures: n² cells per boundary component, n³ interior cells. One boundary term per step, proportional interior layers; partial sums need not agree. Case-specific exact references, not certified error bounds. No free expressions. Steps are not physical time.