Calculus
Stokes: circulation and curl
Compute boundary circulation and surface curl flux independently. Match normal and boundary directions, compare different surfaces and check assumptions.
L=∮∂S F·dr; I=∬S rot F·n̂ dS
Standard normal: upward on planar/curved graphs, outward on hemisphere/sphere. Standard boundary is outer counterclockwise and inner clockwise from above. Controls are independent: reverse both to preserve compatibility.
∂S is empty: there is no path to traverse. Its boundary integral is 0; animation sums cells of the closed surface.
Boundary and field F
Surface and curl of F
Green: added surface; pale: pending; purple: current cell/strip. Outer boundary blue, inner ochre. Left: dark-green F, blue tangent. Right: red unit normal, orange curl. Arrows rescaled separately: do not compare lengths across types. Current cells/arrows remain visible at the back; faces are approximate, weights come from derivatives.
Same boundary, three surfaces
Same R, field F and circular boundary z=0. Areas are analytic references; L and I are independent quadratures. Standard normal is upward (outward on the hemisphere).
Boundary circulation L
Current boundary contribution
Curl flux I
One cell in the current surface portion
Final difference L − I
Boundary contributions
All line contributions
All surface cells
Convergence comparison
Independent midpoint quadratures: n samples per boundary piece and n² surface cells. A boundary step advances proportional strips; partial sums need not agree. On the boundaryless sphere cells are added directly. Guided cases, not a general solver; error comparisons use analytic references, not certified bounds. No free expressions. Steps are not physical time.