Calculus
Green: from boundary to region
Compute boundary and region contributions separately. Check orientation, inner boundaries and regularity before comparing sums.
Positive boundary: region on the left, outer counterclockwise and inner clockwise. Reversing all directions gives s=−1: geometric I stays unchanged, comparable sI changes.
In flux form the normal points to the right of traversal: outward for positive orientation, inward after reversal. The outward normal is not held fixed when reversing.
Boundary and its contributions
Blue: outer boundary; orange: inner boundary; purple: current element. Field arrows share one scale within the view.
Region and area cells
Light cells: region; green: contributions already added; purple: current strip. Drawn cells are approximate; numerical weights use the stated Jacobian.
Boundary integral B
Current contribution
Region integral I
One cell in the current strip
Numerical difference B−sI
Contributions by boundary / side
Boundary contributions table
Region cells table
Comparison as n increases
Independent midpoint quadratures: n samples per boundary piece and n² parameter cells inside. One boundary contribution per step; region strips progress proportionally. Partial sums need not agree: these pieces are not the same closed boundary. Exact references are preset-specific, not certified error bounds. No free expressions are executed.