1. Theorem
Stokes' theorem: circulation around closed curve C equals the flux of curl through spanning surface S: ∮_C F·dr = ∬_S (curl F)·n dS.
2. Symbols
| C | closed boundary |
| S | spanning surface |
| curl F | curl |
| n | unit normal |
3. How it works
- For a flat rectangle with normal k;
- Compute curl_z=Q_x−P_y;
- 2D Simpson integration;
- Equals the boundary circulation.
4. Steps
- Enter P, Q, R;
- Enter the rectangle;
- Click Calculate.
5. Example
Example: F=(−y,x,0) over [0,1]².
Solution: curl_z=2, area=1, so ∬curl·n=2, the boundary circulation.
6. Pitfalls
This tool uses normal k;
Normal and boundary obey the right-hand rule;
Circulation = curl flux.