1. Definition
The vector line integral ∫_C P dx+Q dy = ∫(P x'+Q y')dt is the work done by a force; it depends on orientation and flips sign if the path is reversed.
2. Symbols
| (P,Q) | vector field |
| path | x(t),y(t) |
| dx,dy | coordinate differentials |
| orientation | flips sign |
3. How it works
- Evaluate field and tangent;
- Dot them;
- Integrate by Simpson.
4. Steps
- Enter P, Q and path;
- Enter t range;
- Click Calculate.
5. Example
Example: F=(xy,x²) along x=t,y=t², t:0→1.
Solution: ∫₀¹3t³dt=0.75.
6. Pitfalls
Orientation matters—reversing flips sign;
Positive closed curve usually counterclockwise;
Compare with Green's theorem.