Line integral: ∫C F·dr = ∫ₐᵇ F(r(t))·r'(t)dt. Scalar: ∫C f ds = ∫ₐᵇ f(r(t))|r'(t)|dt. Work done by force along path. Conservative fields: ∫C ∇f·dr = f(B)−f(A) (path independent). Green's theorem converts to double integral.
How to Use
Select curve and field:
- C: Parametric curve r(t)
- F: Vector field
- Output: ∫C F·dr
Conservative Fields
F = ∇f ⟺ ∫C F·dr path-independent ⟺ ∮ F·dr = 0. Test: ∂F₁/∂y = ∂F₂/∂x (2D). If conservative, find potential f and use f(B)−f(A).
Applications
- Work: W = ∫C F·dr
- Circulation: ∮C F·dr
- Arc length: ∫C 1 ds
- Mass of wire: ∫C ρ ds
Step-by-Step Instructions
- 1Select curve.
- 2Select field.
- 3Parameterize.
- 4Evaluate integral.
- 5Get result.