Curl: ∇×F measures rotation of a vector field. In 3D: (∂R/∂y−∂Q/∂z, ∂P/∂z−∂R/∂x, ∂Q/∂x−∂P/∂y). If ∇×F=0, field is irrotational (conservative). Stokes' theorem: ∮F·dr = ∬(∇×F)·dS.
How to Use
Select a vector field:
- F = (P,Q,R): Components
- Point: (x,y,z)
- Output: curl F at point
Physical Meaning
Curl measures local rotation. Imagine a tiny paddle wheel in fluid flow: curl tells how fast and which direction it spins. Zero curl = no rotation (laminar flow).
Key Theorems
- ∇×(∇f) = 0 always
- ∇·(∇×F) = 0 always
- Stokes: ∮F·dr = ∬(∇×F)·dS
- Conservative ⟺ ∇×F = 0
Step-by-Step Instructions
- 1Select a vector field.
- 2Enter point (x,y,z).
- 3View curl components.
- 4Check if irrotational.
- 5Analyze rotation direction.