All Modules
University Math Lab: Systems & Structures FREE
Loading simulation...
Curl & Stokes' Theorem
Learning Objectives
- Compute curl of vector fields
- Apply Stokes' Theorem
- Identify conservative fields
Key Concepts
Curl
curl F⃗ = ∇ × F⃗
Stokes' Theorem
∮_C F⃗·dr⃗ = ∫∫_S (∇×F⃗)·dS⃗
Theory
**Curl:** ∇ × F⃗ = |i j k; ∂/∂x ∂/∂y ∂/∂z; F₁ F₂ F₃|
= ⟨∂F₃/∂y - ∂F₂/∂z, ∂F₁/∂z - ∂F₃/∂x, ∂F₂/∂x - ∂F₁/∂y⟩
**Interpretation:** Measures rotation of field at a point.
**Conservative field:** curl F⃗ = 0⃗ everywhere ⟺ F⃗ = ∇φ for some potential φ.
**Stokes' Theorem:** ∮_C F⃗·dr⃗ = ∫∫_S (∇×F⃗)·dS⃗
Examples
curl of F⃗ = ⟨yz, xz, xy⟩
Solution: ⟨x-x, y-y, z-z⟩ = ⟨0,0,0⟩ — conservative!