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Stokes' Theorem

Source lecture(s): PC604 Lec1-2

Intuition

Circulation around a loop equals the integral of curl over any surface it bounds.

Formal Definition

\[\oint_C \vec{A}\cdot d\vec{r}=\int_S (\nabla\times\vec{A})\cdot d\vec{S}\]

Mathematical Formulation

\[\oint_C \vec{A}\cdot d\vec{r}=\int_S (\nabla\times\vec{A})\cdot d\vec{S}\]

Derivation

Divide the surface into small loops; interior edges cancel, leaving only the outer boundary.

Worked Example

For \(\vec{A}=-y\hat{i}+x\hat{j}\) around the unit circle, \(\oint \vec{A}\cdot d\vec{r}=2\pi\), matching \(\int_S 2\hat{k}\cdot d\vec{S}=2\pi\).

Common Mistakes

  • Using the wrong orientation between loop and surface normal.
  • Applying to surfaces with holes.

Quiz

Q1. Does Stokes' theorem apply to an open curve?

Answer

No—the curve must be closed.