How does Stokes' formula connect the circulation of a vector field around a surface boundary to the flux of its curl through the surface?

Conditions

• The surface $S$ is an oriented piecewise smooth surface.
• The vector field $\vec{F}$ is $C^1$ near the surface.
• The boundary orientation of $C$ is induced by the chosen normal of $S$ via the right-hand rule.

Reasoning, step by step

1. Identify the oriented surface $S$ and its closed boundary curve $C$.
2. Compute the line integral of $\vec{F}$ along $C$ to find the circulation.
3. Compute the curl of $\vec{F}$, denoted $\nabla \times \vec{F}$.
4. Determine the oriented unit normal vector $\hat{n}$ for the surface $S$.
5. Evaluate the surface integral of $(\nabla \times \vec{F}) \cdot \hat{n}$ over $S$.
6. Equate the two results according to Stokes' formula.

Example

The script states: 'Stokes’ formula equates the line integral of F around C with the surface integral of curl F dotted with the oriented unit normal. One accumulates tangential field components; the other accumulates normal components of curl.'

Common misconceptions

• Believing that the base-plane projection shown in visualizations is automatically the surface $S$ used in the theorem.
• Assuming the normal vector is constant across a curved surface.
• Confusing the accumulation of tangential components (circulation) with normal components (flux of curl).

Watch the explanation

Explore next

• Why must the boundary orientation follow the right-hand rule?
• What happens if the surface is not piecewise smooth?
• How does the choice of surface affect the calculation if the boundary remains the same?