Stokes Theorem

Stokes’ Theorem

Stokes theorem generalizes Greens Theorem and offers two approaches to calculate the Circulation of a vector field along a surface.

• Using the boundary-curve itself as parametrized function
• Using a surface integral over the surface with the boundary-curve in question as boundary with the vector fields curl around it.

$$\oint \_CF\cdot dr=\iint \_S(\nabla \times F)\cdot \vec{n}d\sigma$$

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• If two different oriented surfaces $S\_1$ and $S\_2$ have the same boundary $C$, their curl integrals are equal
• The surface normal vector $\vec{n}$ is calculated based on how the function is given
• The surface differential $d\sigma$ is calculated also based on how the function is given

Important Identities

Curl Gradient = 0

$$\text{curl grad}f=0\text{or}\nabla \times \nabla f=0$$

Closed Loop Property If $\nabla \times F=0$ at every point of a simply connected region $D$, then on any piecewise-smooth closed path $C$ in $D$

$$\oint \_CF\cdot dr=0$$