The Curl of a Vector in Terms of a Surface Integral
\[\mathbf{F}\]
, written \[\mathbf{\nabla} \times \mathbf{F}\]
in terms of a surface integral: \[\mathbf{\nabla} \times \mathbf{F}= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{ \delta S} \mathbf{n} \times \mathbf{F} dS\]
, where \[\delta S\]
is the surface of the volume \[\delta V\]
To show this we use the identity
\[\int \int \int_{\delta V} \mathbf{\nabla} \times \mathbf{F} dV = \int \int_{\delta S} \mathbf{n} \times \mathbf{F} dS\]
Take the dot product of this identity with
\[\mathbf{i} , \: \mathbf{j}, \: \mathbf{k}\]
respectively to give\[\int \int \int_{\delta V} (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{i} dV = \int \int_{\delta S} (\mathbf{n} \times \mathbf{F}) \cdot \mathbf{i} dS\]
\[\int \int \int_{\delta V} (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{j} dV = \int \int_{\delta S} (\mathbf{n} \times \mathbf{F}) \cdot \mathbf{j} dS\]
\[\int \int \int_{\delta V} (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{k} dV = \int \int_{\delta S} (\mathbf{n} \times \mathbf{F}) \cdot \mathbf{k} dS\]
The Mean Value Theorem for Volumes states
\[\int \int \int_{\delta V} (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{i} dV = \delta V ((\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{i})_{(x_0 , y_0 , z_0 )}\]
for some point \[(x_0 , y_0 , z_0 ) \in \delta V\]
Le
\[\delta V \rightarrow 0\]
then \[\mathbf{\nabla} \times \mathbf{F}= lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_{ \delta S} \mathbf{n} \times \mathbf{F} dS\]
