Gradient of a Function as a Surface Integral
\[\phi\]
, \[\mathbf{\nabla} \phi\]
at a point is the rate of change of \[\phi\]
in the direction normal to the surface \[S\]
defined by \[\phi = CONSTANT\]
Mathematically,
\[\mathbf{\nabla} \phi = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_S \phi \cdot \mathbf{n} dS\]
To show this use the identity
\[\int \int \int_{V} \nabla \phi dV = \int \int_S \phi \mathbf{n} dS \]
Taking the dot product with
\[\mathbf{i}\]
gives\[\int \int \int_{V} \nabla \phi \cdot \mathbf{i} dV = \int \int_S \phi \mathbf{n} \cdot \mathbf{i} dS \]
By the Mean Value Theorem for volumes,
\[\int \int \int_{\delta V} \mathbf{\nabla} \phi dV =(\mathbf{\nabla} \phi)_{(x_0 , y_0 , z_0)} \delta V \rightarrow (\mathbf{\nabla} \phi)_{(x_0 , y_0 , z_0)} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_S \phi \mathbf{n} dS, \: (x_0 , y_0 , z_0) \in \delta V \]
Hence
\[\mathbf{\nabla} \phi_{(x_0 , y_0 , z_0 )} \cdot \mathbf{i} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_S \phi \mathbf{n} \cdot \mathbf{i} dS, \: (x_0 , y_0 , z_0) \in \delta V\]
Similarly
\[\mathbf{\nabla} \phi_{(x_0 , y_0 , z_0 )} \cdot \mathbf{j} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_S \phi \mathbf{n} \cdot \mathbf{j} dS, \: (x_0 , y_0 , z_0) \in \delta V\]
\[\mathbf{\nabla} \phi_{(x_0 , y_0 , z_0 )} \cdot \mathbf{k} = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_S \phi \mathbf{n} \cdot \mathbf{k} dS, \: (x_0 , y_0 , z_0) \in \delta V\]
Then
\[\mathbf{\nabla} \phi = (\mathbf{\nabla} \cdot \mathbf{i} ) \mathbf{i} + (\mathbf{\nabla} \cdot \mathbf{j} ) \mathbf{j} + (\mathbf{\nabla} \cdot \mathbf{k} ) \mathbf{k}\]
\[\mathbf{n} = (\mathbf{n} \cdot \mathbf{i} ) \mathbf{i} + (\mathbf{n} \cdot \mathbf{j} ) \mathbf{j} + (\mathbf{n} \cdot \mathbf{k} ) \mathbf{k}\]
Hence
\[\mathbf{\nabla} \phi = lim_{\delta V \rightarrow 0} \frac{1}{\delta V} \int \int_S \phi \cdot \mathbf{n} dS\]
