Integrating dot Product of Radial and Normal Vectors Over a Surface

To integrate  
\[\mathbf{r} \cdot \mathbf{n}\]
  over a surface we can use the divergence theorem.
\[\begin{equation} \begin{aligned} \int \int_S \mathbf{r} \cdot \mathbf{n} dS &= \int \int \int_V \mathbf{\nabla} \cdot \mathbf{r} dV \\ &= \int \int \int_V (\frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k} ) \cdot (x \mathbf{i} + y \mathbf{j} + z \mathbf{k} ) dV \\ &= \int \int \int_V 3 dV = 3V \end{aligned} \end{equation}\]

Example: If  
  is a cube of side 3,
\[\int \int_S \mathbf{r} \cdot \mathbf{n} dS = 3 \times 3^3 =81\]

Example: If  
  is a sphere of radius 2,
\[\int \int_S \mathbf{r} \cdot \mathbf{n} dS = 3 \times \frac{4}{3} \pi \times 2^3 = 32 \pi\]

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