Flux of a Fluid Out of a Surface

If the velocity field of a fluid is  
  the volume of fluid flowing through a surface drawn in the fluid per second is  
\[\int_S \mathbf{v} \cdot \mathbf{n} d S\]
, where  
  is a normal to the surface element  
Example: If a fluid has velocity field  
\[\mathbf{v} =\frac{x}{x^2 +y^2+z^2}\mathbf{i} + \frac{y}{x^2 +y^2+z^2}\mathbf{j} + \frac{z}{x^2 +y^2+z^2}\mathbf{k}\]
  the flux out of the surface  
  consisting of a sphere of radius  
  centred at the origin, with normal  
\[\mathbf{n} = \frac{x}{a} \mathbf{i} + \frac{y}{a} \mathbf{j} + \frac{z}{a} \mathbf{k} +\]
\[\begin{equation} \begin{aligned} \int_S \mathbf{v} \cdot \mathbf{n} dS &= \int^{\pi}_0 \int^{2 \pi}_0 (\frac{x}{x^2 +y^2+z^2}\mathbf{i} + \frac{y}{x^2 +y^2+z^2}\mathbf{j} + \frac{z}{x^2 +y^2+z^2}\mathbf{k}) \cdot (\frac{x}{a} \mathbf{i} + \frac{y}{a} \mathbf{j} + \frac{z}{a} \mathbf{k}) a^2 sin \theta d \phi d \theta \\ &= \int^{\pi}_0 \int^{2 \pi}_0 \frac{x^2 + y^2 +z^2}{a^3} a^2 sin \theta d \phi d \theta \\ &= 2 \pi \int^{\pi}_0 sin \theta d \theta \\ &= 2 \pi a [-cos \theta]^{\pi}_0 \\ &= 2 \pi a (-cos \pi - (-cos 0)) \\ &= 4 \pi aa \end{aligned} \end{equation} \]

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