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Divergence and Curl

Divergence #

$$ \text{div }\vec v=\vec \nabla \cdot \vec v $$

If $\text{div }\vec v$=0, then $\vec v$ is called solenoidal

Physical Interpretation #

Consider a fluid flow with $\vec v=v_x(x,y,z)+v_y(x,y,z)+v_z(x,y,z)$

Let’s calculate the change in fluid flux in unit volume

$$ \begin{align*} \Delta_x=(v_x(x+\delta x,y,z))dydz-(v_x(x,y,z))dydz\\ =(v_x(x,y,z)+\frac{\partial v_x}{\partial x}\delta x)dydz-(v_x(x,y,z))dydz\\ =\frac{\partial v_x}{\partial x}dxdydz\\ \Delta_y=\frac{\partial v_y}{\partial y}dxdydz\\ \Delta_z=\frac{\partial v_z}{\partial z}dxdydz\\ \\ \Delta=\Delta_x+\Delta_y+\Delta_z\\ =(\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial ŷ}+\frac{\partial v_z}{\partial z})dxdydz\\ \frac{\Delta}{dV}=\text{div }\vec v \end{align*} $$

If $(\text{div }\vec v)_P$

  • > 0: P is a source
  • < 0: P is a sink

Curl #

$$ \text{curl }\vec v=\vec \nabla \times \vec v $$

If curl $\vec v=\vec 0$ then $\vec v$ is called irrotational

If a vector field is irrotational, then $\exists \phi \text{ such that }\vec v=\vec \nabla \phi$ where ϕ is a scalar potential field. Such a vector field is called conservative.

∴ curl of a gradient of a scalar function= $\vec 0$