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= $\vec 0$