Divergence Free Vector Field
- Divergence of \(u\): \(div(u)=\nabla\cdot u=\frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial u_y}+\frac{\partial u_z}{\partial u_z}\)
- Divergence free / solenoidal field : \(div(u)=0\)
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- Ex. Magnetic fields, incompressible fluids velocity, are divergent free
- \(\Rightarrow\) Volume preserving deformation
Dem. (in 2D)
- \(f = I + u\)
- \(\Rightarrow J_f = \left(\begin{array}{ccc} \frac{\partial f_x}{\partial x} & \frac{\partial f_x}{\partial y} \\ \frac{\partial f_y}{\partial x} &
\frac{\partial f_y}{\partial y} \end{array}\right) = \left(\begin{array}{ccc} 1+\frac{\partial u_x}{\partial x} & \frac{\partial u_x}{\partial y} \\
\frac{\partial u_y}{\partial x} & 1+\frac{\partial u_y}{\partial y} \end{array}\right) \)
- Volume preserving deformation \(det(J_f)=1\)
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- \(\Rightarrow\left(1+\frac{\partial u_x}{\partial x}\right)\left(1+\frac{\partial u_y}{\partial y}\right)-\frac{\partial u_x}{\partial
y}\frac{\partial u_y}{\partial x}=1\)
- \(\Rightarrow \frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}=0\)
