Del

#Math

$\displaystyle \nabla={\left\langle{\partial_{x},\partial_{y},\partial_{z}}\right\rangle}$

• $\displaystyle \partial_{x}=\frac{\partial }{\partial x}$

$\displaystyle \nabla={\left\langle{\partial_{r},\frac{\partial _{\theta}}{r},\frac{\partial _{\theta}}{r\sin \theta}}\right\rangle}$

• Spherical coordinates version

Topics

• Gradient, Divergence, and Curl in General Coordinates
• Gradient
• Divergence
• Curl
• Advection

Properties

$\displaystyle \nabla(\vec{a}\cdot \vec{b})=\vec{a}\times(\nabla \times \vec{b})+\vec{b}\times(\nabla \times \vec{a})+(\vec{a}\cdot \nabla)\vec{b}+(\vec{b}\cdot \nabla)\vec{a}$

$\displaystyle \nabla(f\vec{a})=f(\nabla\cdot \vec{a})+\vec{a}\cdot(\nabla f)$