Del

$\nabla =⟨{\partial }_x,{\partial }_y,{\partial }_z⟩$

• ${\partial }_x=\frac{\partial }{\partial x}$

$\nabla =⟨{\partial }_r,\frac{{\partial }_{\theta }}{r},\frac{{\partial }_{\theta }}{r\sin \theta }⟩$

• Spherical coordinates version

Topics

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

Properties

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

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