Cross, Del and Dot-Product Identities

Del Multiplications

• $\nabla (fg)=f\nabla g+g\nabla f$
• $\nabla (A\cdot B)=A\times (\nabla \times B)+B\times (\nabla \times A)+(A\cdot \nabla )B+(B\cdot \nabla )A$
• $\nabla (fA)=f(\nabla \cdot A)+A\cdot (\nabla f)$
• $\nabla (A\times B)=B(\nabla \cdot A)-A(\nabla \times B)$

Cross

• $\nabla \times (fA)=f(\nabla \times A)-A\times (\nabla f)$
• $\nabla \times (A\times B)=(B\cdot \nabla )A-(A\cdot \nabla )B+A(\nabla \cdot B)-B(\nabla \cdot A)$

Divisions

• $\nabla \left(\frac{f}{g}\right)=\frac{(\nabla g)f-(\nabla f)g}{g^2}$
• $\nabla \left(\frac{A}{g}\right)=\frac{\nabla g\cdot A-(\nabla \cdot A)g}{g^2}$
• $\nabla \times \left(\frac{A}{g}\right)=\frac{g(\nabla \times A)+A\times (\nabla g)}{g^2}$