Chapter 15.1, Problem 77E

Proof In parts (a) - (h), prove the property for vector fields F and G and scalar function f. (Assume that the required partial derivatives are continuous.)

$\text{curl(}F+G\text{)}=\text{curl }F+\text{curl}G$

$\text{curl}(\nabla f)=\nabla \times (\nabla f)=0$

$\text{div(}F+G\text{)}=\text{div}F+\text{div}G$

$\text{div(}F\times G\text{)}=(\text{curl}\text{}F)·G-F·(\text{curl}G)$

$\nabla \times [\nabla f+(\nabla \times \text{F})]=\nabla \times (\nabla \times \text{F})$

$\nabla \times (fF)=f(\nabla \times F)+(\nabla f)\times F$

$\text{div}(fF)=f\text{div}\text{F}+\nabla f·F$

$\text{div}(\text{curl}F)=0$