Chapter 15.1, Problem 77E

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.)

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

b) $curl(\nabla f)=\nabla x(\nabla f)=0$

c) $\text{div(}\nabla f)=\nabla x(\nabla f)=0$

d) $\text{div(FxG})=(\text{curl}G-F(\text{curlG})$

e) $\nabla x[\nabla f+(\nabla xF)]=\nabla x(\nabla xF)$

f) $\nabla x(fF)=f(\nabla xF)+(\nabla f)xF$

g) $div(fF)=f(divF)+(\nabla f).F$

h) $div(curlF)=0$