Are the following statements true or false?

1. ∫cf⃗ ⋅dr⃗ =f(q)−f(p)∫cf→⋅dr→=f(q)−f(p) whenever pp and qq are the endpoints of the curve cc.
2. if ∫cf⃗ ⋅dr⃗ =0∫cf→⋅dr→=0 for one particular closed path, then f⃗ f→ is path-independent.
3. if f⃗ f→ is path-independent, then there is a potential function for f⃗ f→.
4. if ∫cf⃗ ⋅dr⃗ ≠0∫cf→⋅dr→≠0 for some closed path cc, then f⃗ f→ is path dependent.
5. if the vector fields f⃗ f→ and g⃗ g→ have ∫cf⃗ ⋅dr⃗ =∫cg⃗ ⋅dr⃗ ∫cf→⋅dr→=∫cg→⋅dr→ for a particular path cc, then f⃗ =g⃗ f→=g→.
6. if f⃗ f→ is path-independent and cc is any closed curve, then ∫cf⃗ ⋅dr⃗ =0∫cf→⋅dr→=0.
7. if f⃗ =∇ff→=∇f, then f⃗ f→ is conservative.
8. the circulation of any vector field f⃗ f→ around any closed curve cc is zero.