True or false questions about Divergence and Curl

[LCSphysicist]

Homework Statement

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Relevant Equations

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$F = (P,Q,R)$ is a field of vector C1 defined on $V = R3-{0,0,0}$

There are a lot of true or false statement here. I am a little skeptical about my answer because it contains a lot of F, but let's go.

1 Rot of F is null in V iff $\int \int_{S} P dx + Q dy + R dz = 0$ for all sphere S belonging to V oriented outside.

2 F is conservative in V iff rot of F is null in V

3 P,Q and R are positive iff $\int \int_{S} P dx + Q dy + R dz > 0$ for all sphere S belonging to V oriented outside.

4 F is gradient in V iff $\int \int_{S} P dx + Q dy + R dz = 0$ for all sphere S belonging to V oriented outside.

5 F is null in V iff $\int \int_{S} P dx + Q dy + R dz = 0$ for all sphere S belonging to Voriented outside.

6 rot of F is null in V iff $\int \int_{C} P dx + Q dy + R dz = 0# for all circumference oriented with C belonging to V 7 P,Q,R are positie iff$\int \int_{C} P dx + Q dy + R dz>0$for all circumference C belonging to V 8 F is gradient in V iff$\int \int_{C} P dx + Q dy + R dz = 0$for all circumference C belonging to V 9 F is null in V iff$\int \int_{C} P dx + Q dy + R dz = 0## for all circumference C belonging to V

To be pretty honest, i am not sure if the author really want to means "if" instead of "iff", or if he commited a typo writing "circumference C" instead of "closed curve C". But in the way it is, i found too much F! see

1 F
2 T
(3 ... 9) F

So only the 2 seems true to me, since R3 minus origin is simply connected and the vector is C1.
All the others answer would be really different if C were just a closed curve, not a circumference. Or if S were a closed surface, not a sphere.
I have some doubts, are my answers right? Or there is some special proeprty about circumference that i am missing that changes all the answer.

 

[Delta2]

First of all for statements 1,3,4,5 do you mean the surface integral $$\iint_S \vec{F}\cdot \vec{n} dS$$ and for statements 6,7,8,9 do you mean the line integral $$\oint_C\vec{F}\cdot d\vec{r}$$?

Given that you mean the above, I agree with your answers and also I agree that some would be True if instead of "any sphere S" we had "any closed surface S" or instead of "any circumference C" we had "any closed curve C"