The following are equivalent if the space is simply connected (no holes, one piece) and $U$ is not a function of time:

1. The force $\vec F$ is conservative.

2. $\nabla\times \vec F=0$

3. $W=\oint_\mathcal{C}\vec F\cdot\mathrm{d}\vec r=0$

4. $\vec F=-\nabla U$

Hope that it helps. (Might want to consider using another way than calculating integrals.)

The following are equivalent if the space is simply connected (no holes, one piece) and $U$ is not a function of time:

1. The force $\vec F$ is conservative.

2. $\nabla\times \vec F=0$

3. $W=\oint_\mathcal{C}\vec F\cdot\mathrm{d}\vec r=0\quad \forall\> \mathcal{C}$

4. $\vec F=-\nabla U$

Hope that it helps. (Might want to consider using another way than calculating integrals.)

The following are equivalent:

1. The force $\vec F$ is conservative.

2. $\nabla\times \vec F=0$

3. $W=\oint_\mathcal{C}\vec F\cdot\mathrm{d}\vec r=0$

4. $\vec F=-\nabla U$

Hope that it helps. (Might want to consider using another way than calculating integrals.)

The following are equivalent if the space is simply connected (no holes, one piece) and $U$ is not a function of time:

1. The force $\vec F$ is conservative.

2. $\nabla\times \vec F=0$

3. $W=\oint_\mathcal{C}\vec F\cdot\mathrm{d}\vec r=0$

4. $\vec F=-\nabla U$

Hope that it helps. (Might want to consider using another way than calculating integrals.)