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There are several things I need to clarify on Curl.

1)
Is the conservativeness of a gradient field only applicable for a Closed curve?
If the field is gradient and if c (curve) is not closed then $\int F.dr$ may not be zero? So I can't say a gradient field is always conservative.can I?
What it says is that if $\overrightarrow F$ is a gradient Field then work done along any closed curve will be zero.

2) It says that,
Let Vector Field F =$ M \hat i+N \hat j$ where $M= \partial f/\partial x =f_x $ and $ N=\partial f/\partial y =f_y $. Thus if $\overrightarrow F= \nabla f$ (F is a gradient field) then $M_y = N_x$. ----(1)

By definition curl $\overrightarrow F$ = $N_x - M_y$ . ---(2)

My question is By (1) and (2) does it imply that if $\overrightarrow F$ is a gradient field then curl $\overrightarrow F$ = 0.
But curl $\overrightarrow F$ =0 desn't imply that field is conservative(gradient ) does it? Because to say that when curl $\overrightarrow F$=0 that the field is conservative , the field should be defined everywhere in the region ,

3) Curl is used on vector fields not necessarily gradient fields right ?

Please tell me if I have made any wrong conclusions.

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please note:
1.if field is conservative, then $\int F.dr$ is path independent and depends only on end points, now this implies that if path is closed then obviously $\int F.dr$=0
2. if $\overrightarrow F$ is conservative then curl$\overrightarrow F$ = 0 and vice versa is also true
3. since curl is cross product , so curl is only defined for vector fields

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