Zero curl and gradient of some scalar potential

[redredred]

Can someone help me intuitively understand why if a field has zero curl then it must be the gradient of a scalar potential?

Thanks!

 

[lavinia]

Can someone help me intuitively understand why if a field has zero curl then it must be the gradient of a scalar potential?

Thanks!

If the curl is zero the field does not rotate so the field lines look like gradient flows.