Are Both Conditions Necessary to Confirm a Conservative Electric Field?

[Spoony]

I'm being taught electromagnetism at university, but there's one definition that has been left slightly ambiguous for an electric field to be conservative I've been taught that
1) \nablaxE = 0
But I've also been taught that
2) \oint E.dl = 0

But I am not sure if 1) & 2) have to be true for it to be conservitive.
OR that if 1) is true then 2) is true (and visa versa) ie, 1) \Leftrightarrow 2)

So do i have to test for both to check the field is conservitive, or just one and say it implies the other.

 

[cristo]

2. is a definition, which is equivalent to saying a vector field E is conservative if is can be written as the gradient of some scalar field.
1. can be shown to be equivalent to 2. by writing \mathbf{E}=\nabla\phi and taking the curl.

So, in short, no you don't need to test both. Most of the time it's a lot easier to calculate 1. to check whether the vector field is conservative.

 

[Spoony]

Thanks dude :)

 

[Hootenanny]

In addition to what cristo said, knowing that (2) is true for a conservative field can make later calculations easier, but as cristo said, (1) is usually easier to calculate. So if you're asked to prove whether a vector field is conservative it is usually best to use (1), but it is also useful to know (2) as well.