I can do part a)
Part b) says hence or otherwise, do I have to use my results from part a somehow?
and part c) I am not familiar with these theorems so I have no idea.
Some help would be appreciated, thanks.
Do you know what the integral over a closed loop in a conservative field is? That will probably simplify some of (b)
Other wise just do the line integral the old fashioned way.
Are you currently in a class, or self studying? I think you may be expected to know what Greens and Stokes theorems are. Look em up, they are your friends as far as simplifying problems like this.
I know the integral over a closed loop in a conservative field is zero! Sorry I'm not making these connections in my head for some reason until someone points them out, I've been redlining my brain with stress trying to get this work finished.
Well as your problems are in 3D Green's theorem won't apply but Stoke's theorem is a 3D form of Green's theorem which could be of use here.
Basically what it lets you do is convert a closed line integral in a field to the surface integral of the surface enclosed by that loop of the divergence of your field.
You're not going to be able to see where it would help until you start to push the pencil to solve the loop integral. You might notice your loop is independent of z and thus your surface will be as well.
Another thing to quickly do is take the divergence of the non-conservative fields above. That will help you see where Stoke's Theorem can help. The fact that he's talking about using Green's theorem at all makes me think he's using the independence of z to reduce the problem to a 2 dimensional one. You might look at that as well.
The only field that is not conversative is F2, but I just done the integral f.dr for F1 and it turned out to be -2pi? Did I do an error in my calculations or is not always 0 for a conservative field?
Then by the loop integral I assume you meant that same thing f.dr in part B? So does that mean that Stokes or Greens can only apply to F2?
If you got a non-zero result for a closed loop integral in a conservative field then yes you made a calculation error somewhere.
Stokes/Green theorem will apply for your conservative fields but why bother? You know the answer is zero there. So yes you wouldn't break out the Stokes/Green machinery until you have to do the integral in a non-conservative field.
For that matter..... (you'll need to learn this eventually) If a field is conservative it can be written as the gradient of a scalar potential function. This is done over and over again in physics. Stokes theorem (and I made a rather bad error in my earlier post, it's not the divergence you use, it's the curl) says to integrate the curl of your field across the surface enclosed by your loop. Well the curl of the gradient of a scalar potential is always 0. So immediately Stokes theorem gets you the result about closed loop integrals in conservative fields.
sorry for the long winded reply and for the error.
That's fine no need to apologise, I'm just grateful that someone is kind enough to help me out, thanks a lot.
I have another question on surface integrals but I really need some sleep, would it be ok if I PM you the question tomorrow to see if you can help me out? (I think we are in different time zones it's currently 00:12am here)