00:01
So for this exercise will mostly be just eyeballing the vector fuse, but a very useful tool to have in mind when you deal with questions associated to curve and divergence are the following theorems.
00:14
If you have a closed curve, let's call it gamma -oriented, say counterclockwise, then we have the divergence theorem, which tells us that the flow.
00:36
Of a vector field v that is going through gamma like this or well like this so like anything else really the flow of v will be given by the integral on this interior region which we call the interior of gamma of the divergence of the vector field v and likewise the code is going to be related to how much this vector field is spinning as it goes around gamma so basically we know that the the spin which we could denote by the line integral of the vector gamma prime cartesian v along gamma is going to be equal to the integral, the curl of v in the interior of gamma.
02:30
So essentially, now that we have that in mind, we're going to try to apply that to the examples that we have here.
02:41
So first, in case a, we have a vector field that looks like this, and we put in our, let's say, test curve gamma, inside and we see that anything that goes inside say on this end of gamma is going to come out on the other side so there's no net flow of the vector field through gamma and likewise the vector isn't really spinning because if at this point it's going in this way then at this point it's going to go that way encounter that...