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Help me out here...what exactly is the difference between these 2 integrals? I'm not asking about the answer, but rather the notation. I've seen this "circle inside an integral" notation a lot, but it's never been explained. Is there actually a difference, or do mathematicians just put that circle there to look "fancy"?
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The circle indicates that the surface fully encloses a volume, i.e, when Gauss's Law for magnetism applies. Top integral always equates to zero. The bottom integral refers to any old surface, doesn't equate to zero in general.
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Ok, that makes sense. Thanks
yw bby, just please don't use the nasty [math]\text{d}\mathbf{s}[/math] when you can write [math]\text{d}\mathbf{A}[/math]
I think the ds means its a line integral though? Anyway, the circle means youre integrating along a closed loop.
The integral is over "S" which I assumed was a surface. In any case, just try to be unambiguous.
oh yeah you're right, it's over a closed surface.

Follow up question (Im not OP): If I integrated the curl of a vector field over a closed surface, will that integral always be zero?
Yes, that comes from Stoke's theorem. The curl of a vector field integrate over a surface is equal to the circulation of the closed path that bounce the surface. If the surface encloses a volume, that path does not exist.
Mew knot surface integral current density dot dee ay
Might be helpful to look at their stoke alternative to see kinda what it might imply. This can also be done for gauss if you look at divergenve theorem and it's implications.
The circle means that you are integrating over a enclosed region. Sort of like a cyclic process in a PV diagram.
First one's on a loop. Second holds value.
the circle means if you cant solve it in less than 10 min you are going to get fucked in your ass
... no wonder calc turned me gay

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