Maths Notation: What Does \oint Mean?

[Defennder]

This is probably a dumb question but what does the circle through the integral sign mean?
$$\oint$$

I was thinking perhaps it could denote either the line integral over a closed curve, or the surface integral over a closed surface, depending on the context. But it seems the textbook I use employ that notation even when the line/surface integral is not over a closed curve/surface.

 

[HallsofIvy]

Yes, that symbol means the integral over a closed curve. I've never seen it used for the surface integral over a closed surface but I guess it could be. I would be very surprised if a textbook used that to indicate an integral over a curve that was NOT closed. Could you give an example of that?

 

[Defennder]

Ok this is from the 7th Edn of Engineering Electromagnetics by William Hayt and John Buck, pg 212 in the chapter on time-invariant magnetic fields:

It follows that only the integral form of the Biot-Savart law can be verified experimentally,
$$\textbf{H} = \oint \frac{Id\textbf{L} \times \textbf{a}_R}{4\pi R^2}$$

I suppose that it might be explained that the current I would be flowing in a closed circuit and hence a closed path. But what is the interpretation of $$\oint$$ as applied to surface integrals? Must it be a closed surface?

 

[Knissp]

I've seen that in Gauss's Law:

$$\oint E\! dA = Q / \epsilon_o$$

And, yes, to my understanding it must be a closed surface.