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I am trying to understand the concepts of surface and line integrals better, and since I'm not a mathematician I have only come across explanaitons which are a bit over my head. I have a few fairly basic questions.

1. Can you give me simple definitions of surface and line integrals?

I am farly sure I understand them, but sometimes it's good to see how somebody else might explain it.

2. Why is a surface integral a double integral?

I understand the surface integral gives you either scalar or vector values that are distributed on a surface. Let's say I have an surface defined by $f(x,y)$. Is it a double integral because I have to integrate over $dx$ and $dy$?

3. Is doing the surface integral just doing a line integral twice?

It seems to me that if the line integral is summing the points on a curve, then the surface integral is doing the same thing but over all the infinitesimal curves on the surface (i.e: all the curves form a surface).

I hope my ignorance will not offend you, mathematicians :)

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    $\begingroup$ To your second two questions, in laymans terms, yes and yes. Which reflects you understand the concept, which means the answer to the first is also yes. P.S: You don't seem very ignorant, you understand a lot better than most apparently. :) $\endgroup$
    – Guy
    Commented Apr 1, 2014 at 14:39

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A line integral is essentially like a regular integral but instead of taking little $\Delta x$'s along the x-axis and multiplying them by f(x), you take little segments of the curve (i.e., arc length - which is why you integrate ds) and multiply that by f(x) and then do the usual limit thing and let the norm of the partition tend to zero etc. Another way to think of a line integral is as a regular integral taken or twisted along a curve. Thus a line integral gives you the area of the "curtain" formed between the surface and the curve (Wikipedia has a good animation).

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  • $\begingroup$ This is true for a $ds$ integral, but is not a good mental picture for a $Pdx +Qdy$ line integral. $\endgroup$ Commented Jan 14, 2015 at 2:46
  • $\begingroup$ True. Wikipedia has another good way to picture the relation of line integrals of vector fields to area, but these are generally easier understood using physical definitions (such as work), which is why I focused on the line integrals of scalar fields for an intuitive understanding still within the realm of math. $\endgroup$
    – Gabriel
    Commented Jan 14, 2015 at 2:53

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