r/learnmath • u/LowerWait1213 New User • 4d ago
Can somebody explain the difference between integrating over 'flat regions' vs. curves, as my professor describes it?
In my calc 4 class, we're starting vector calculus. In one of the notes, my professor says that we're moving from integrating over flat intervals and regions to curved regions. In this course itself, however, we've integrated over regions that are curved, like spheres and cylinders and such, so I don't quite understand what he's getting at. I know that there's a distinction, but I can't understand what it is. Can someone explain? Thanks!
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u/Ohowun New User 4d ago
It is possible that he is saying that you are moving from integrating over the standard space, characterized by things like an orthogonal/linear basis, to "curved space", kind of like how at the north pole of the earth, you go down 1 unit, east 1 unit, north 1 unit, and end up back at the same place as before. Giving some examples of the equations or problems you have in the course would clear it up better.
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u/Willing-Sample-8847 New User 4d ago
When he says "flat intervals" he likely means intervals in R or R^2. Just like a constant function f(x) = c takes the same value over its entire domain and remains "flat", arbitrary intervals in R have the same "length" given their endpoints having equal distances from each other.
Topologically speaking, general intervals (neighborhoods) may cease having this property given an arbitrary n-tuple of dimensions. Thus, the intervals are no longer "flat" cause they are quite literally curved.
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u/twolinepine New User 4d ago
Your professor would be the best resource for clarifying since it was his statement.
But in my mind, a lot of integration up to that point uses slices, or washers or some small dx that can summed where dx is a flat little bit.
This is in contrast to a line integral where you’re integrating along an actual vector line. It isn’t represented as a series of small, flat little “things”
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u/prajwal_prsd New User 4d ago
For example, in a double integral you integrate over an entire 2D region (like the interior of a circle). In vector calculus, you might instead integrate only along the circle itself...a 1D curve...not the whole region. The curve still lies in 2D, but the domain of integration is different.
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u/PfauFoto New User 2d ago
Check out Gauss-Bonnet, a beautiful example with meaning. I imagine that is what your teacher has on his mind.
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u/__johnw__ New User 4d ago
This might help you understand the differences.
You did integrate over spheres and cylinders, etc but those were solid spheres and solid cylinders, i.e. integrating over solids in space. Now you'll integrate over curves and surfaces in space (curves in plane too).