Line integral curiosity/confusion..

[optinyx]

Heya!

I was hoping someone could clear this up for me: how would a line integral be represted graphically? I've always liked calculus because it's easy to visualize (almost all the problems have graphs associated with them) - but I don't quite get how to visualize a line integral. Or is it a concept that can't be easily visualized?

any help would, well.. help. :redface:

I've also been pondering something my teacher brought up last week:
Let's say the line integral of f(x,y,z) over a curve C is equal to zero, then would it have to be true that f(x,y,z) = 0 on C?

Thank you! I eagerly await a response. :biggrin:

[ReyChiquito]

its easier to visualize it with physics... think of it in terms of Work done under a force. On the other q, you mean for ALL C's?, because if C is closed, then is not necesarily true, think of it in terms of the work done :)

[HallsofIvy]

No, the fact that the line integral of f over a curve is 0 does NOT mean that the function must be identically 0 for two reasons:

First it is possible that f is positive on part of the curve and negative on another part so that the two parts cancel.
Also, knowing the integral on a particular curve only gives you information about f ON THAT CURVE. Even if f were 0 on the curve itself, it might be non-zero elsewhere.

[optinyx]

okay, I think I understand now. thinking of it in terms of work done makes it easier, like in physics.. when you push a block ten feet in one direction and then ten feet back to where you started, they say the total work done was zero because the total distance was zero. Or am I completely off my rocker? (it's been a while since general physics).

If I understand though, then that's what hallsofivy was saying right? - about it being positive on one part and negative on the other? I think I get it.

Could anyone give me an example function, f(x,y,z), and a curve where this would be the case?

Thank you for all the help.

[Warr]

okay, I think I understand now. thinking of it in terms of work done makes it easier, like in physics.. when you push a block ten feet in one direction and then ten feet back to where you started, they say the total work done was zero because the total distance was zero. Or am I completely off my rocker? (it's been a while since general physics).

displacement*

[matt grime]

try to stop thinking visually, it doesn't help in the real world, or the world of mathematics, just work with the definitions.