Line Integrals & Green's Theorem

[DisaFear]

Hi guys,

Just seeking some clarification on some points:

1. This is my definition of a line integral, is it correct?

Say we have a function that acts as a 'ceiling' and a curve that is a 'path'. A line integral will find the area of the 'curtain' that connects this 'path' to the 'ceiling' function.

2. "Conservative vector fields have the property that the line integral from one point to another is independent of the choice of path connecting the two points: it is path independent"

How exactly does this work? I have two points, but I could make one path extremely long, the other short. They would give different answers, right? Am I mixing up the definition of 'conservative'?

3. I've seen the relation between line integrals and Green's Theorem and all, but don't understand how to interpret the results. So, line integral gives us an area. I was watching some videos on Khan Academy, and Sal kept referring to 'volume' between the region R made by the path curves, and the ceiling function (or vector field). Are they somehow the same answer? Are they meant to be the same?

4. F= grad_f - What is the point of this? They mention it a lot of times, that blah equals blah - of course it does, doesn't it always have to? F is the vector field, right? What I take can be the 'ceiling' function of sorts. I know that grad_f is the gradient vector.

Sorry for the stupid questions! If you can answer in dot points, that would be great. Thought processes, clarifications, anything! Thanks

[kamil9876]

How exactly does this work? I have two points, but I could make one path extremely long, the other short.

So you expect the longer path to give a greater integral? Let's look at gravitational potential for instance. If you start at height=0 and go to height=10 then height =5, then your change of gravitational potential is the same as going from height=0 to height =100 to height =5. Even though the latter is a longer path. So only depends on endpoints of for such a vector field. (Here F is the gravitational force vector field). (In this case, extra 90metres going up is cancelled out by the extra 90 going down).

On the other hand, friction doesn't behave this way: Going 10 metres left then 5 metres right causes less "energy loss due to friction" than going going 100 metres left and then 95 metres right. This is not really integrating a vector field though, but rather integrating a scalar field. Note the difference as explained here: http://en.wikipedia.org/wiki/Line_integral#Line_integral_of_a_scalar_field

Edit: Replace "5" with "95", obviously.

[Ahmad]

There are two kinds of line integral. Your interpretation in terms of area of a "curtain" is one kind of line integral, the so called line integral with respect to "ds" the infinitesimal arclength (not the same ds in kamil's post, which is an infinitesimal vector, and is the 2nd kind of line integral). Your questions all implicitly refer to the 2nd kind of line integral, which is causing the confusion. The 2nd kind of line integral is the more important type.

For (4), if you start with a vector field F, it may or may not be the gradient of some function f. If it is, then there's a result which says that F is a conservative field.

[DisaFear]

Hm, starting to make a bit more sense.
Can you give me an example of a non-conservative vector field?

[Thu Thu Train]

Is a non-conservative vector field (I'll let you work out why because I'm lazy and assume you know the definition of a conservative vector field)

Yay, I'm helping

[DisaFear]

I wrote a reply but it wouldn't post >_>

For the post above, it's because dP/dy =/= dQ/dx right?
I emailed my tutor who just replied, feeling a bit more confident about this. So basically, we can get a non-conservative vector field when we lose a variable, right?

As F(x,y,z) = grad_f(x,y,z), if we differentiate and lose a variable, would that mean we get a non-conservative field?

Or am I rambling :O