Closed Curves.
Hey, I am wondering if anyone can help me understand a mathematical explanation as to how they work.
From what I understand, the area under a closed curve is the same, independent of the path taken. So when doing an integral you only need to take the initial and final into account. There have been 2 a few situations so far when I have come across these curves. Particularly when dealing with conservative forces, i.e. Gravity, Electricity. So with these forces, it is saying the work done is the same independent of the part taken. But what actually makes it a closed curve? What part of it all makes it so the path does not matter when getting the area under the curve? 
Re: Closed Curves.
It looks like you are talking about line integrals. But, line integrals do not find the area under the curve that you integrate over. One interpretation (the one it looks like you are talking about) is that if f is a force field, then the line integral of f over a curve is the amount of work done in transporting a particle along the curve.
Line integrals are independent of parameterization of the curve. If f is a complexanalytic function (a conservative force field), then the line integral does depend only on the end points, but if the curve is closed, the integral is zero. For instance: [tex]\int_C f = F(b)F(a)[/tex] where F'=f and a is the starting point of C and b is the ending point. So if C is closed, a=b and the integral is zero. 
Re: Closed Curves.
Ah, so how does a line integral work then, if it does not find the area under the curve? Does it use the fact that the integral is always equal to zero in a closed curve?

Re: Closed Curves.
Do you have a particular example that you want clarified? Wikipedia's article on line integrals seems to do a decent job of explaining it.

Re: Closed Curves.
Yeah, that Article does a pretty good job of explaining it. It made a lot of sense once I saw it. So from what I understand at the moment, it sounds like a line integral, in a vector field, is when you do the dot product between the 2 vectors at each little point then add them all up. Is that the right idea? or have I missed a detail out?

Re: Closed Curves.
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(And, if you are integrating around a closed path, the integral of an exact differential is 0 definitely NOT the area!) Quote:

Re: Closed Curves.
Hahaha it seems that I do not know what I am talking about when I say area under the curve. I have only seen closed curves from a physics point of view, but in terms of maths, I am not really sure what it actually means. I understand if I was to talk about a closed path, it is a path that meets back where it started. But other than that, I am not too sure about how they work.
Also when I am integrating with this closed curve, I have not been told if I am finding the area, I just have a habit of saying that when I do integrals. I guess since I am generally dealing with a vector field, I would use a line integral? I am sure I am being confusing and unspecific but that is because I am not completely sure about what is going on, when dealing with closed paths and integrating. Oh and if a path is different to a curve, please let me know. I probably should be using path. 
Re: Closed Curves.
It seems HallsofIvy just repeated exactly what I had said. In terms of math, line integrals are defined. Then they are given physical interpretations. They can have different meanings based upon the interpretation. See my post here (and even look at the general thread there too). I'm sure there are others as well.
For instance. If f is a complex valued function and [itex]\gamma(t)[/itex] is a curve where [itex]t\in[a,b][/itex], then the line integral is defined to be [tex]\int_\gamma f(z)\,dz = \int_a^b f(\gamma(t)) \gamma'(t) \,dt . [/tex] This is just the definition. As I explained above and in the post I linked to, this can be given different interpretations depending on what you treat f as. The line integral is just another version of an integral, and you don't want to always think of integrals computing areas. I think if you view them as summing something up, then this works better for the physical interpretations. You just need to determine what you are summing. I used a complex function for my definitions because it is the simplest in notation. You can think of a complex function as a vector field (it takes a point in [itex]\mathds{R}^2[/itex] and assigns a point in [itex]\mathds{R}^2[/itex]). Curve and path are often used interchangeably. There are some books though that make a small technical distinction between them. 
Re: Closed Curves.
Cool thanks, I have a good idea of what is going on. I think it is kind of deceptive that I had never been told about the line integral, no wonder it did not make sense to me.
So with a complex function, why would you not use a matrix? Is it not always linear or something? Yeah, I had the idea they were almost the same, but wanted to be careful because I remember reading a text book I had awhile back, and it talked about how they were different, but could not remember how. 
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I am also told that I can think of a complex function as a vector field. Which means I am probably over complicating things by bring in the idea of matrices. But to me they seem related in the fact that they manipulate vectors. I guess another way of dealing with how they move is to use a parametric system, in which you have the components of the vectors as a function of t. 
Re: Closed Curves.
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Complex functions (and thus the vector fields associated with them) can be manipulated by fractionallinear transformations, which can be represented by matrices. They can perform translations, dilations, rotations, and inversions (or any combination thereof) which is what you are referring to matrices changing vector fields. These are complex valued functions, but not all complex valued functions are fractionallinear transformations and thus can't always be represented by matrices. I don't know what class you are taking, but this is why complex variables is one of the most applicable areas of math. If you view complex functions as vector fields or some other physical interpretation, the theory gives you powerful results. 
Re: Closed Curves.
I am a math and physics major. But first year in both. I end up learning most of this stuff next year so I am in no hurry to become an expert in it all, how ever it is nice to understand how things work, and properly. I have definitely dealt with the complex plain before, and used complex numbers as vectors. Hahaha I just did not think that you would use vectors with complex numbers. I know complex numbers have lots of useful properties to do with waves and so on, but did not expect vectors to find them useful.
I have been told before that you can treat complex numbers like vectors but there are some small things that are not the same. For example I am not sure if you can use a vectors reflection along the y axis like you can with complex numbers and their conjugate. So I understand that you have to be careful with small things. 
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