General meaning of line integral in vector fields

[Siddartha]

So, as i understand, the geometrical meaning of this type of integral should still be the area under the curve, however, I really do not see how you can obtain each infinitesimal rectangle from the dot product.

I have understood the typical work example, that is, the line integral as the sum of all the work done by each force in the direction of its respective displacement vector, but this only make sense in such a context, and not as a general mathematical conception.

However, influenced by such idea i tried to understand it as the influence of a specific field vector over each displacement vector that composes the curve, and given that the particle or object can not move outside of the curve's path, we ignore the parts of the field vector that are not in the same direction as the displacement vector, however (again) I do not see why would we define such influence as the product of the component of the field vector in the same direction as the displacement one with the displacement one itself (well their norms of course).

Can anyone help me out explaining the reasoning of this equation, specifically, why is the dot product in it?

 

[jedishrfu]

Welcome to PF!

The geometrical interpretation of single variable integral is the area under the cover.

In this case, you have a different kind of integral. I don't think there's a general geometrical interpretation of it in a mathematical context.

However there is one in a physical context when dealing with work over a path.

Taking the work example F.dr is the work done to move the particle a distance |dr| and the F.dr is the force parallel to dr. Hence, when you evaluate the line integral you do so over a specified path and so you can ignore forces perpendicular to the particle's path (you chose the path so the only force that matters is the one parallel to dr).

The wikipedia article may help you with more detail:

http://en.wikipedia.org/wiki/Line_integral

They have a couple of graphics that show geometrical understanding of line integrals over a scalar field and over a vector field.

 

[Siddartha]

Thank you for your reply! I was anxious for one hahaha, been stuck in this topic for a couple of days now.

As I said, I have understood the "work" conception of the line integral, but such conception is of course, limited to that specific context (line integral as a mean of obtaining work), however, my desire is to understand it in a more mathematical way (and thus, more general) to be able to know why it is used in other contexts and, if possible, know for myself when to apply it in various situation, that is why I consider this "work" exmaple insuficient.
Or is it that i have misjudged the line integral on vector field, and in reality, it can only be used in the context of forces and work? if this is the case, i would like anyone to be able to comfirm it to me.

I have already looked on wikipedia, however, it does not resolve my main doubt, why is the dot product used?
on one hand, the "derivation" paragraph just reafirms us that the dot product "gives us the infinitesimal contribution of each partition of F on C" which only makes sense in the specific "work" context, but in generalized way, I do not see why the dot product would be any kind of "contribution" or why do we use that specific kind of contribution (i could as simply say that summing only the components of the field vectors in their respective displacement vector, ie: F.dr/|dr|, is the total contribution too, as unconvinient as it could be operation-wise).
The graph, while mesmerizing, is more confusing still, it just affirms that the sums of all the dot products gives us another area that is equivalent to the one that is under the particle's path, but, why is it equivalent? the new curve itself has a very different shape to the path's curve to be visually obvious, and I see no kind of demostration of the equivalence of both areas under each curve, so wikipedia still leaves me in the dark.

so my question remains, what is the reasoning behind the dot product in some kind of purely mathematical understanding of the line integral? or am I wrong on my intentios, and the line integral (on vector fields) can only be applied in the "work" context? or maybe is it that each application of the line integral (in vector fields) is reached by completely saparated logics, in independent contexts, and therefore there is no unifying concept, making it a group of unrelated equations that just share coincidentally a form and so there is no "line integral in vector fields" as such?

 

[jedishrfu]

You could say that you have two choices the dot product to get the field component parallel to dr and the cross product to get the field component normal to dr.

I imagine if you considered a particle sliding on a wire then the cross product could be used to add friction into the mix.

This may be way beyond what you're looking for but it's something I've been studying recently with respect to vector calculus.

Have you looked at differential forms?

The differential form view of a line integral would that the path through the vector field punctures a bunch of layered surfaces where the F vector is normal to the surface being punctured. Differential forms unify a lot vector calculus and extend well into multidimensional spaces.

http://www.math.boun.edu.tr/instructors/ozturk/eskiders/fall04math488/bachman.pdf

See page 87 in the pdf

 

[Siddartha]

Well that is a very interesting read, and one that opens a whole other can of worms for me to study. while I am familiar with many of the concepts used, and could (midly) understand the read, I'm not ready to be comfortable with the use of differential forms, less for such an examination of line integrals hahaha.
However, i kind of understood the idea, that is that as a curves moves in a tridimensional space (in this case) each unidimensional dimension punctures various planes, and as such, an unidimensional integratio would give us the quantity of punctured planes. I am probably losing a lot of important facts on this (and am pretty sure that i am just seeing it as a normal integration and not a line one...).

But in any case, it does seems to be just a passing example to explore the incapacity to generalize derivate-forms integration, however if you have an idea of how the use of this tools could explain the equation of the line integral (and specially, the use of the dot product in that case) I would really appreciate a more profundized exploration of it, but in any case, thanks for the awesome text that i will surely read after finishing some topics!

 

[jedishrfu]

They are described as equipotential surfaces that the path punctures. Interestingly, if the surfaces touch at any point then the field is not a conservative field.

It looks like only you will be able to define and answer your question so study things some more and keep thinking about it.

 

[Siddartha]

You are probably right, I underestimated the complexity of the equation given it's age, ignoring the fact that mathematicians then had access to very complex tools like differential equations and many other things.

However I had been lead to a possible right direction to manage to define this in the future, so thanks a lot!