| Posted on Sunday, 18 January, 2004 - 09:28 pm: |
what is a flux integral? definition?
| Posted on Monday, 19 January, 2004 - 03:45 pm: |
A flux integral is an integral of a vector field over an oriented surface.
If F is a continuous vector field defined on an oriented surface S with unit normal vector N, then the surface integral (or flux) of F over S is given by:
This can be interpreted physically of the rate of flow of F through S.
Post back if you want more information.
| Posted on Monday, 19 January, 2004 - 10:20 pm: |
Yes please?
| Posted on Tuesday, 20 January, 2004 - 03:59 pm: |
Okay, I'll attempt to explain the principles of it in more detail. I'll use the analogy of a fluid throughout this.
Firstly, the surface S. Imagine that S is something covering the flow of the fluid which doesn't impede the flow in any way (rather like a fishing net). Then, you can define the flow by a velocity field
and
a density 
at any point. The rate
of flow is then given by 
.Divide S into individual patches Sij . If they are made small enough, the patches are nearly planar, and the mass of fluid crossing the patch per unit time in the direction of the unit normal n is approximated by
, where
A(Sij ) is the area of the patch. It should be
fairly obvious that to get the total fluid flow across
the surface, you need to sum these individual rates of
flow across the entire surface. Summing and taking the
limit as the number of patches approaches infinity gives
the surface integral:
The definition in my post above is given by simply combining the mass with the velocity as a vector field
. Then, the integral
becomes:
I won't go into depth on surface integrals, but I'll give a quick overview of the simplest form. If our surface S is defined by the equation
then the integral over
S is, in this case, given by:
where D is the parameter domain over which you are integrating, and gx and gy are the partial derivatives of g w.r.t. x and y, respectively.
Hope that clarifies things a bit. Post back if you're unsure on anything or would like clarification or an example.
| Posted on Tuesday, 20 January, 2004 - 09:18 pm: |
Properly, the number of integral signs should match up with the number of d(-)'s that you have explicitly written. So for example, the last equation: You should have only one integral sign, or should change the 'dA' to 'dxdy' (not both, or neither).
Kerwin
| Posted on Wednesday, 21 January, 2004 - 03:25 pm: |
Well, my 'dA' was shorthand for 'dx dy' or whatever ones are required (it could be 'dy dx' or 'dx dy', depending on the order of integration - D isn't necessarily rectangular, and you should only have an independant variable in the limits of the inner integral, as I'm sure you know).
| Posted on Wednesday, 21 January, 2004 - 04:21 pm: |
But
is still a measure,
so the second integral sign is redundant. You wouldn't
write
instead of
,
| Posted on Wednesday, 21 January, 2004 - 04:24 pm: |
Let us not quibble over notation.My notaton is the notation employed in my calculus textbook, so I apologise if it is misleading.
| Posted on Wednesday, 21 January, 2004 - 07:58 pm: |
I am aware that a lot of book authors decide to write things as the way your calculus book did, but it is not a good notation - just as
and 
make sense (for a function f mapping [a,b] to the reals, say) but
and 
are meaningless, but unfortunately most people make that mistake.
The reason why the number of d's and the number of integral should match up is that otherwise confusion arises as to what we are integrating (properly the degree of the differential forms, but I won't go into that) and over which space (i.e. whether we are integrating locally or globally).
Kerwin
| Posted on Thursday, 22 January, 2004 - 07:05 am: |
Hrm, that's interesting Kerwin because I see everywhere (and even use it myself). I'll try avoid it in the future
Marcos
| Posted on Thursday, 22 January, 2004 - 11:21 am: |
Thanks Francis!
| Posted on Thursday, 22 January, 2004 - 09:29 pm: |
For my part, I don't think much confusion really does arise. A good notation has more to do with the way the mind works than with mathematical formalism, and -- though I can only speak for myself -- a double integral with respect to dS is more suggestive than a single integral, even if the second is redundant (or worse still, technically without a measure to be integrated with respect to). This sort of case-specific notation arises all over mathematics, and insofar as people learn math in pieces, not all at once (e.g. vector calculus is usually learned before a measure theoretic approach to integration: rightly, I think, since that's the historical progression), I don't see the problem with notation not being fully general.
Brad