Correct notation for some functional expressions

[Hypatio]

Having some trouble determining the most mathematically correct way to express something I understand only numerically and physically. Basically I am modeling radiation within a volume.

1. For each point dV within a volume V there is a scalar value e(dV), which is an amount of radiation emitted from the point.
2. For each point dV there is also a corresponding function q(x,y,z), which is the energy emitted from dV and absorbed throughout the volume. The volume integral of q(x,y,z) is equal to e(dV).
3. For each point dV, an important value is the net energy change from emission and absorption. For each point dV emitted energy is equal to e(dV), but absorbed energy at the point takes the sum of an infinite number of functions q(x,y,z), one for each point dV, and summation occurs at the position dV=(x,y,z). If we call this total absorbed energy Q(dV), then how is it properly expressed? Is Q a functional?

Perhaps it is something like

[itex]Q(dV) = \int_V[q(x,y,z)\cup dV] dV[/itex]

but I don't know.

 

[haruspex]

each point dV

dV will not be a point; it will be an element of volume.

value e(dV), which is an amount of radiation emitted from the point.

It would make more sense to say there is a density function of position, ##e(\vec x)##, so that the rate of radiation emitted by an element dV at ##\vec x## is ##e(\vec x)dV##.

function q(x,y,z), which is the energy emitted from dV and absorbed throughout the volume.

I thought that was how e was defined. I think you mean that q is a function of two position vectors, ##\vec x, \vec y##, such that if dV is a volume element at ##\vec x## and dW is a volume element at ##\vec y## then the rate at which radiation is emitted by dW and absorbed by dV is ##q(\vec x, \vec y)dVdW##.
Does that make sense!

 

[Hypatio]

Thanks for the help. So, there is a function for the emission rate. Shouldn't this be [itex]e(x,y,z)[/itex], not [itex]e(\vec{x})[/itex], [itex]x,y,z[/itex] are positions (not position vectors?). The transport of emitted radiation itself will of course have a vector, but my approach ignores those details. Instead, you just know the emission field [itex]e(x,y,z)[/itex] and an infinite number of consequent absorption fields [itex]q(x,y,z)[/itex] for each [itex]dV[/itex] of [itex]e(x,y,z)[/itex]. Moreover, positions should preferably be given in an absolute reference frame, not relative to the emission point. I need an expression for the amount of radiation absorbed by a given volume element [itex]dV[/itex] from all [itex]dV[/itex] of the volume (the sum, or integral, of an infinite number of functions [itex]q(x,y,z)[/itex]).

 

[Mark44]

Shouldn't this be [itex]e(x,y,z)[/itex], not [itex]e(\vec{x})[/itex], [itex]x,y,z[/itex] are positions (not position vectors?).

I believe that haruspex's notation is equivalent to what you wrote. By ##\vec{x}## he means the point (x, y, z).

 

[Hypatio]

Could I define energy emitted by [itex]dW[/itex] and absorbed by [itex]dV[/itex] as [itex]q(V,W)dWdV[/itex] so that that there is a function for the total absorbed energy as [itex]Q(V) = \int_V q(V,W)dWdV[/itex], and thus absorbed energy is [itex]QdV[/itex]? Does that make sense?

This is so much easier with a discrete notation:

[itex] Q(x,y) = \sum_{i=dV}^{V/dV}q_i(x,y)[/itex]

 

[Mark44]

Could I define energy emitted by [itex]dW[/itex] and absorbed by [itex]dV[/itex] as [itex]q(V,W)dWdV[/itex] so that that there is a function for the total absorbed energy as [itex]Q(V) = \int_V q(V,W)dWdV[/itex], and thus absorbed energy is [itex]QdV[/itex]? Does that make sense?

@haruspex wrote

the rate at which radiation is emitted by dW and absorbed by dV is ##q(\vec x, \vec y)dW dV##, not ##q(V,W)dWdV##.

For the total absorbed energy you would need to integrate over the entire volume, something like this:##Q(V) = \int_V q(\vec x, \vec y)dWdVdV##
Since dV is used for two different purposes here, let's call ##\rho_x## the rate at which energy is emitted by a the volume element around ##\vec x## and ##\rho_y##, the rate at which energy is absorbed by the volume element around ##\vec y##.
Then the equation above would be :##Q(V) = \int_V q(\vec x, \vec y)\rho_x(\vec x) \rho_y(\vec y) dV##
This would necessarily be a triple integral, since you have to integrate over all of the points within the volume.

I don't guarantee that the integral above is correct. I'm just going by the information provided in this thread.

This is so much easier with a discrete notation:

[itex] Q(x,y) = \sum_{i=dV}^{V/dV}q_i(x,y)[/itex]

This doesn't make a lot of sense. For a summation, the index generally ranges over the integers, or a subset of it. dV isn't a number -- it's an increment of volume.