Re: Simple integration

From: Kalle Rutanen (kalle_rutanen_at_nospam.hotmail.com)
Date: 07/15/04 

Date: Thu, 15 Jul 2004 10:07:14 +0300

Thank you for a long answer.
Trying to use the things I learned from your post, let me try to solve my
real problem. Would you be so kind to check for notation and errors ?

Problem: Solve for F(x, w)

A = all positions on a planar surface
omega = all directions on a hemisphere w.r.t a surface normal n
x = position on planar surface
w = direction on a hemisphere w.r.t a normal n
n = surface normal

L(x,w) = d^2F(x, w) / [(w dot n) * dA * dw]

<=>

L(x, w)(w dot n) = d^2F(x, w) / (dA * dw)

<=>

I(omega)L(x, w)(w dot n)dw = I(omega)[d^2F(x, w) / (dA * dw)]dw =
I(omega)[d^2F(x, w) / dA] = d/dA * I(omega)[dF(x, w)] = d/dA * I(omega)F'(x,
w)dw = d/dA * F(x, w) = dF(x, w) / dA

<=>

I(A)I(omega)L(x, w)(w dot n)dwdx = I(A)[dF(x, w) / dA]dA = I(A)dF(x, w) =
I(A)F'(x, w)dx = F(x, w)

What we get is

F(x, w) = I(A)I(omega)L(x, w)(w dot n)dwdx

Which is what my book also says. But.. The way I got this is propably not
correct because I left away the integration constants.

So the problems with this is:

1) Is it correct to take out the d/dA out of the integral ? If yes, then in
what expections ?
2) It seems that I got the correct answer with false calculation, without
integration constants... This must mean the integration constants are either
zero or add to zero. In what circumstances does this happen ?
3) I used F' for both the derivative w.r.t to w and x, clearly incorrect,
what notation should I use to mark the derivate w.r.t arbitrary variable ?