Different formulations for multiple integral equation

Question

On several papers, I found the following model for a multiple integral equation:

$$g(s)=\int\limits_{\Omega} h(s,t)f(t)\,\mathrm{d}t$$

where $s,t \in \mathbb{R}^3$, and $\Omega \subseteq \mathbb{R}^3$.

I would like to know wether it is possible to express the above equation as:

$$g(x,y,z)=\iiint\limits_{\Omega} h\left(x,y,z,u,v,w\right) f\left(u,v,w\right) \, \mathrm{d}u\, \mathrm{d}v\, \mathrm{d}w$$

or are two different problems.

Answer 1

The question if these are two different problems or not, will depend upon whether and how $s$ and $t$ might be dependent on $\Bbb R^3$. There is no dependency between your expressions yet if you dont clearly express a dependency between $s$ and $t$.

Answer 2

Yes. All you did was to rename the components of $s,t$ as $x,y,z$ and $u,v,w$. Therefore the two expressions are completely identical, differing only in that you define $g(x,y,z):=g(\,(x,y,z)\,)$ and so on.