Integration scalar or vector variables confusion

Does $d P_x d P_y = d^2 \vec P$?

I thought not because $P_x$ is a scalar , a component of the vector, whereas $\vec P$ is a vector?

However someone said it is just a matter of convention and you can write $d P_x d P_y = d^2 \vec P$

But, if I consider this example:

Say I have $\int dP_x dP_y e^{-\vec{P}^2}$ if $dP_x dP_y =d^2\vec P$(*)then this is $\int d^2\vec{P}e^{-\vec{P}^2}=\sqrt{\pi}$

Since $\vec P^2$ is a scalar I can do the integral working with the components $dP_x$ and $dP_y$ however had it not been and just been $\vec{P}$

I don't know how you convert the integral between $dP_x dP_y$ and $d\vec P$ So $\int dP_x dP_y e^{-\vec{P}^2}=\int dP_x dP_y e^{-P_x^2-P_y^2}=\int dP_x e^{-P_x^2}\int dP_x e^{-P_y^2}=\sqrt{\pi}\sqrt{\pi}=\pi$

so using * these don't agree

I'm not familiar with this notation, but the final statement looks vaguely plausible. Where has this arisen? What is the topic? P looks like momentum, I guess?

More likely probability measures or joint probability measures?

But nothing has been defined so I am not sure what is being argued.

Does $d P_x d P_y = d^2 \vec P$?

I'm not familiar with this notation, but the final statement looks vaguely plausible. Where has this arisen? What is the topic? P looks like momentum, I guess?



I don't follow this.



This isn't a complete sentence.


What is $d\vec P$ in your notation?

This is largely impossible to follow without more background info, I think.The final calculation looks plausible, I suppose, if the limits are +/- infinity.

Maybe one for RichE?

it's common for physicists to write

$dx dy dz = d^3{\bf r}$

or similar. All this is just a matter of notation.

momentum.
statistical mechanics.