A delta function can be represented as an integral using its Fourier-representation:
\begin{equation} \delta\left(\vec{s}_j^2-1\right)=\int \frac{\mathrm{d} m}{2 \pi} \exp \left\{-\mathrm{i} m\left(\vec{s}_j^2-1\right)\right\} \end{equation}
I have $N$ vectors: $\left\{\vec{s}_1,\vec{s}_2,\dots,\vec{s}_N \right\}$ and would like to use the Fourier representation of: $$\prod_j\delta\left(\vec{s}_j^2-1\right)$$
Can I use one integral variable like this: \begin{align} \prod_i\delta\left(\vec{s}_i^2-1\right)&=\int \prod_j\left(\frac{\mathrm{d} m}{2 \pi} \exp \left\{-\mathrm{i} m\left(\vec{s}_j^2-1\right)\right\}\right)\\ &=\int \frac{\mathrm{d} m}{(2 \pi)^N} \exp \left\{-\mathrm{i} m\sum_j\left(\vec{s}_j^2-1\right)\right\} \end{align}
Or do I need to introduce $N$ distinct integration variables like this:
\begin{align} \prod_j\delta\left(\vec{s}_j^2-1\right)&=\int \prod_j\left(\frac{\mathrm{d} m_j}{2 \pi} \exp \left\{-\mathrm{i} m_j\left(\vec{s}_j^2-1\right)\right\}\right)\\ &=\int \prod_j\left(\frac{\mathrm{d} m_j}{2 \pi}\right) \exp \left\{-\mathrm{i} \sum_jm_j\left(\vec{s}_j^2-1\right)\right\} \end{align}
Are both representations correct?