Multi-dimensional chi-square distance

The chi-square distance between continuous distributions $p(x)$ and $q(x)$ is, $$\int \frac{(p(x)-q(x))^2}{p(x)}dx$$ My question: what is the chi-square distance between multivariate continuous distributions $p(x_1,\ldots,x_n)$ and $q(x_1,\ldots,x_n)$? Do we simply have, $$\int\cdots\int \frac{\big(p(x_1,\ldots,x_n)-q(x_1,\ldots,x_n)\big)^2}{p(x_1,\ldots,x_n)}dx_1\ldots dx_n$$ or we should take some other considerations into account?