Hello

Suppose $C(U_1,...U_d)$ is a d dimensional copula and $C_k(U_1,...U_k)=C(U_1,...U_k,1,...,1)$ is the k dimensional marginal copula.

I am trying to prove the relationship

$\mathbb{P}\left(U_{j}\leq u_{j},u_{1}\leq U_{1}\leq u_{1}+\triangle u_{1},\ldots,u_{j-1}\leq U_{j-1}\leq u_{j-1}+\triangle u_{j-1}\right)=\frac{\partial^{j-1}C_{j}\left(u_{1},\ldots,u_{j}\right)}{\partial u_{1},\ldots,\partial u_{j-1}} $

Maybe something along the lines of,

$\mathbb{P}\left(U_{j}\leq u_{j},u_{1}\leq U_{1}\leq u_{1}+\triangle u_{1},\ldots,u_{j-1}\leq U_{j-1}\leq u_{j-1}+\triangle u_{j-1}\right)$

$=\lim_{\triangle u_{1},\ldots,\triangle u_{j-1}\rightarrow0}\mathbb{P}\left(U_{j}\leq u_{j},u_{1}\leq U_{1}\leq u_{1}+\triangle u_{1},\ldots,u_{j-1}\leq U_{j-1}\leq u_{j-1}+\triangle u_{j-1}\right) $

then the paper i'm reading tells me if this is right it should equal

$=\lim_{\triangle u_{1},\ldots,\triangle u_{j-1}\rightarrow0}\frac{C_{j}\left(u_{1}+\triangle u_{1},\ldots,u_{j-1}+\triangle u_{j-1},u_{j}\right)-C\left(u_{1},\ldots,u_{j}\right)}{\triangle u_{1}\cdots\triangle u_{j-1}}$ ... (*)

Where the fact the copula $C_j$ is the joint distribution of $j$ uniform random variables, $U_1,...,U_j$ is used. But I dont quite understand where the denominator in (*) $\triangle u_{1}\cdots\triangle u_{j-1}$ comes from??