It is shown that under some restrictions (see the conditions $C_\Phi$ , $C_\xi$ , $C_{\xi\xi}$ ) the moments of the number of crossing of a set $\Gamma$ with a smooth boundary $S_\Phi=\{\mathbf x\colon\Phi(\mathbf x)=0\}$ . $\mathbf x\in R^m$ , by a continually differentiable vector stochastic process $\xi_i$ can be found explicitly. For example, the intensity $\mu^+(\Gamma,t)$ of the number of out-crossings of $\Gamma$ from the region $\Phi(x)<0$ at time $t$ is expressed by a surface integral of the first kind: $$ \mu^+(\Gamma,t)=\int_{x\in\Gamma}\mathbf M\{(\mathbf n_\Phi(\mathbf x)'\xi_t)^+\mid\xi_t'=\mathbf x\}p_t(\mathbf x)\,ds(\mathbf x). $$At the end of the paper examples are given, which illustrate advantages of the obtained formulas.