Suppose I have a function $F:\mathbb{R}^n \rightarrow \mathbb{R}$, and $F$ is absolutely continuous function on $\mathbb{R}^n$.
Then, Can I say
partial derivative $\frac{\partial^n F(x_1,x_2,\cdots,x_n)}{\partial x_1\cdots\partial x_n}$, and $\int_{\times_{i=1}^n[a_i,b_i]}\frac{\partial^n F(x_1,x_2,\cdots,x_n)}{\partial x_1\cdots\partial x_n}\,dx_1\cdots,dx_n=F(b_1,\cdots,b_n)-F(a_1,\cdots,a_n)$?
If not, what additional conditions do I need to say the result holds true?