Let $C=(0,1)^{N}$ be the $N$-dimensional unit cube. Let $f:\mathbb{R}^{N}\to\mathbb{R}$ be a sufficiently regular function for what follows. What does the following notation mean:
$$\int_{C}\nabla f(x)\text{d}x$$
How can I interpret the integral of a vector field over a volume? As it may help, here is the context: let $\mathbf{a}\in\mathbb{R}^{N}$ be a constant vector. We are interested in the following integral (where $\cdot$ denotes the natural scalar product on $\mathbb{R}^{N}$):
$$\int_{C}\mathbf{a}\cdot\nabla f(x)\text{d}x$$
And it is written that
$$\int_{C}\mathbf{a}\cdot\nabla f(x)\text{d}x=\mathbf{a}\cdot\int_{C}\nabla f(x)\text{d}x=-\mathbf{a}\cdot\int_{\partial C}f(x)\mathbf{n}\,\text{d}\sigma(x)$$
(where $\mathbf{n}$ is the normal) which I understand intuitively as a variant of
$$\int_{C}\left(g\text{ div}\mathbf{F}+\mathbf{F}\cdot\nabla g\right)\text{d}V=\int_{\partial C}g\mathbf{F}\cdot\mathbf{n}\,\text{d}S$$
where $g:\mathbb{R}^{N}\to\mathbb{R}$ and $\mathbf{F}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is a vector field.
