Consider the following version of the divergence theorem (taken from $\text{M. E. GURTIN. An Introduction to Continuum Mechanics. Academic Press. New York, 1981}$). Let $R$ be a regular region in a 3-dimensional affine space over a real vector space $V$. Let $\phi:R\rightarrow \mathbb R$, $\mathbf{v}:R\rightarrow V$ and $\mathbf T: R\rightarrow Lin$, where Lin are the linear mappings (tensors) in $Hom_{\mathbb R}(V,V)$. The three fields are smooth. Then:
$\int_{\partial R}\phi\mathbf n dA=\int_R\nabla\phi dV$
$\int_{\partial R}\mathbf v\otimes\mathbf n dA=\int_R\nabla\mathbf v dV$
$\int_{\partial R}\mathbf v\cdot\mathbf n dA=\int_R\operatorname{div}\mathbf v dV$
$\int_{\partial R}\mathbf n\times\mathbf v dA=\int_R\mathbf{\operatorname{curl}}\mathbf v dV$
$\int_{\partial R}\mathbf T\mathbf n dA=\int_R\operatorname{div}\mathbf T dV$
My problem here is that I'm not used to the integral notation of this author at all and I don't know how to define some of the integrals above. So, assuming that I have a parametrization $X$ of $S=\partial R$:
$S=\{X(u,v):(u,v)\in D\}$,
for some $D\subset\mathbb R^2$, how can I compute these integrals?
I think I understand the classical one (number 3), which I would write as: $\int_S vdA\equiv\iint_D\langle v(X(u,v)),X_u\times X_v\rangle dudv=\int_R \operatorname{div} vdV$. But how can I write the other left-hand integrals in terms of an integral in $D$?
In addition, I know how to integrate a scalar field in $R$ and when it comes to a vector field I simply integrate every component, but what happens with $\int_R\nabla\mathbf v dV$? Do I integrate every component too and get a tensor as a result?
Any help will be welcome, especially if it contains any link to a book or something that helps me to understand it better. Thank you in advance