# Integral over surface vector in the language of differential geometry?

I would like to relate the idea of taking an integral over a surface vector in the language of differential geometry.

For example, the integral (in $$\mathbb{R}^3$$) $$\vec{A} = \int_{d\Omega} f d\vec{S}$$ where $$f: \mathbb{R}^3 \rightarrow \mathbb{R}$$ is a function of position. In standard vector calculus, this is interpreted as the integral over surface vectors (resulting in a vector). Is there a way to write this in terms of differential forms?

If we had instead $$A = \int_{d\Omega} \vec{f} \cdot d\vec{S}$$ where $$f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$$ is a vector function of position. We could write this as a $$2$$-form $$df = f_x dy \wedge dz + f_y dz \wedge dx + f_z dy \wedge dz$$. I'm trying to relate it to something like this.