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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.

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