$x=(x_{1},...,x_{n})$. If $\frac{\partial g(x)}{\partial x_{l}}=f(x_{l})$ for $l=1,...,n$, should we have $$g(x)=\sum_{l=1}^{n}\int f(x_{l})dx_{l}+c\ ?$$
If yes, what's the theorem or proposition behind this?
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Yes, this is true. It looks better in vector notation: if $\nabla g = \mathbf F$, then $g(\mathbf r) = \int^{\mathbf r} \mathbf F(\mathbf r')\cdot \mathbf {dr'}+c$. This is known as the fundamental theorem of calculus for line integrals.
Answering follow-up question:
How can $g(x)=\sum_{l=1}^{n}\int f(x_{l})dx_{l}+c$ correspond to $g(\mathbf r) = \int^{\mathbf r} \mathbf F(\mathbf r')\cdot \mathbf {dr'}+c$ ?
The first integral is what you get by writing out the dot product in the second integral in coordinates.