Having recently studied magnetostatics, I came across the Biot-Savart law, which is based on the line integal over a current distribution in a curve $C$:
$$\mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\int_C\frac{I\text{d}\mathbf r'\times\mathbf{\hat r'}}{|\mathbf r'|^2},$$
which, generally, contains an integral in the form
$$\mathbf I=\int_C\mathbf F(\mathbf r)\times\text{d}\mathbf r$$
where $\mathbf F:\;U\subseteq\mathbb R^3\to\mathbb R^3$ is an arbitrary function on $U$, and $\mathbf r$ is the vector of coordinates in this space. This bears a similarity to the conventional line integrals of the form
$$I=\int_C\mathbf F(\mathbf r)\cdot \text d\mathbf r$$
which is a scalar, whereas the former generates a vector. Is there any relation between the two? Although there is a lot of literature on the conventional line integrals, I could not find any on its 'crossed' variant. Could it be that one can transform one into another? Also, are there any useful relations (such as Stoke's theorem with the conventional line integral)? And how does this extend to surface integrals in the form
$$\mathbf \Phi=\iint_S\mathbf F(\mathbf r)\times\text d\mathbf S$$
compared to the common
$$\Phi=\iint_S\mathbf F(\mathbf r)\cdot\text d\mathbf S.$$
[Please excuse the multiple questions but it seemed a better option compared to writing different posts.]