How to evaluate a line integral? In some physical applications I found things like $\int_{\gamma} r \times dl$ and was wondering what this means? see this one for example In principle it should be some sort of line integral. 
So let' say that I have a parametrization $\gamma : [0,1] \rightarrow \mathbb{R}^n$ of my curve, how would I evaluate this formally?
 A: We begin by noting that as we have the cross product of two vectors we must restrict ourselves to the vector space $\mathbb{R}^{3}$ (and also $\mathbb{R}^{7}$, but I will leave it to someone else to answer that if they so wish).
We also note that in general if we have $\vec{\gamma}:[0,1]\to\mathbb{R}^{3}$:
$$\vec{\gamma}(t)=\begin{pmatrix}\gamma_{1}(t) \\ \gamma_{2}(t)\\ \gamma_{3}(t)\end{pmatrix}$$
We thus have:
$$\mathrm{d}\vec{\ell}=\begin{pmatrix}\gamma_{1}'(t) \\ \gamma_{2}'(t) \\ \gamma_{3}'(t)\end{pmatrix}\:\mathrm{d}t$$
We also note that $\vec{r}$ is in fact the difference between the current position of the line element $\mathrm{d}\vec{\ell}$ along the curve and some point $\left(\begin{smallmatrix}x \\ y \\ z\end{smallmatrix}\right)$, i.e.:
$$\vec{r}=\begin{pmatrix}x - \gamma_{1}(t) \\ y-\gamma_{2}(t) \\ z-\gamma_{3}(t)\end{pmatrix}$$
Thus we have the Biot-Savart Law in full:
$$\vec{B}(x,y,z)=\int_{\gamma}\frac{I\:\mathrm{d}\vec{\ell}\times \vec{r}}{\|\vec{r}\|^{3}}=I\int_0^{1}\left.\left(\begin{pmatrix}\gamma_{1}'(t) \\ \gamma_{2}'(t) \\ \gamma_{3}'(t)\end{pmatrix} \times \begin{pmatrix}x-\gamma_{1}(t) \\ y-\gamma_{2}(t) \\ z- \gamma_{3}(t)\end{pmatrix}\right)\right/\left\|\begin{pmatrix}x-\gamma_{1}(t) \\ y-\gamma_{2}(t) \\ z-\gamma_{3}(t)\end{pmatrix}\right\|^{3}\:\mathrm{d}t$$
And you can just compute the cross product before evaluating the integral with respect to $t$. 
I hope this helps, if you have any further questions don't hesitate to ask in the comments!

Note that you may also find this Math.SE question interesting.
