Line integral in a vector field I have the following question: 
Determine the total flux of the velocity field
$$\vec{v} = \begin{pmatrix}
1 \\
x + y
\end{pmatrix}$$
out of the disk $D$, a circle with radius 2 and center at the origin
The answer is: 
the Boundary of this disk $D \in \mathbb{R}^{2}$ may be parametrized by
$$\begin{pmatrix}
x(t) \\
y(t)
\end{pmatrix} = \begin{pmatrix}
2 \cos t \\
2 \sin t
\end{pmatrix} \quad \text{for} \quad 0\leq t \leq 2\pi$$
Thus we have $$
\begin{pmatrix}
-2 \sin t \\
2 \cos t
\end{pmatrix}dt$$
and
$$ds = ||\vec{ds}|| = 2dt \tag{1}\label{eq1}$$
The outer unit normal vector a time (or angle) $t$ is given by 
$$\vec{n}(t) = \begin{pmatrix}
\cos t \\
\sin t
\end{pmatrix} \tag{2}\label{eq2}$$
Which mathematical axiom / theorem do we have to invoke to be able to state (1) and (2)
 A: The flux of a closed curved $\boldsymbol{\gamma}:[a,b] \rightarrow \mathbb{R}^2$ is given by the formula:
$$\oint_{\gamma} \left < \mathbf{v}, \mathbf{\hat{n}}\right> ds \;\;\;(1)$$
Where $\mathbf{v}$ is the velocity field, $\mathbf{\hat{n}}$ is the unit normal of $\boldsymbol{\gamma}$ and $<.,.>$ is the dot product.
As the comments suggested $ds$ is the infinitesimal arc-length of the curve, which is given by:
$$ds = |\dot{\boldsymbol{\gamma}}| dt$$
And $\mathbf{\hat{n}}$ can be thought as the left/right rotation of the unit tangent vector of the curve:
$$\mathbf{\hat{n}} = \mathbf{R}\mathbf{\hat{t}}$$
Where $\mathbf{\hat{t}}$ is the unit tangent vector, and  $\mathbf{R}$  is the rotation matrix, that rotates a vector to the right.
So,
$$\mathbf{\hat{t}} = \frac{\boldsymbol{\dot\gamma}}{|\boldsymbol{\dot\gamma}|}$$
and
$$\mathbf{R} = \begin{bmatrix}
 0& 1\\ 
 -1&0 
\end{bmatrix}$$
Therefore (1) becomes:
$$\oint_{\gamma} \left < \mathbf{v}, \mathbf{\mathbf{R}\mathbf{\hat{t}}}\right> |\dot{\boldsymbol{\gamma}}| dt$$
$$\oint_{\gamma} \left < \mathbf{v}, \mathbf{\mathbf{R}\frac{\boldsymbol{\dot\gamma}}{|\boldsymbol{\dot\gamma}|}}\right> |\dot{\boldsymbol{\gamma}}| dt$$
Taking the scalar inside the dot product:
$$\oint_{\gamma} \left < \mathbf{v}, \mathbf{\mathbf{R}\boldsymbol{\dot\gamma}}\right> |\dot{\boldsymbol{\gamma}}|/ |\boldsymbol{\dot\gamma}| dt$$
$$\oint_{\gamma} \left < \mathbf{v}, \mathbf{\mathbf{R}\boldsymbol{\dot\gamma}}\right> dt$$
Using the info above, I can answer your questions:
Note that $\boldsymbol{\gamma}(t) = \begin{bmatrix}
 2\cos(t)\\ 
 2\sin(t) 
\end{bmatrix}$ then:
$$ds = |\dot{\boldsymbol{\gamma}}| dt = \sqrt{\left(2\cos(t)\right)^2 + \left(2\sin(t)\right)^2} dt = \sqrt{4\cos^2(t) + 4\sin^2(t)}dt = 2dt$$
And $\mathbf{n}(t)$:
$$\mathbf{n}(t) = \mathbf{R}\mathbf{t} = \mathbf{R}\dot{\boldsymbol{\gamma}}(t) =\begin{bmatrix}
 0 & 1\\ 
 -1 & 0 
\end{bmatrix} 
\begin{bmatrix} 
-2\sin(t)\\
2\cos(t)
\end{bmatrix} = \begin{bmatrix} 
2\cos(t)\\
2\sin(t)
\end{bmatrix}$$
Hope this answer your question, in some way :)
