Past exam of the fluids and waves course of the University of Oxford / 2017 / 2 / a / i / Solution

Orient the planar material curve $C(t)$ counterclockwise, and let $\mathbf{t}$ and $\mathbf{n}$ be its unit tangent and outward unit normal vector. The circulation and outward volume flux areFor $\mathbf{u}=(u,v)$ and this orientation, $\mathbf{t}ds=(dx,dy)$ and $\mathbf{n}ds=(dy,-dx)$, givingHere $M$ is the instantaneous flux of the velocity field, rather than transport relative to the moving material curve; that relative transport is zero. If the enclosed region contains no source and the velocity field is smooth there, incompressibility and the planar divergence theorem imply $M=0$.