Applet: Fluid flow through a point of oriented helicoid
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The function $\dlsp(\spfv,\spsv) = (\spfv\cos \spsv, \spfv\sin \spsv, \spsv)$ parametrizes a helicoid when $(\spfv,\spsv) \in \dlr$, where $\dlr$ is the rectangle $[0,1] \times [0, 2\pi]$ shown in the first panel. The cyan vector at the blue point $\dlsp(\spfv,\spsv)$ is the upward pointing unit normal vector at that point. The magenta vector at that point represents fluid flow that passes through the surface. In this case, the fluid flow is the constant $\dlvf=(0,1,1)$ at every point. Even though the fluid flow is constant, the flux through the surface changes, as it is the component of the flow normal to the surface. At the location of the blue point, the flux through the surface, $\dlvf \cdot \vc{n}$, is shown in the lower right corner. You can drag the blue point in $\dlr$ or on the helicoid to specify both $\spfv$ and $\spsv$.
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