An Example with Explicit Velocity Vectors 

Although completely general solutions are not forthcoming at this point, we can still evaluate some commonly used expressions that are often needed.  To demonstrate this, we first need to define the velocity vector v(x,y,z).  I use completely arbitrary definitions for the velocity components in terms of x, y, z and the unknown function Δ(x).  These functions were inspired by the Maple code of professor Davis as seen in ex1241.mw although they do not correspond exactly. 

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We can see that the components of v have now been assigned 

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Now we can see what some of the terms will look like at an arbitrary point (x,y,z) 

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These results, although lacking full generality, are still useful in many applications, especially since the results can be easily mapped to other coordinate systems as already demonstrated.