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I typed up one example on wolfram to see if this could be visualized

https://www.wolframalpha.com/input?i=plot+f(x,y)+=+(x+y,xy)

which was inspired by this question

https://math.stackexchange.com/questions/350963/inverse-of-a-vector-valued-function

and it returned a result that looks a lot like the kinds of trajectories that arise in systems of ordinary differential equations, though instead of lines, they're "vectors".

What are the applications of vector-valued functions of this form and their inverses functions to physics (or general science)? What is the relationship of such vectors to differential equations?

If I did the math correctly, the inverse function of this, assuming ##(u,v) = f(x,y)## then we solve for ##x## and ##y## in terms of ##u## and ##v## to obtain

##y = u/2 \pm \frac{1}{2} \sqrt{u^2 - 4v}## and ##x = u - u/2 \pm \frac{1}{2} \sqrt{u^2 - 4v}## though I'm confused as to what the final vector is now.

What is the correct inverse vector function, and why might someone take interest in it?