Given an 3-d object defined by a set of parametric equations (x, y, z), can you write a formula expressing the rate that a liquid rises as it's poured into this object at a constant flow rate?
Assume the object is an enclosed solid of revolution with a hole at the top.
(I have seen multiple questions like this asked before but they are not nearly as general as this one.)
Hint?
You can find a unique path/parameterization for a any curve that would yield constant speed with respect to the re-parameterization (using arc length as one example).
Let the volume occupied by the liquid be your curve, with the original parameter, and differentiate $z$ with respect to the arc-length (times your flow rate).
Alternatively a general expression would be the reciprocal of the $z$ derivative of the volume (times your constant).