does the volume of a ball remain constant under deformation? I'm a psychology student and was reading Piaget, he says that the volume of a sphere (ball of clay) remains constant if we deform the sphere into a roll for example, If you take the limit case of the roll its volume goes to zero as one of its dimensions goes to zero, I mean, its length increases but the diameter decrases. Using the intermediate value theorem could I say that when the sphere is deformed its volume is not conserved?. I am not a mathematics student, I guess I should say it is continue uosly deformed and add some more information, I did the best I could.
 A: Say, the volume is $V$ and the roll is a cylinder of height $h$ and radius $r$. Then 
$\pi r^2h=V $
hence $h = V/(\pi r^2)$. As the radius $r$ approaches $0$, the height $h$ grows indefinitely. One expresses this as
$$
\lim_{r\to 0} V/(\pi r^2) = \infty
$$
The volume does remain constant in the process.
The apparent contradiction arises when you imagine that $r$ actually becomes $0$ in the process, and then conclude that the volume suddenly drops from $V$ to $0$. But $r$ never becomes $0$ in reality. It can be arbitrarily small, but not zero. 
To speculate about the volume of a "cylinder" with zero radius and infinite height is about as productive as to debate what would happen if an unstoppable object collided with an immovable one.
A: I shall answer this question to myself five years ago:
It depends on what transformations you are considering, if you are defining volume preserving transformations then the volume will be preserved (I tell you now that I know it is symplectic geometry what studies those transformations that you thought were a huge theorem for one of Piaget's children to naturally realize). Although you were intuitively thinking about something called a homotopy from the ball, to the cylinder to the line segment and there volume is not preserved. The interplay of topological transformations and geometrical transformations of a space is probably the most beautiful thing I have seen. This is proof for the power of curiosity.
