In general, the volume
has total derivative
If the cylinder's height is kept constant, then
and we have
which is to say,
and
are directly proportional by a factor equivalent to the lateral surface area of the cylinder (
).
Meanwhile, if the cylinder's radius is kept fixed, then
since
. In other words,
and
are directly proportional by a factor of the surface area of the cylinder's circular face (
).
Finally, the general case (
and
not constant), you can see from the total derivative that
is affected by both
and
in combination.