Surface Area and Volume Derivative

Date: 10/30/2000 at 23:58:36
From: Mike
Subject: Derivatives of volume equations

Hi. I needed to find the equations for volumes of a sphere, cone, 
ellipsoid and cylinder. Then I needed to take the first derivative of 
these equations. I have the equations for volume, and I got the first 
derivative for the volume equation of a sphere is 4*pi*r^2, which is 
the surface area. I was wondering if the derivatives of the other 
three volume equations are also the surface area equations.
  
What would be the first derivative of these volumes?

     Cone:       V = 1/3*pi*r^2*h
     Ellipsoid:  V = 4/3*pi*a*b*c
     Cylinder:   V = pi*r^2*h

Thank you.

Date: 10/31/2000 at 08:48:15
From: Doctor Peterson
Subject: Re: Derivatives of volume equations

Hi, Mike.

Yes, this is true in many cases, but you have to be careful. I wrote 
about this (for students who haven't seen calculus yet) here:

  Area, Surface Area, and Volume: How to Tell One Formula from Another
  http://mathforum.org/dr.math/problems/leslies1.18.99.html   

The trick in most cases is that there is more than one variable; if 
you take the derivative with respect to one variable, AND if variation 
in that variable is perpendicular to the surface, you get the area of 
the surface generated.

For example, if you differentiate the volume of a cylinder with 
respect to the radius, you are looking at the rate of change in volume 
when you expand the cylinder radially, as if by painting the lateral 
surface. The volume of paint, for a thin layer, is the surface area 
times the thickness; so the derivative will in fact be the area of 
this lateral surface (the volume of paint divided by the thickness.) 
If, on the other hand, you differentiate with respect to the height, 
you are increasing the height by "painting" one end surface, and the 
derivative will be the area of that circle (NOT both ends).

The cone and ellipsoid are different, because in these cases adding a 
delta to r, h, a, b, or c doesn't add an even "coat of paint" to the 
shape, so you won't get the surface area. You may want to consider 
whether there is a way to change r and h at the same time in a cone in 
order to "paint" it evenly, and so find the surface area; but that's 
pretty advanced. For the ellipsoid, there isn't even a formula (using 
standard functions) for the surface area, so I know you won't be able 
to come up with anything in that case.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/