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Derivative of area =circumference

  • #1
For a circle, the derivative of the formula for area is the formula for circumference:

d/dr([pi*r^2)=2*pi*r.

Similarly, the derivative of the formula for the volume of a sphere is equal to the formula for surface area: d/dr(4/3*pi*r^3)=4*pi*r^2.

I'm positive that these are consequences of Stokes's (Green's) and Divergence theorems, but I don't know how to show this. Could someone please explain? Thanks.
 

Answers and Replies

  • #2
mathman
Science Advisor
7,758
414
Looking at it as an integration exercise makes things clearer. Add up circular lamina of length 2πr and width dr to get area. Similarly shells of area 4πr2 and width dr to get volume.
 

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