Differentiate the formula for the volume of a sphere.?

Question

When I differentiate the formula for the volume of a sphere I get the surface area of a sphere. Similarly for the area and circumference of a circle. Is this more than just a coincidence. Does anyone know?

Answer 1

It works for some coordinate systems but not for others; for example, it does not work for rectangular or cylindrical coordinates.

Answer 2

This is not a coincidence. You can see this as follows.

For a sphere, what is the volume of the sphere in terms of an integral? Well, it is a sum (integral) of thin spherical shells. The volume of a spherical shell is ~ 4 pi r^2 dr.

So the volume of a sphere is int 4 pr r^2 dr = 4/3 pi r ^3.

So you see the integral of surface area is volume and hence derivative of volume is surface area.

Same with circle.

Answer 3

given quantity is a function of radius (a)V = V(R) = 4/3piR^3 fee of replace of quantity wrt radius = dV/dR = V'(R) = 4/3pi*(3R^2) => dV/dR = 4piR^2 (b)the quantity V'(R) provides the floor component to the sphere pondering the gadgets V = m^3 and R =m so for dV/dR we've m^3/m = m^2 the unit m^2 corresponds to section(ie., floor component to a sphere)

Answer 4

This is not just a coincidence. The differentiation calculations are correct.

Answer 5

No, it's not a pure coincidence. This page explains it to some extent:
http://mathforum.org/library/drmath/view/53661.html