# Differentials of spherical surface area and volume

## Main Question or Discussion Point

please tell me if i did this correctly:

task: i'm trying to divide the differential dA by dV

where.. dA = differential surface area of a sphere, dV = differential volume of a sphere

dA=8$\pi$rdr
dV=4$\pi$r2dr

so then dA/dV= 2/r

Also, if i treat this as a derivative, then would dA/dV = 1/dr?

Simon Bridge
Homework Helper
dA/dV would be how the area changes with the volume.
So you can check it out from the known relations:

##A(R)=4\pi R^2## is the surface area of a sphere radius R, and
##V(R)=\frac{4}{3}\pi R^3## is it's volume, then

So find A(V) and differentiate.

lurflurf
Homework Helper
dA/dV= 2/r
this is right
dA/dV = 1/dr
this is not, how did you arrive at it?

HallsofIvy
For a sphere of radius r, $A= 4\pi r^2$ and $V= (4/3)\pi r^3[itex]. So [itex]dA/dr= 8\pi r$ and $dV/dr= 4\pi r^2$.
Then $dV/dA= (dV/dr)/(dA/dr)= (4\pi r^2)/(8\pi r)= (1/2)r$
and $dA/dV= (dA/dr)/(dA/dr)= (8\pi r)/(4\pi r^2)= 2/r$