Differentials of spherical surface area and volume

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  • #1
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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[itex]\pi[/itex]rdr
dV=4[itex]\pi[/itex]r2dr

so then dA/dV= 2/r


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

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  • #2
Simon Bridge
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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.
 
  • #3
lurflurf
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dA/dV= 2/r
this is right
dA/dV = 1/dr
this is not, how did you arrive at it?
 
  • #4
HallsofIvy
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For a sphere of radius r, [itex]A= 4\pi r^2[/itex] and [itex]V= (4/3)\pi r^3[itex].

So [itex]dA/dr= 8\pi r[/itex] and [itex]dV/dr= 4\pi r^2[/itex].

Then [itex]dV/dA= (dV/dr)/(dA/dr)= (4\pi r^2)/(8\pi r)= (1/2)r[/itex]
and [itex]dA/dV= (dA/dr)/(dA/dr)= (8\pi r)/(4\pi r^2)= 2/r[/itex]

"dA/dV= 1/dr" makes no sense because you have a differential on the right side and not on the left.
 

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