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Differentials of spherical surface area and volume

  1. Apr 13, 2014 #1
    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?
     
  2. jcsd
  3. Apr 13, 2014 #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.
     
  4. Apr 13, 2014 #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?
     
  5. Apr 13, 2014 #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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