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Thread: Differential Formulas

  1. #1
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    Differential Formulas

    I have a question that says:

    Write a differential formula of a sphere when the radius of a balloon goes from r0 to r0+ dr.

    Can you walk me through this?
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  2. #2
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    Re: Differential Formulas

    For a cube:
    The volume of a cube of edge a is V=a^3

    Thus the change in volume \mathrm dV as the edge goes from a to a + \mathrm da is given by:

    \mathrm dV = (a + \mathrm da)^3 - a^3

    Expanding and simplifying we get

    \mathrm dV = 3a^2\mathrm da + 3a(\mathrm da)^2 + (\mathrm da)^3

    Ignoring non-linear terms of differentials (squares, cubes, etc.) we end up with

    \mathrm dV = 3a^2\mathrm da
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  3. #3
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    Re: Differential Formulas

    Ok, so for a sphere. The formula is V = 4/3pia^3 . Would i set it up like this:

    dV = 4/3pi(a+da)3 - 4/3pi(a)3
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  4. #4
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    Re: Differential Formulas

    Wait. Is this question asking for dy? I think I might just be confused over the wording.

    I know dy = f`(a)dx

    Is that what is happening when they ask for a differential?
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  5. #5
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    Re: Differential Formulas

    Yeah, that'd be it.
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  6. #6
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    Re: Differential Formulas

    When the radius of the sphere "goes from r to r+ dr" the volume goes from \frac{4}{3}\pi r^3 to \frac{4}{3}\pi (r+ dr)^3= \frac{4}{3}\pi(r^3+ 3r^2dr+3rdr^2+ dr^3). The difference is \Delta y= \frac{4}{3}\pi(3r^2 dr+ 3r dr^2+ dr^3). If we take dr sufficiently small that we can ignore the much smaller dr^2 and dr^3 that would be the differential dy= 3r^2 dr.
    Thanks from topsquark
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  7. #7
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    Re: Differential Formulas

    Quote Originally Posted by HallsofIvy View Post
    When the radius of the sphere "goes from r to r+ dr" the volume goes from \frac{4}{3}\pi r^3 to \frac{4}{3}\pi (r+ dr)^3= \frac{4}{3}\pi(r^3+ 3r^2dr+3rdr^2+ dr^3). The difference is \Delta y= \frac{4}{3}\pi(3r^2 dr+ 3r dr^2+ dr^3). If we take dr sufficiently small that we can ignore the much smaller dr^2 and dr^3 that would be the differential dy= 3r^2 dr.
    Well said, but for clarity, I just want to correct the small typo at the end. dy = 4\pi r^2 dr.
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  8. #8
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    Re: Differential Formulas

    Yes, I accidently left out the " \pi". Thanks for the correction.
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