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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 $\displaystyle a$ is $\displaystyle V=a^3$

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

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

    Expanding and simplifying we get

    $\displaystyle \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

    $\displaystyle \mathrm dV = 3a^2\mathrm da$
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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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    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 $\displaystyle \frac{4}{3}\pi r^3$ to $\displaystyle \frac{4}{3}\pi (r+ dr)^3= \frac{4}{3}\pi(r^3+ 3r^2dr+3rdr^2+ dr^3)$. The difference is $\displaystyle \Delta y= \frac{4}{3}\pi(3r^2 dr+ 3r dr^2+ dr^3)$. If we take dr sufficiently small that we can ignore the much smaller $\displaystyle dr^2$ and $\displaystyle dr^3$ that would be the differential $\displaystyle dy= 3r^2 dr$.
    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 $\displaystyle \frac{4}{3}\pi r^3$ to $\displaystyle \frac{4}{3}\pi (r+ dr)^3= \frac{4}{3}\pi(r^3+ 3r^2dr+3rdr^2+ dr^3)$. The difference is $\displaystyle \Delta y= \frac{4}{3}\pi(3r^2 dr+ 3r dr^2+ dr^3)$. If we take dr sufficiently small that we can ignore the much smaller $\displaystyle dr^2$ and $\displaystyle dr^3$ that would be the differential $\displaystyle dy= 3r^2 dr$.
    Well said, but for clarity, I just want to correct the small typo at the end. $\displaystyle dy = 4\pi r^2 dr$.
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  8. #8
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    Re: Differential Formulas

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