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Math Help - Using diferentials

  1. #1
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    Using diferentials

    Let there be 3 resistors in parallel. We have that the total resistance R satisfies:

     1/R=1/R_1+1/R_2+1/R_3 . Find dR.

    When I tried to do this, I ended up getting a ridicilously long answer. I was just wondering if someone wouldn't mind helping me out with this. Thanks
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  2. #2
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    Quote Originally Posted by HelloWorld2 View Post
    Let there be 3 resistors in parallel. We have that the total resistance R satisfies:

     1/R=1/R_1+1/R_2+1/R_3 . Find dR.

    When I tried to do this, I ended up getting a ridicilously long answer. I was just wondering if someone wouldn't mind helping me out with this. Thanks
    Dear HelloWorld2,

    dR=\frac{\partial R}{\partial R_1}dR_1+\frac{\partial R}{\partial R_2}dR_2+\frac{\partial R}{\partial R_3}dR_3

    Since, \frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R  _3}

    -\frac{1}{R^2}\frac{\partial R}{\partial R_1}=-\frac{1}{R_{1}^{2}}\Rightarrow{\frac{\partial R}{\partial R_1}=\frac{R^2}{R_{1}^{2}}}

    Similarly, \frac{\partial R}{\partial R_2}=\frac{R^2}{R_{2}^{2}}}

    \frac{\partial R}{\partial R_3}=\frac{R^2}{R_{3}^{2}}}

    Therefore, dR=R^2\left(\frac{dR_1}{R_{1}^{2}}+\frac{dR_2}{R_{  2}^{2}}+\frac{dR_3}{R_{3}^{2}}\right)

    Hope this will help you.
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