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Math Help - Double differential / differential squared

  1. #1
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    Double differential / differential squared

    If r and \theta are funtions of t, does \frac{d^2 r}{dt^2}\bigg / \left ( \frac{d\theta}{dt}\right ) ^2 equal \frac{d^2 r}{d\theta ^2}?
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    Quote Originally Posted by wglmb View Post
    If r and \theta are funtions of t, does \frac{d^2 r}{dt^2}\bigg / \left ( \frac{d\theta}{dt}\right ) ^2 equal \frac{d^2 r}{d\theta ^2}?
    No. While the first derivative can be "treated like" a fraction, the second derivative cannot.

    For example, take r= t^2 and \theta= t^4. Then \frac{dr}{dt}= 2t, \frac{d\theta}{dt}= 4t^3, \frac{d^2r}{dt^2}= 2 and \frac{d^2\theta}{dt^2}= 12t^2.

    Now it is easy to see that \theta= (t^2)^2= r^2 so \frac{d\theta}{dr}= 2r= 2t^2= \frac{4t^3}{2t} = \frac{\frac{d\theta}{dt}}{\frac{dr}{dt}}.

    But \frac{d^2\theta}{dr^2}= 2 while \frac{\frac{d^2\theta}{dt^2}}{\frac{d^2r}{dt^2}}= \frac{12t^2}{2}= 6t^2.
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    Thanks a lot
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    Quote Originally Posted by wglmb View Post
    If r and \theta are funtions of t, does \frac{d^2 r}{dt^2}\bigg / \left ( \frac{d\theta}{dt}\right ) ^2 equal \frac{d^2 r}{d\theta ^2}?
    To find  \frac{d^2r}{d \theta^2} you would need to calculate  <br />
\frac{\displaystyle \frac{d}{dt}\left(\displaystyle \displaystyle \frac{dr}{dt}\bigg /\displaystyle \frac{d \theta}{dt}\right)}{\displaystyle \frac{d \theta}{dt}}<br />
.
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