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Find the function phi(r,t) given its partial derivatives.

  1. May 23, 2015 #1
    I would like to define [itex]t^*= \phi(r, t)[/itex] given [itex]dt^* = \left( 1-\frac{k}{r} \right) dt + 0dr[/itex] where k is a constant.

    Perhaps it doesn't exist. It appears so simple, yet I've been running around in circles. Any hints?
     
  2. jcsd
  3. May 23, 2015 #2

    fzero

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    You want ##dt^*## to be an exact form, so that you can integrate it. An exact form can be expressed as

    $$ df = M dx + N dy,~~~M = \frac{\partial f}{\partial x},~~~N = \frac{\partial f}{\partial y},$$

    therefore a necessary condition that ##df## be exact is that

    $$ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.$$

    The corresponding relation for ##dt^*## fails, so we conclude that it is not exact and the corresponding ##t^*(r,t)## does not exist.
     
  4. May 23, 2015 #3
    Nicely done. Thank you!
     
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