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I Find potential integrating on segments parallel to axes

  1. Nov 17, 2016 #1
    A simple method to find the potential of a conservative vector field defined on a domain ##D## is to calculate the integral
    $$U(x,y,z)=\int_{\gamma} F \cdot ds$$

    On a curve ##\gamma## that is made of segments parallel to the coordinate axes, that start from a chosen point ##(x_0,y_0,z_0)##.

    I would like to know what are precisely the conditions that ##D## must satisfy to use this method. ##D## should be made in such way that "any point can be connected to ##(x_0,y_0,z_0)## with, indeed, a segment parallel to the coordinate axes".

    But what are the sufficient mathematical conditions for $D$ in order to have this property?**

    I would say that it surely has to be connected, but that seems not to be enough. For example taking
    $$D= \{ (x,y) : y>x-1\} \,\,\,\, \,\,\,\,\,(x_0,y_0)=(0,0)$$
    ##D## is connected but I do not think that any point can be connected to ##(0,0)## via a segment parallel to the coordinate axes.
     
  2. jcsd
  3. Nov 17, 2016 #2

    haruspex

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    It does not need to be. It only has to be connected by some path consisting of segments parallel to the axes. Of course, it could get quite messy.
    A simple example which does not work is two regions connected only by a diagonal line.
     
    Last edited: Nov 18, 2016
  4. Nov 18, 2016 #3

    Svein

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    In my mind it seems like a compactness argument should be useful, but I do not presently have the energy to follow up.
     
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