# I Find potential integrating on segments parallel to axes

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1. Nov 17, 2016

### crick

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. Nov 17, 2016

### haruspex

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
3. Nov 18, 2016

### Svein

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