I would need help for a complete and detailed demonstration of this theorem if it is possible. I found several handouts on the net but I wondered if there was a demonstration not too long and complicated.

Theorem: Let $D\subseteq \mathbb R^3$ a non-empty open and $\overline F:D\rightarrow \mathbb R^3$ a vector field of class $\mathcal C^1$ with $\overline F=(F_x,F_y, F_z)=f(r)\widehat{\bf r}$ where:

$$F_x=f(r) \frac xr, \quad etc.$$

If $D$ is a domain simply connected - the differential form is therefore exact - and for every $\overline r=(x,y,z)\in D$ are valid the symmetry conditions

\begin{equation} \frac{\partial F_x}{\partial y}=\frac{\partial F_y}{\partial x}, \quad \frac{\partial F_x}{\partial z}=\frac{\partial F_z}{\partial x}, \quad \frac{\partial F_y}{\partial z}=\frac{\partial F_z}{\partial y} \end{equation}

hence the vectorial function $\overline F$ is conservative. In addition $\overline \nabla \times F=0$, $\forall\,\, \overline r\ne 0$.

  • $\begingroup$ Isn't the mathematics section of SE a better place to ask this pure mathematics question? $\endgroup$ – DanielC Dec 8 '18 at 23:50
  • $\begingroup$ @DanielC With so much honesty, I didn't get along so well. Free offenses of all kinds, and afterwards I was wiped out of the community of mathematicians. I feel great among LaTeX users and here. $\endgroup$ – Sebastiano Dec 8 '18 at 23:57

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