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 $\overline \nabla \times F=0$, $\forall\,\, \overline r\ne 0$.

  • Isn't the mathematics section of SE a better place to ask this pure mathematics question? – DanielC Dec 8 at 23:50
  • @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. – Sebastiano Dec 8 at 23:57

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.