\(F(\vec{r})\)
is de grootte van de kracht in radieele richting. Hieruit volgt dat:
\(F_x(\vec{r}) = F(\vec{r}) \cdot \frac{x}{|\vec{r}|}\)
dus:
\(\frac{\partial F_x(\vec{r})}{\partial y} = \frac{\partial F(\vec{r}) \cdot \frac{x}{|\vec{r}|} }{\partial y} = \frac{\partial F(\vec{r})}{\partial y} \cdot \frac{x}{|\vec{r}|} + F(\vec{r}) \cdot \frac{\partial \frac{x}{|\vec{r}|} }{\partial y}\)
Voor de lengte van de vector r geldt:
\(|\vec{r}| = \sqrt{x^2 + y^2 + z^2}\)
dus:
\(= \frac{\partial F(\vec{r})}{\partial y} \cdot \frac{x}{|\vec{r}|} + F(\vec{r}) \cdot \frac{\partial \frac{x}{\sqrt{x^2 + y^2 + z^2}} }{\partial y} = \frac{\partial F(\vec{r})}{\partial y} \cdot \frac{x}{|\vec{r}|} - F(\vec{r}) \cdot \frac{x \cdot y}{(x^2 + y^2 + z^2)^\frac{3}{2}}\)
Analoog kun je
\(\frac{\partial F_y(\vec{r})}{\partial x}\)
afleiden. Deze moet je, vanwege de rotatie = nul, aan elkaar gelijk stellen:
\(\frac{\partial F(\vec{r})}{\partial y} \cdot \frac{x}{|\vec{r}|} - F(\vec{r}) \cdot \frac{x \cdot y}{(x^2 + y^2 + z^2)^\frac{3}{2}} = \frac{\partial F(\vec{r})}{\partial x} \cdot \frac{y}{|\vec{r}|} - F(\vec{r}) \cdot \frac{y \cdot x}{(x^2 + y^2 + z^2)^\frac{3}{2}}\)
ofwel:
\(\frac{\partial F(\vec{r})}{\partial y} \cdot \frac{x}{|\vec{r}|} = \frac{\partial F(\vec{r})}{\partial x} \cdot \frac{y}{|\vec{r}|}\)
en dus:
\(\frac{1}{y} \frac{\partial F(\vec{r})}{\partial y} = \frac{1}{x} \frac{\partial F(\vec{r})}{\partial x}\)