PlanetPhysics/Position Vector

In the space ${\displaystyle \mathbb {R} ^{3}}$, the vector ${\displaystyle {\vec {r}}\;:=\;(x,\,y,\,z)\;=\;x{\vec {i}}+y{\vec {j}}+z{\vec {k}}}$ directed from the origin to a point \,${\displaystyle (x,\,y,\,z)}$\, is the position vector of this point.\, When the point is variable, ${\displaystyle {\vec {r}}}$ represents a vector field and its length ${\displaystyle r\;:=\;{\sqrt {x^{2}+y^{2}+z^{2}}}}$ a scalar field.

The simple formulae

• ${\displaystyle \nabla \!\cdot {\vec {r}}=3}$
• ${\displaystyle \nabla \!\times \!{\vec {r}}={\vec {0}}}$
• ${\displaystyle \nabla r={\frac {\vec {r}}{r}}={\vec {r}}^{0}}$
• ${\displaystyle \nabla {\frac {1}{r}}=-{\frac {\vec {r}}{r^{3}}}=-{\frac {{\vec {r}}^{0}}{r^{2}}}}$
• ${\displaystyle \nabla ^{2}{\frac {1}{r}}=0}$

are valid, where ${\displaystyle {\vec {r}}^{0}}$ is the unit vector having the direction of ${\displaystyle {\vec {r}}}$.

If\, $\displaystyle \vec$ \, is a constant vector,\, ${\displaystyle {\vec {U}}\!\!:\mathbb {R} ^{3}\to \mathbb {R} ^{3}}$\, a vector function and\, ${\displaystyle f\!\!:\mathbb {R} \to \mathbb {R} }$\, is a twice differentiable function, then the formulae

• $\displaystyle \nabla(\vec\cdot\!\vec{r}) = \vec$
• ${\displaystyle \nabla \cdot ({\vec {\times }}{\vec {r}})=0}$
• ${\displaystyle ({\vec {U}}\!\cdot \!\nabla ){\vec {r}}={\vec {U}}}$
• ${\displaystyle ({\vec {U}}\!\times \!\nabla )\!\cdot \!{\vec {r}}=0}$
• ${\displaystyle ({\vec {U}}\!\times \!\nabla )\!\times \!{\vec {r}}=-2{\vec {U}}}$
• ${\displaystyle \nabla f(r)=f'(r)\,{\vec {r}}^{0}}$
• ${\displaystyle \nabla ^{2}f(r)=f''(r)\!+{\frac {2}{r}}f'(r)}$

hold.

[1]

References

1. {\sc K. V\"ais\"al\"a:} Vektorianalyysi . \,Werner S\"oderstr\"om Osakeyhti\"o, Helsinki (1961).