PlanetPhysics/Position Vector

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

The simple formulae

• $\mathrm{\nabla }\cdot \overrightarrow{r}=3$
• $\mathrm{\nabla }\times \overrightarrow{r}=\overrightarrow{0}$
• $\mathrm{\nabla }r=\frac{\overrightarrow{r}}{r}={\overrightarrow{r}}^0$
• $\mathrm{\nabla }\frac{1}{r}=-\frac{\overrightarrow{r}}{r^3}=-\frac{{\overrightarrow{r}}^0}{r^2}$
• ${\mathrm{\nabla }}^2\frac{1}{r}=0$

are valid, where ${\overrightarrow{r}}^0$ is the [[../PureState/|unit vector]] having the direction of $\overrightarrow{r}$.

If\, Failed to parse (syntax error): {\displaystyle \vec} \, is a constant vector,\, $\overrightarrow{U}:{ℝ}^3\to {ℝ}^3$\, a [[../VectorFunctions/|vector function]] and\, $f:ℝ\to ℝ$\, is a twice differentiable [[../Bijective/|function]], then the formulae

• Failed to parse (syntax error): {\displaystyle \nabla(\vec\cdot\!\vec{r}) = \vec}
• $\mathrm{\nabla }\cdot (\overrightarrow{\times }\overrightarrow{r})=0$
• $(\overrightarrow{U}\cdot \mathrm{\nabla })\overrightarrow{r}=\overrightarrow{U}$
• $(\overrightarrow{U}\times \mathrm{\nabla })\cdot \overrightarrow{r}=0$
• $(\overrightarrow{U}\times \mathrm{\nabla })\times \overrightarrow{r}=-2\overrightarrow{U}$
• $\mathrm{\nabla }f(r)=f^{\prime }(r){\overrightarrow{r}}^0$
• ${\mathrm{\nabla }}^{2}f(r)=f^{″}(r)+\frac{2}{r}{f}^{\prime }(r)$

hold.