# position vector

In the space $\mathbb{R}^{3}$, the vector

 $\vec{r}\;:=\;(x,\,y,\,z)\;=\;x\vec{i}+y\vec{j}+z\vec{k}$

directed from the origin to a point  $(x,\,y,\,z)$  is the position vector of this point. When the point is , $\vec{r}$ a vector field and its

 $r\;:=\;\sqrt{x^{2}\!+\!y^{2}\!+\!z^{2}}$

a scalar .

The

• $\nabla\!\cdot\vec{r}\;=\;3$

• $\nabla\!\times\!\vec{r}\;=\;\vec{0}$

• $\displaystyle\nabla r\;=\;\frac{\vec{r}}{r}\;=\;\vec{r}^{\,0}$

• $\displaystyle\nabla\frac{1}{r}\;=\;-\frac{\vec{r}}{r^{3}}\;=\;-\frac{\vec{r}^{% \,0}}{r^{2}}$

• $\displaystyle\nabla^{2}\frac{1}{r}\;=\;0$

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

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

• $\nabla(\vec{c}\cdot\!\vec{r})\;\;=\vec{c}$

• $\nabla\cdot(\vec{c}\times\vec{r})\;=\;0$

• $(\vec{U}\!\cdot\!\nabla)\vec{r}\;=\;\vec{U}$

• $(\vec{U}\!\times\!\nabla)\!\cdot\!\vec{r}\;=\;0$

• $(\vec{U}\!\times\!\nabla)\!\times\!\vec{r}\;=\;-2\vec{U}$

• $\nabla f(r)\;=\;f^{\prime}(r)\,\vec{r}^{\,0}$

• $\displaystyle\nabla^{2}f(r)\;=\;f^{\prime\prime}(r)\!+\frac{2}{r}f^{\prime}(r)$

hold.

## References

• 1 K. Väisälä: Vektorianalyysi.  Werner Söderström Osakeyhtiö, Helsinki (1961).
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