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).

Title	position vector
Canonical name	PositionVector
Date of creation	2013-03-22 15:25:15
Last modified on	2013-03-22 15:25:15
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	18
Author	pahio (2872)
Entry type	Definition
Classification	msc 15A72
Synonym	radius vector
Related topic	ExampleOfCurvatureSpaceCurve
Related topic	DyadProduct
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