position vector | Physics Library

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

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

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

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

a scalar field.

The simple formulae

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

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

If $\vec{c}$ is a constant 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

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

hold.

Bibliography

1: K. V�AIS�L�: Vektorianalyysi. Werner S�derstr�m Osakeyhti�, Helsinki (1961).

"position vector" is owned by pahio. [ full author list (2) ]
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