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[parent] position vector (Definition)

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¨AISÄLÄ: Vektorianalyysi. Werner Söderström Osakeyhtiö, Helsinki (1961).



"position vector" is owned by pahio.
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See Also: example of curvature (space curve), dyad product, tilt curve

Other names:  radius vector

This object's parent.

Attachments:
solid angle (Definition) by pahio
direction cosines (Definition) by pahio
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Cross-references: differentiable function, function, unit vector, scalar, vector field, point, origin, vector
There are 19 references to this entry.

This is version 14 of position vector, born on 2005-07-25, modified 2007-09-22.
Object id is 7265, canonical name is PositionVector.
Accessed 3153 times total.

Classification:
AMS MSC: 15A72 (Linear and multilinear algebra; matrix theory :: Vector and tensor algebra, theory of invariants)
Pending Errata and Addenda
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Browsing in PM is very slow by perucho on 2006-04-15 02:42:25
Since I can to browse fine in another sites, I assume that issue is in PM. I can't look TeX source entries because of excesive delay.
Also I got either SERVER ERROR or the page can't be displayed.
Connection transfer=768Kbps.
perucho
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