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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. [ full author list (2) ]

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See Also: Functorial Algebraic Geometry and Physics, vector

Other names:  radius vector

This object's parent.

Cross-references: function, vector function, unit vector, scalar, vector field, vector
There are 14 references to this object.

This is version 2 of position vector, born on 2009-04-17, modified 2009-04-18.
Object id is 641, canonical name is PositionVector.
Accessed 587 times total.

Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
Pending Errata and Addenda
None.
Discussion
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PositionVector by bci1 on 2009-04-17 21:32:47
Confirmed
[ reply | up ]

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