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'position vector'
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| Title of object: |
position vector |
| Canonical Name: |
PositionVector |
| Type: |
Definition |
| Created on: |
2005-07-25 17:40:54 |
| Modified on: |
2005-07-25 18:08:06 |
| Classification: |
msc:15A72 |
Revision comment (for changes between this and next version):
| Changes for correction #6924 ('entry broken'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem} |
Content:
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 variable point\, $(x,\,y,\,z)$\, \PMlinkescapetext{represents} a vector field and its \PMlinkescapetext{length}
$$r := \sqrt{x^2+y^2+z^2}$$
a scalar \PMlinkescapetext{field}.
The simple formulae}
\begin{itemize}
\item $\nabla\cdot\vec{r} = 3$
\item $\nabla\times\vec{r} = \vec{0}$
\item $\nabla r = \frac{\vec{r}}{r} = \vec{r}^0$
\item $\nabla\frac{1}{r} = -\frac{\vec{r}}{r^3} = -\frac{\vec{r}^0}{r^2}$
\item $\nabla^2\frac{1}{r} = 0$
\end{itemize}
are valid, where $\vec{r}^0$ is the unit vector having the direction of $\vec{r}$.
If\, $\vec{c}$\, is a \PMlinkescapetext{constant} vector,\, $\vec{V}\!:\mathbb{R}\to\mathbb{R}^3$\, a vector function and\, $F\!:\mathbb{R}\to\mathbb{R}$\, is a twice differentiable function, then the formulae
\begin{itemize}
\item $\nabla(\vec{c}\cdot\vec{r}) = \vec{c}$
\item $\nabla\cdot(\vec{c}\times\vec{r}) = 0$
\item $(\vec{V}\cdot\nabla)\vec{r} = \vec{V}$
\item $\nabla F(r) = F'(r)\vec{r}^0$
\item $\nabla^2F(r) = F''(r)+\frac{2}{r}F'(r)$
\end{itemize}
hold.
\begin{thebibliography}{9}
\bibitem{VV}{\sc K. V\"ais\"al\"a:} {\em Vektorianalyysi}. \,Werner S\"oderstr\"om Osakeyhti\"o, Helsinki (1961).
\end{thebibliography} |
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