| Version 5 |
Version 4 |
| In the space $\mathbb{R}^3$, the vector |
In the space $\mathbb{R}^3$, the vector |
| $$\vec{r} := (x,\,y,\,z) = x\vec{i}+y\vec{j}+z\vec{k}$$ |
$$\vec{r} := (x,\,y,\,z) = x\vec{i}+y\vec{j}+z\vec{k}$$ |
| directed from the origin to a variable point\, $(x,\,y,\,z)$\, \PMlinkescapetext{represents} a vector field and its \PMlinkescapetext{length} |
directed from the origin to a variable point\, $(x,\,y,\,z)$\, \PMlinkescapetext{represents} a vector field and its \PMlinkescapetext{length} |
| $$r := \sqrt{x^2+y^2+z^2}$$ |
$$r := \sqrt{x^2+y^2+z^2}$$ |
|
a scalar \PMlinkescapetext{field}.
|
a scalar \PMlinkescapetext{field.
|
|
|
| The simple formulae} |
The simple formulae} |
| \begin{itemize} |
\begin{itemize} |
| \item $\nabla\cdot\vec{r} = 3$ |
\item $\nabla\cdot\vec{r} = 3$ |
| \item $\nabla\times\vec{r} = \vec{0}$ |
\item $\nabla\times\vec{r} = \vec{0}$ |
| \item $\nabla r = \frac{\vec{r}}{r} = \vec{r}^0$ |
\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 $\nabla\frac{1}{r} = -\frac{\vec{r}}{r^3} = -\frac{\vec{r}^0}{r^2}$ |
| \item $\nabla^2\frac{1}{r} = 0$ |
\item $\nabla^2\frac{1}{r} = 0$ |
| \end{itemize} |
\end{itemize} |
| are valid, where $\vec{r}^0$ is the unit vector having the direction of $\vec{r}$. |
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 |
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} |
\begin{itemize} |
| \item $\nabla(\vec{c}\cdot\vec{r}) = \vec{c}$ |
\item $\nabla(\vec{c}\cdot\vec{r}) = \vec{c}$ |
| \item $\nabla\cdot(\vec{c}\times\vec{r}) = 0$ |
\item $\nabla\cdot(\vec{c}\times\vec{r}) = 0$ |
| \item $(\vec{V}\cdot\nabla)\vec{r} = \vec{V}$ |
\item $(\vec{V}\cdot\nabla)\vec{r} = \vec{V}$ |
| \item $\nabla F(r) = F'(r)\vec{r}^0$ |
\item $\nabla F(r) = F'(r)\vec{r}^0$ |
| \item $\nabla^2F(r) = F''(r)+\frac{2}{r}F'(r)$ |
\item $\nabla^2F(r) = F''(r)+\frac{2}{r}F'(r)$ |
| \end{itemize} |
\end{itemize} |
| hold. |
hold. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{VV}{\sc K. V\"ais\"al\"a:} {\em Vektorianalyysi}. \,Werner S\"oderstr\"om Osakeyhti\"o, Helsinki (1961). |
\bibitem{VV}{\sc K. V\"ais\"al\"a:} {\em Vektorianalyysi}. \,Werner S\"oderstr\"om Osakeyhti\"o, Helsinki (1961). |
| \end{thebibliography} |
\end{thebibliography} |
