| Version 12 |
Version 11 |
| 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}. |
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|
| The \PMlinkescapetext{simple formulae} |
The \PMlinkescapetext{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 $\displaystyle\nabla r = \frac{\vec{r}}{r} = \vec{r}^0$
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\item $\\nabla r = \frac{\vec{r}}{r} = \vec{r}^0$
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\item $\displaystyle\nabla\frac{1}{r} = -\frac{\vec{r}}{r^3} = -\frac{\vec{r}^0}{r^2}$
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\item $\\nabla\frac{1}{r} = -\frac{\vec{r}}{r^3} = -\frac{\vec{r}^0}{r^2}$
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\item $\displaystyle\nabla^2\frac{1}{r} = 0$
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\item $\\nabla^2\frac{1}{r} = 0$
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| \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 $\vec{r}$. |
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| If\, $\vec{c}$\, is a \PMlinkescapetext{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 |
If\, $\vec{c}$\, is a \PMlinkescapetext{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 |
| \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 $(\vec{U}\!\cdot\!\nabla)\vec{r} = \vec{U}$ |
\item $(\vec{U}\!\cdot\!\nabla)\vec{r} = \vec{U}$ |
| \item $(\vec{U}\!\times\!\nabla)\!\cdot\!\vec{r} = 0$ |
\item $(\vec{U}\!\times\!\nabla)\!\cdot\!\vec{r} = 0$ |
| \item $(\vec{U}\!\times\!\nabla)\!\times\!\vec{r} = -2\vec{U}$ |
\item $(\vec{U}\!\times\!\nabla)\!\times\!\vec{r} = -2\vec{U}$ |
| \item $\nabla f(r) = f'(r)\,\vec{r}^0$ |
\item $\nabla f(r) = f'(r)\,\vec{r}^0$ |
|
\item $\displaystyle\nabla^2f(r) = f''(r)\!+\frac{2}{r}f'(r)$
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\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} |
