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Merge spherical coordinate figure
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oToToT committed Oct 20, 2023
1 parent eb7ff44 commit 23e34c6
Showing 1 changed file with 14 additions and 7 deletions.
21 changes: 14 additions & 7 deletions codes/Misc/Theorems.tex
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\subsubsection{Spherical Coordinate}

\begin{center}
\begin{tabular}{ccc}
{$\begin{aligned}
x = r\sin\theta\cos\phi \\
y = r\sin\theta\sin\phi \\
z = r\cos\theta
\end{aligned}$}
&
\includegraphics[width=25mm]{../codes/Misc/sphericalCoordinates.pdf}
\end{center}
\[\begin{array}{cc}
x = r\sin\theta\cos\phi & r = \sqrt{x^2+y^2+z^2}\\
y = r\sin\theta\sin\phi & \theta = \textrm{acos}(z/\sqrt{x^2+y^2+z^2})\\
z = r\cos\theta & \phi = \textrm{atan2}(y,x)
\end{array}\]
&
{$\begin{aligned}
r = \sqrt{x^2+y^2+z^2} \\
\theta = \textrm{acos}(z/\sqrt{x^2+y^2+z^2}) \\
\phi = \textrm{atan2}(y,x)
\end{aligned}$}
\end{tabular}

\subsubsection{Sherman-Morrison formula}
$\left(A + uv^\textsf{T}\right)^{-1} = A^{-1} - \frac{A^{-1}uv^\textsf{T}A^{-1}}{1 + v^\textsf{T}A^{-1}u}$
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