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How do we show on a picture that a solid angle equals to this equation i found on a Wikipedia:

\begin{align} \Omega =\!\!\!\int\limits_{\vartheta =0}^{\pi}\int\limits_{\varphi=0}^{2\pi}\underbrace{\sin\vartheta\, \textrm{d}\vartheta\, \textrm{d}\varphi}_{\textrm{d}\Omega} \end{align}

71GA
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1 Answers1

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Draw a "square" on a sphere from latitude $\frac \pi 2- \vartheta$ to $\frac \pi 2- \vartheta + d \vartheta$ and longitude $\varphi$ to $\varphi + d\varphi$. Note that the linear distance along the line of latitude is $\sin \vartheta d\varphi$ and the distance along the line of longitude is $d\vartheta$

Ross Millikan
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    Could you please draw the picture ... i am a bit confused only listening to an explaination :) – 71GA Jun 12 '13 at 19:39
  • @71GA: I don't have a good way to draw a 3d sketch. You could look at a globe, say from 35N to 40N and 0W to 5W. Note that the distance around the line of latitude is shorter than the distance along the line of longitude. The factor is about $\cos 37.5^\circ$. The change between $\sin$ and $\cos$ is because latitude is measured up from the equator, while $\vartheta$ is measured down from the pole. – Ross Millikan Jun 12 '13 at 19:47