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A-R-Q
 Post subject: Evaluate the Integral A.S. (1)
PostPosted: Fri, 23 Mar 2012 03:22:56 UTC 
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This is just one question. The typed-in version is this: http://i1084.photobucket.com/albums/j40 ... 0PicP4.png

The written attempt is below:
I would like to know where I messed up. :confused:
Part 1: http://i1084.photobucket.com/albums/j40 ... 0PicP1.png
Part 2: http://i1084.photobucket.com/albums/j40 ... 0PicP2.png
Part 3: http://i1084.photobucket.com/albums/j40 ... 0PicP3.png
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outermeasure
 Post subject: Re: Evaluate the Integral A.S. (1)
PostPosted: Fri, 23 Mar 2012 05:16:41 UTC 
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A-R-Q wrote:
This is just one question. The typed-in version is this: http://i1084.photobucket.com/albums/j40 ... 0PicP4.png

The written attempt is below:
I would like to know where I messed up. :confused:
Part 1: http://i1084.photobucket.com/albums/j40 ... 0PicP1.png
Part 2: http://i1084.photobucket.com/albums/j40 ... 0PicP2.png
Part 3: http://i1084.photobucket.com/albums/j40 ... 0PicP3.png


If F is an antiderivative of f, then \int_{\pi/4}^{\pi/3}f(x)\,\mathrm{d}x is F(\frac{\pi}{3})-F(\frac{\pi}{4}), not F(\frac{\pi}{4})-F(\frac{\pi}{3}).

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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