2010 - Solutions
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Transcript of 2010 - Solutions
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7/28/2019 2010 - Solutions
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2. (b)
Let X= Xii and Y = Yii and f, g C(M), then
X2f = Xjj(Xiif)
= Xj(jXiif+ X
i2jif)
= XjjXiif+ X
jXi2ijf
hence
X2(fg) = XjjXii(fg) + X
jXi2ij(fg)
= XjjXi((if)g + fig) + X
iXj(j(ifg + fig))
= XjjXi((if)g + fig) + X
iXj(2jifg + ifjg + jfig + f2
jig)
= (XjjXiif+ X
jXi2jif)g + f(XjjX
iig + XjXi2jig) + 2X
jXiifjg
= (X2f)g + f(X2g) + 2(Xf)(Xg)
the extra term doesnt always vanish, so X2 is not a derivation.