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.