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I'm interested in part iv) on the attachment. This is my work so far:
e=(1,0) and e'=(0,1) form a basis of the tangent space at any point z=(x,y). Making the identification (x,y)>x+iy, we get g(e,e')=0 and g(e,e)=g(e',e')=$\frac{1}{im(z)^2}$.
a(t)=z+t and b(t)=z+it are generating curves for e,e' respectively.
(lets call the function f)
$f(z)=\frac{z}{z^2}$ so $f(a(t))=\frac{z+t}{z+t^2}$. I need to find f'(a(t)) to proceed. How can I cope with differentiating the modulus of a complex number z? Thanks
e=(1,0) and e'=(0,1) form a basis of the tangent space at any point z=(x,y). Making the identification (x,y)>x+iy, we get g(e,e')=0 and g(e,e)=g(e',e')=$\frac{1}{im(z)^2}$.
a(t)=z+t and b(t)=z+it are generating curves for e,e' respectively.
(lets call the function f)
$f(z)=\frac{z}{z^2}$ so $f(a(t))=\frac{z+t}{z+t^2}$. I need to find f'(a(t)) to proceed. How can I cope with differentiating the modulus of a complex number z? Thanks
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