Math and stuff
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Zariski topology
Proposition Let A be a ring and let X be the set of all prime ideals of A. For each subset E of A, let V(E) denote the set of all prime ideals of A which contain E. Prove that if a is the ideal generated by E, then $V(E)...
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Necessary and sufficient conditions for A/R to be a field
Proposition Let A be a ring, R its nilradical. Show that the following are equivalent: A has exactly one prime ideal; every element of A is either a unit or nilpotent; A/R is a field. Solution By Proposition 1.8[Atiyah], R is the intersection of all the prime ideals of A....
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Path integrals
Proposition Evaluate the integrals ∫γxdz,∫γydz,∫γzdz and ∫γ¯zdz along each of the following paths. γ is the line segment from 0 to 1−i. γ=C[0,1]. γ=C[a,r] for some a∈C. Solution 1 γ(t)=t−it with $0...
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Change of variables
Proposition Prove Proposition 4.2 (A first course in complex analysis) and the fact that the length of γ does not change under reparametrization. Solution Let τ:[c,d]→[a,b] be given such that σ=γ∘τ. \[\begin{align*} \int_{c}^{d} f(\sigma(t))\sigma'(t) dt &= \int_{c}^{d} f((\gamma \circ \tau)(t))(\gamma \circ \tau)'(t)...
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Basic properties of push-forward and pullback
Proposition If f:Rn→Rm and g:Rm→Rp, show that (g∘f)∗=g∗∘f∗ and (g∘f)∗=f∗∘g∗. If f,g:Rn→R, show that d(f⋅g)=f⋅dg+g⋅df. Solution 1 Let...