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  • Perpendicular tangent vector

    Proposition Let c:[0,1]Rn be a curve such that |c(t)|=1 for all t. Show that c(t)c(t) and the tangent vector to c at t are perpendicular. Solution \[\begin{align*} \ev{(c^1(t), \cdots, c^n(t))_{c(t)}, ((c^1)'(t), \cdots, (c^n)'(t))_{c(t)}} &= \ev{(c^1(t), \cdots, c^n(t)), ((c^1)'(t), \cdots, (c^n)'(t))} \\ &= c^1(t)(c^1)'(t) + \cdots +...


  • The set of all ideals in which every element is a zero divisor

    Proposition In a ring A, let Σ be the set of all ideals in which every element is a zero divisor. Show that the set Σ has maximal elements and that every maximal element of Σ is a prime ideal. Hence the set of zero-divisors in A is a union...


  • Pictures of Spec

    Proposition Draw pictures of Spec(Z),Spec(R),Spec(C[x]),Spec(R[x]),Spec(Z[x]). Solution I am not sure how to draw pictures of these spaces, so I will just describe them in words. Spec(Z) Spec(Z)={(0)}{(p)p prime}. Let EZ. If E is...


  • Tangent vector

    Proposition Let c be a differentiable curve in Rn, that is, a differentiable function c:[0,1]Rn. Define the tangent vector v of c at t as c((e1)t)=((c1)(t),,(cn)(t))c(t). If f:RnRm, show that the tangent vector to fc at t is...


  • Tangent vectors and tangent lines

    Proposition Let f:RR and define c:RR2 by c(t)=(t,f(t)). Show that the end point of the tangent vector of c at t lies on the tangent line to the graph of f at (t,f(t)). Solution Let t0R be given....