Math and stuff
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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 +...
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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...
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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 E⊂Z. If E is...
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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:Rn→Rm, show that the tangent vector to f∘c at t is...
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Tangent vectors and tangent lines
Proposition Let f:R→R and define c:R→R2 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 t0∈R be given....