Transcription of Chapter 2. Parameterized Curves in R - University of Miami
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Chapter 2. Parameterized Curves in R - University of Miami
Chapter 2. Parameterized Curves smooth curve inR3is a smooth map : (a, b) eacht (a, b), (t) R3. Astincreases fromatob, (t) traces out a curve inR3. In terms of components, (t) = (x(t), y(t), z(t)),(1)or :x=x(t)y=y(t)z=z(t)a < t < b ,velocity at timet:d dt(t) = (t) = (x (t), y (t), z (t)) .speed at timet: d dt(t) =| (t)|Ex. :R R3, (t) = (rcost, rsint,0) - the standard parameterization of theunit circle, :x=rcosty=rsintz= 0 (t) = ( rsint, rcost,0)| (t)|=r(constant speed)1Ex. :R R3, (t) = (rcost, rsint, ht),r, h >0 constants (helix).
Chapter 2. Parameterized Curves in R3 Def. A smooth curve in R3 is a smooth map σ : (a,b) → R3. For each t ∈ (a,b), σ(t) ∈ R3.As t increases from a to b, σ(t) traces out a curve in
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