Engineering Calculus Notes 203

Engineering Calculus Notes 203 - 191 2.4. REGULAR CURVES...

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Unformatted text preview: 191 2.4. REGULAR CURVES with dx = 0 at t = t0 , and hence nearby (i.e., on J ). Now consider dt the two plane curves parametrized by → − (t) = (x(t) , y (t)) py → − (t) = (x(t) , z (t)). p z These are the projections of C onto, respectively the xy -plane and the xz -plane. Mimic the argument for Proposition 2 to show that each of these is the graph of the second coordinate as a function of the first. Challenge problems: 10. Arcs: In this exercise, we study some properties of arcs, as defined in Definition 2.4.6. → (a) Suppose − : I → R3 is a continuous, one-to-one vector-valued p function on the closed interval I = [a, b] with image the arc C , → → and suppose − (ti ) → − (t0 ) ∈ C . Show that ti → t0 in I . (Hint: p p Show that the sequence ti must have at least one accumulation point t∗ in I , and that for every such accumulation point t∗ , we → → → must have − (t∗ ) = − (t0 ). Then use the fact that − is p p p one-to-one to conclude that the only accumulation point of ti is t∗ = t0 . But a bounded sequence with exactly one accumulation point must converge to that point.) (b) Show that this property fails for the parametrization of the → circle by − (θ ) = (cos θ, sin θ ), 0 ≤ θ < 2π . p (c) Show that the graph of a continuous function f (x) defined on a closed interval [a, b] is an arc. (d) Give an example of an arc in space whose projections onto the three coordinate planes are not arcs. (This is a caution concerning how you answer the next question.) (e) In particular, it follows from Proposition 2.4.7 that every regular curve in the plane is locally an arc. Show that every regular curve in space is also locally an arc. (f) Show that, as in Lemma 2.4.2, any two regular parametrizations of the same arc are reparametrizations of each other. (Hint: First, prove that if one of the parametrizations is assumed to be globally one-to-one, and defined on [a, b], then there is a unique recalibration function relating the two parametrizations; then ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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