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Calculus IIChapter 6: 3-Dimensional SpaceSection 6-12

Cylindrical Coordinates — Question 10

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Question 10

Explain why cylindrical coordinates are not unique if negative values of rr are allowed. Prove the identity relating (r,θ,z)(r,\theta,z) and (−r,θ+π,z)(-r,\theta+\pi,z), then state a standard uniqueness convention and its remaining exception.

Original worksheet page 1: question and worked solution for 6-12-010
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Question 10 – Solution

Strategy Compare the rectangular coordinates produced by the two triples.

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Identity (−r)cos⁡(θ+π)=(−r)(−cos⁡θ)=rcos⁡θ,(-r)\cos(\theta+\pi)=(-r)(-\cos\theta)=r\cos\theta, (−r)sin⁡(θ+π)=(−r)(−sin⁡θ)=rsin⁡θ.(-r)\sin(\theta+\pi)=(-r)(-\sin\theta)=r\sin\theta. The zz-coordinate is unchanged, so (r,θ,z)∼(−r,θ+π,z).\boxed{(r,\theta,z)\sim(-r,\theta+\pi,z)}.

Convention Requiring r≥0r\ge 0 and 0≤θ<2π0\le\theta<2\pi makes coordinates unique whenever r>0r>0. On the zz-axis, r=0r=0 and every angle represents the same point, so θ\theta remains intrinsically undetermined there.

Original worksheet page 2: question and worked solution for 6-12-010

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