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Calculus IIIChapter 6: Surface IntegralsSection 6-2

Parametric Surfaces — Question 4

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Question 4

Consider the standard spherical parametrization 𝒓(ϕ,θ)=⟨3sin⁡ϕcos⁡θ,3sin⁡ϕsin⁡θ,3cos⁡ϕ⟩,\mathbf r(\phi,\theta)=\langle 3\sin\phi\cos\theta,3\sin\phi\sin\theta,3\cos\phi\rangle, where 0≤ϕ≤π0\le\phi\le\pi and 0≤θ≤2π0\le\theta\le 2\pi.

Tasks

  1. Show that the image is the sphere of radius 33.

  2. Compute an outward normal vector from the tangent vectors.

  3. Identify where the parametrization fails to be regular or one-to-one.

Original worksheet page 1: question and worked solution for 6-2-004
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Question 4 – Solution

Strategy. Verify the implicit equation, then use the ordered product 𝒓ϕ×𝒓θ\mathbf r_\phi\times\mathbf r_\theta.

Step 1: Identify the image x2+y2+z2=9sin⁡2ϕ+9cos⁡2ϕ=9.x^2+y^2+z^2 =9\sin^2\phi+9\cos^2\phi=9. Thus the image is x2+y2+z2=9x^2+y^2+z^2=9.

Step 2: Normal Differentiation and crossing give 𝒓ϕ×𝒓θ=9sin⁡ϕ⟨sin⁡ϕcos⁡θ,sin⁡ϕsin⁡θ,cos⁡ϕ⟩.\boxed{\mathbf r_\phi\times\mathbf r_\theta =9\sin\phi\langle\sin\phi\cos\theta,\sin\phi\sin\theta,\cos\phi\rangle}. For 0<ϕ<π0<\phi<\pi, this is a positive multiple of the outward radial direction.

See the diagram in the original worksheet below.

Step 3: Degeneracies Its magnitude is 9sin⁡ϕ9\sin\phi, which vanishes at ϕ=0\phi=0 and ϕ=π\phi=\pi. Every value of θ\theta maps to the same pole there. Also θ=0\theta=0 and θ=2π\theta=2\pi describe the same meridian seam.

Verification Away from the poles, 9sin⁡ϕ>09\sin\phi>0, so the tangent vectors are independent and the patch is regular locally.

Original worksheet page 2: question and worked solution for 6-2-004

Original worksheet layout. Use Enlarge or open the PDF for a closer view.

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