Astroidal Sphere An equation of an astroidal sphere inand isA graph of an astroidal

Chapter 15, Problem 51

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Astroidal Sphere   An equation of an astroidal sphere in x, y, and z is

\(x^{2 / 3}+y^{2 / 3}+z^{2 / 3}=a^{2 / 3}\).

A graph of an astroidal sphere is shown below. Show that this surface can be represented parametrically by

\(\mathbf{r}(u, v)=a \sin ^{3} u \cos ^{3} v \mathbf{i}+a \sin ^{3} u \sin ^{3} v \mathbf{j}+a \cos ^{3} u \mathbf{k}\)

where \(0 \leq u \leq \pi\) and \(0 \leq v \leq 2 \pi\).

Text Transcription:

x^2/3+y^2/3+z^2/3=a^2/3

r(u,v)=a sin^3 u cos^3 vi+a sin^3 u sin^3 vj+a cos^3 uk

0<=u<=pi

0<=v<=2 pi