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Chapter 13: Problem 24

Find a vector-valued function whose graph is the indicated surface. The ellipsoid \(\frac{x^{2}}{9}+\frac{y^{2}}{4}+\frac{z^{2}}{1}=1\)

Short Answer

Expert verified
The vector-valued function representing the ellipsoid is \(r(u,v) = (3cos(u)cos(v), 2sin(u)cos(v), sin(v))\), where 0 \(\leq u \leq 2\pi\) and \(-\frac{\pi}{2} \leq v \leq \frac{\pi}{2}\)

Step by step solution

01

Identify the Constants

The constants a, b, and c from the equation of the ellipsoid are 3, 2, and 1 respectively. These constants give us the radii of the ellipsoid on the x, y, and z axes.
02

Write the Vector-Valued Function

The vector-valued function representing the ellipsoid with these constants is given by \(r(u,v) = (acos(u)cos(v), bsin(u)cos(v), csin(v))\), where 0 \(\leq u \leq 2\pi\) and \(-\frac{\pi}{2} \leq v \leq \frac{\pi}{2}\). So in this case, we fill in the constants to get: \(r(u,v) = (3cos(u)cos(v), 2sin(u)cos(v), sin(v))\)

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