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Chapter 14: Q. 9 (page 1119)

Give a smooth parametrization, in terms of u and v, of the sphere of radius k and centered at the origin.

Short Answer

Expert verified

$T he required parametrization is r (u, v) = <ksinv cosu, ksinv sinu, k \cos v> 0 \leq u \leq 2 π and 0 \leq v \leq π$

Step by step solution

01

Step 1. Given

The given equation of the sphere of radius r and centered at the origin is

$x^{2} + y^{2} + z^{2} = k^{2} . . . . . . . . . (1)$

02

Step 2. Parametrization of sphere

$L e t x^{2} + y^{2} = a^{2} be the parametrization of the circle \{\begin{cases} x = a \cos u \\ y = a \sin u \end{cases} where, 0 \leq u \leq 2 π Then (1) becomes x^{2} + y^{2} + z^{2} = k^{2} a^{2} + z^{2} = k^{2} The parametrization of the circle a^{2} + z^{2} = k^{2} is \{\begin{cases} a = ksinv \\ z = k \cos v \end{cases} The parametrization of the sphere x^{2} + y^{2} + z^{2} = k^{2}, is \{\begin{cases} x = a cosu = ksinv cosu \\ y = a sinu = ksinv sinu \\ z = k \cos v \end{cases} where, 0 \leq u \leq 2 π and 0 \leq v \leq π Therefore the required parametrization is r (u, v) = <ksinv cosu, ksinv sinu, k \cos v> 0 \leq u \leq 2 π and 0 \leq v \leq π$

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Most popular questions from this chapter

Compute the curl of the vector fields:

$F (x, y) = \cos (x + y) i + \sin (x - y) j$ .

What is the difference between the graphs of

$G (x, y) = x i + y j a n d F (x, y) = - x i + (- y) j$

Use the same vector field as in Exercise 13, and compute the k-component of the curl of F(x, y).

$f (x, y, z) = <\frac{1}{x \ln (z + 2 y)}>$ , where Sis the surface given by $r (u, v) = 2 u i + (u + v) j + (2 u - 2 v) k$ for $3 \leq u \leq 7$ and $2 \leq v \leq 10$ .

Let a, b, and c be nonzero constants. Find a general formula for the area of the portion of the plane with equation $a x + b y + c z = k$ that lies above a rectangle $[α, β] \times [γ, δ]$ in thexy-plane.

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