Math Problem Statement

Find a parametrization for the portion of the sphere x^2 + y^2 + z^2 = 3 between the planes z = -sqrt(3)/2 and z = sqrt(3)/2. Also, find a parametrization for the portion of the plane x + y + z = 1 inside the cylinder x^2 + y^2 = 9.

Solution

The problem involves finding parameterizations for specific surfaces, and it suggests using cylindrical or spherical coordinates.

Let's break it down:

(a) The portion of the sphere x2+y2+z2=3x^2 + y^2 + z^2 = 3 between the planes z=−32z = -\frac{\sqrt{3}}{2} and z=32z = \frac{\sqrt{3}}{2}.

Since we're working with a sphere, spherical coordinates are useful. The conversion to spherical coordinates is:

x=ρsin⁡(ϕ)cos⁡(θ),y=ρsin⁡(ϕ)sin⁡(θ),z=ρcos⁡(ϕ)x = \rho \sin(\phi) \cos(\theta), \quad y = \rho \sin(\phi) \sin(\theta), \quad z = \rho \cos(\phi)

For the given sphere x2+y2+z2=3x^2 + y^2 + z^2 = 3, the radius is ρ=3\rho = \sqrt{3}.

The planes are given as z=±32z = \pm \frac{\sqrt{3}}{2}, which corresponds to specific values of ϕ\phi. Since z=ρcos⁡(ϕ)z = \rho \cos(\phi), we can solve for ϕ\phi:

\quad \Rightarrow \quad \cos(\phi) = \frac{1}{2} \quad \Rightarrow \quad \phi = \frac{\pi}{3}$$ Similarly, for $$z = -\frac{\sqrt{3}}{2}$$: $$\cos(\phi) = -\frac{1}{2} \quad \Rightarrow \quad \phi = \frac{2\pi}{3}$$ Thus, the parameterization in spherical coordinates is: $$x = \sqrt{3} \sin(\phi) \cos(\theta), \quad y = \sqrt{3} \sin(\phi) \sin(\theta), \quad z = \sqrt{3} \cos(\phi)$$ Where $$\theta \in [0, 2\pi]$$ and $$\phi \in \left[\frac{\pi}{3}, \frac{2\pi}{3}\right]$$. ### (b) The portion of the plane $$x + y + z = 1$$ inside the cylinder $$x^2 + y^2 = 9$$. For this, cylindrical coordinates $$(r, \theta)$$ are convenient, where: $$x = r \cos(\theta), \quad y = r \sin(\theta), \quad z = 1 - r \cos(\theta) - r \sin(\theta)$$ The condition $$x^2 + y^2 = r^2 \leq 9$$ means $$r \in [0, 3]$$. The range of $$\theta$$ is $$[0, 2\pi]$$. Thus, the parameterization is: $$x = r \cos(\theta), \quad y = r \sin(\theta), \quad z = 1 - r \cos(\theta) - r \sin(\theta)$$ Where $$r \in [0, 3]$$ and $$\theta \in [0, 2\pi]$$. --- Would you like more details on any part of the solution? ### Related Questions: 1. How do you convert between spherical and Cartesian coordinates in general? 2. What is the significance of the range of parameters in a parameterization? 3. Can you explain how cylindrical coordinates simplify certain surfaces? 4. What would be the parameterization of a cone using spherical coordinates? 5. How does the choice of coordinates affect the ease of solving surface parametrizations? **Tip:** When working with parameterizations, always check that the ranges of your parameters match the limits imposed by the problem.

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Math Problem Analysis

Mathematical Concepts

Parametrization
Spherical Coordinates
Cylindrical Coordinates
Surfaces

Formulas

x = ρsin(ϕ)cos(θ), y = ρsin(ϕ)sin(θ), z = ρcos(ϕ) for spherical coordinates
x = rcos(θ), y = rsin(θ), z = f(r,θ) for cylindrical coordinates

Theorems

Coordinate Transformations
Parametrization of Surfaces

Suitable Grade Level

Undergraduate Level

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