Math Problem Statement

Convert the equation x^2 + y^2 = 9 to spherical coordinates and write in standard form, solving for ρ.

Solution

To convert the equation x2+y2=9x^2 + y^2 = 9 to spherical coordinates, we first need to express the variables xx and yy in terms of spherical coordinates using the following relations:

In spherical coordinates, the relations for xx, yy, and zz are:

x=ρsinϕcosθx = \rho \sin \phi \cos \theta y=ρsinϕsinθy = \rho \sin \phi \sin \theta z=ρcosϕz = \rho \cos \phi

Given the equation x2+y2=9x^2 + y^2 = 9, substitute xx and yy with their spherical coordinate equivalents:

(ρsinϕcosθ)2+(ρsinϕsinθ)2=9(\rho \sin \phi \cos \theta)^2 + (\rho \sin \phi \sin \theta)^2 = 9

Simplifying the left-hand side:

ρ2sin2ϕ(cos2θ+sin2θ)=9\rho^2 \sin^2 \phi (\cos^2 \theta + \sin^2 \theta) = 9

Since cos2θ+sin2θ=1\cos^2 \theta + \sin^2 \theta = 1, the equation becomes:

ρ2sin2ϕ=9\rho^2 \sin^2 \phi = 9

Now, solving for ρ\rho:

ρ2=9sin2ϕ\rho^2 = \frac{9}{\sin^2 \phi}

ρ=3sinϕ\rho = \frac{3}{|\sin \phi|}

This is the equation in spherical coordinates. It shows the relation between ρ\rho and ϕ\phi, and represents a cylindrical surface in spherical coordinates.

Would you like any further details on this? Here are some related questions to explore:

  1. How do you convert the full 3D equation x2+y2+z2=9x^2 + y^2 + z^2 = 9 to spherical coordinates?
  2. What is the significance of the variables ρ\rho, ϕ\phi, and θ\theta in spherical coordinates?
  3. How do you convert between spherical and cylindrical coordinate systems?
  4. How do you interpret this equation geometrically in spherical coordinates?
  5. Can you derive a similar equation for a cone or other shapes in spherical coordinates?

Tip: Remember that in spherical coordinates, ϕ\phi is the angle from the positive zz-axis, while θ\theta is the angle in the xyxy-plane from the positive xx-axis.

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Math Problem Analysis

Mathematical Concepts

Spherical Coordinates
Coordinate Systems
Algebra

Formulas

x = ρ sin(φ) cos(θ)
y = ρ sin(φ) sin(θ)
z = ρ cos(φ)
cos^2(θ) + sin^2(θ) = 1

Theorems

Trigonometric identity
Equation conversion to spherical coordinates

Suitable Grade Level

Grades 10-12