Find an equation for the paraboloid z=x2+y2z=x2+y2 in spherical coordinates. (enter rho, phi and theta for ρρ, ϕϕ and θθ, respectively.)

Accepted Solution

Answer:[tex]\displaystyle \rho \sin^2 \phi - \cos \phi = 0[/tex]General Formulas and Concepts:Multivariable CalculusCylindrical Coordinate Conversions:[tex]\displaystyle x = r \cos \theta[/tex][tex]\displaystyle y = r \sin \theta[/tex][tex]\displaystyle z = z[/tex][tex]\displaystyle r^2 = x^2 + y^2[/tex][tex]\displaystyle \tan \theta = \frac{y}{x}}[/tex]Spherical Coordinate Conversions:[tex]\displaystyle r = \rho \sin \phi[/tex][tex]\displaystyle x = \rho \sin \phi \cos \theta[/tex][tex]\displaystyle z = \rho \cos \phi[/tex][tex]\displaystyle y = \rho \sin \phi \sin \theta[/tex][tex]\displaystyle \rho = \sqrt{x^2 + y^2 + z^2}[/tex]Step-by-step explanation:Step 1: DefineIdentify.[tex]\displaystyle z = x^2 + y^2[/tex]Step 2: Convert[Equation] Substitute in Cylindrical Coordinate Conversions:
[tex]\displaystyle z = r^2[/tex]Substitute in Spherical Coordinate Conversions:
[tex]\displaystyle \rho \cos \phi = ( \rho \sin \phi )^2[/tex]Simplify:
[tex]\displaystyle \rho \cos \phi = \rho^2 \sin^2 \phi[/tex]Rewrite:
[tex]\displaystyle \rho \cos \phi - \rho^2 \sin^2 \phi = 0[/tex]Simplify:
[tex]\displaystyle \cos \phi - \rho \sin^2 \phi = 0[/tex]Rewrite:
[tex]\displaystyle \rho \sin^2 \phi - \cos \phi = 0[/tex]∴ we have written the rectangular equation into spherical coordinates.---Learn more about spherical coordinates: more about multivariable calculus: : Multivariable CalculusUnit: Triple Integrals Applications