Write the equation in spherical coordinates. 3x + 6y + 7z = 1

Answer:[tex]\displaystyle \rho = \frac{1}{3 \sin \phi \cos \theta + 6 \sin \phi \sin \theta + 7 \cos \phi}[/tex]General Formulas and Concepts:
Multivariable CalculusSpherical 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 3x + 6y + 7z = 1[/tex]Step 2: Convert[Equation] Substitute in Spherical Coordinate Conversions:
[tex]\displaystyle 3 \rho \sin \phi \cos \theta + 6 \rho \sin \phi \sin \theta + 7 \rho \cos \phi = 1[/tex]Factor:
[tex]\displaystyle \rho \bigg( 3 \sin \phi \cos \theta + 6 \sin \phi \sin \theta + 7 \cos \phi \bigg) = 1[/tex]Isolate ρ:
[tex]\displaystyle \rho = \frac{1}{3 \sin \phi \cos \theta + 6 \sin \phi \sin \theta + 7 \cos \phi}[/tex]∴ we have written the given equation in spherical coordinates.---Learn more about multivariable calculus: : Multivariable CalculusUnit: Triple Integrals Applications