Write the equation in spherical coordinates. (a) 4x2 − 3x + 4y2 + 4z2 = 0

Answer:[tex]\displaystyle 4 \rho - 3 \sin \phi \cos \theta = 0[/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: Define[tex]\displaystyle 4x^2 - 3x + 4y^2 + 4z^2 = 0[/tex]Step 2: Convert[Equation] Rewrite:
[tex]\displaystyle 4x^2 + 4y^2 + 4z^2 - 3x = 0[/tex][Equation] Factor:
[tex]\displaystyle 4(x^2 + y^2 + z^2) - 3x = 0[/tex][Equation] Substitute in Spherical Coordinate Conversions:
[tex]\displaystyle 4 \rho^2 - 3 \rho \sin \phi \cos \theta = 0[/tex]Factor:
[tex]\displaystyle \rho \big( 4 \rho - 3 \sin \phi \cos \theta \big) = 0[/tex]Simplify:
[tex]\displaystyle 4 \rho - 3 \sin \phi \cos \theta = 0[/tex]∴ we have converted the rectangular equation into spherical coordinates.---Learn more about spherical coordinates: more about multivariable calculus: : Multivariable CalculusUnit: Triple Integrals Applications