Answer:
[tex]\displaystyle \rho \sin^2 \phi - \cos \phi = 0[/tex]
General Formulas and Concepts:
Multivariable Calculus
Cylindrical 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: Define
Identify.
[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.
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Learn more about spherical coordinates: https://brainly.com/question/9728819
Learn more about multivariable calculus: https://brainly.com/question/4746216
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Topic: Multivariable Calculus
Unit: Triple Integrals Applications
