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 Calculus
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 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.
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Learn more about multivariable calculus: https://brainly.com/question/4746216
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Topic: Multivariable Calculus
Unit: Triple Integrals Applications
