Math 233 - December 1, 2009 Spherical coordinates 1. Find the - PowerPoint PPT Presentation

Math 233 - December 1, 2009 ◮ Spherical coordinates

1. Find the determinant: � � sin φ cos θ ρ cos φ cos θ − ρ sin φ sin θ � � � � sin φ sin θ ρ cos φ sin θ ρ sin φ cos θ = � � � � cos φ − ρ sin φ 0 � �

1. Find the determinant: � � sin φ cos θ ρ cos φ cos θ − ρ sin φ sin θ � � = ρ 2 sin φ � � sin φ sin θ ρ cos φ sin θ ρ sin φ cos θ � � � � cos φ − ρ sin φ 0 � �

Lecture Problems 2. Convert the spherical equation to a Cartesian equation: ρ = sec φ 3. Convert the Cartesian equation to a spherical equation: x 2 + y 2 + 4 z 2 = 10 4. Convert the Cartesian equation to a spherical equation: x 2 + y 2 − 2 z 2 = 0 5. Convert the Cartesian equation to a spherical equation: x + y + z = 1 6. Convert the spherical equation to a Cartesian equation: ρ sin φ = 1

Lecture Problems 2. Convert the spherical equation to a Cartesian equation: ρ = sec φ Solution: z = 1 3. Convert the Cartesian equation to a spherical equation: x 2 + y 2 + 4 z 2 = 10 Solution: ρ = (3 / 2) sec φ 4. Convert the Cartesian equation to a spherical equation: x 2 + y 2 − 2 z 2 = 0 √ Solution: tan φ = 1 / 2 5. Convert the Cartesian equation to a spherical equation: x + y + z = 1 Solution: ρ sin φ cos θ + ρ sin φ sin θ + ρ cos φ = 1 6. Convert the spherical equation to a Cartesian equation: ρ sin φ = 1 Solution: x 2 + y 2 = 1

7. Find the volume of the ice cream cone–the solid below the sphere x 2 + y 2 + z 2 = 1 and above the cone z 2 = x 2 + y 2 . V = 8. Find the mass of the solid inside the sphere ρ = 3 and outside the sphere ρ = 2 with density equal to the distance from the origin. m = 9. � 3 � √ � √ 9 − x 2 9 − x 2 − z 2 9 − x 2 − z 2 ( x 2 + y 2 + z 2 ) 3 / 2 dy dz dx √ √ 9 − x 2 − 3 − −

7. Find the volume of the ice cream cone–the solid below the sphere x 2 + y 2 + z 2 = 1 and above the cone z 2 = x 2 + y 2 . � π/ 4 � 2 π � 1 √ ρ 2 sin φ d ρ d φ d θ = π (2 − 2) V = 3 0 0 0 8. Find the mass of the solid inside the sphere ρ = 3 and outside the sphere ρ = 2 with density equal to the distance from the origin. � π � 2 π � 3 ρρ 2 sin φ d ρ d φ d θ = 65 π m = 0 0 2 9. � 3 � √ � √ 9 − x 2 9 − x 2 − z 2 9 − x 2 − z 2 ( x 2 + y 2 + z 2 ) 3 / 2 dy dz dx √ √ 9 − x 2 − 3 − − � π � 2 π � 3 ρ 3 / 2 ρ 2 sin φ d ρ d φ d θ = 486 π = 2 0 0