Study: http://en.wikiversity.org/wiki/Physics_equations/Sheet/All_chapters 1 A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.35 + 2.57 z ) ρ 3 ρ ^ + 7.45 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.35+2.57z)\rho ^{3}{\hat {\rho }}+7.45z^{3}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 2 A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.05 + 2.59 z ) ρ 2 ρ ^ + 7.4 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.05+2.59z)\rho ^{2}{\hat {\rho }}+7.4z^{2}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 3 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.12 + 1.85 z ) ρ 3 ρ ^ + 8.88 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.12+1.85z)\rho ^{3}{\hat {\rho }}+8.88z^{2}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 4 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. 5 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.14 + 2.8 z ) ρ 2 ρ ^ + 9.94 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.14+2.8z)\rho ^{2}{\hat {\rho }}+9.94z^{2}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 6 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. 7 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.07 + 2.87 z ) ρ 2 ρ ^ + 9.56 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.07+2.87z)\rho ^{2}{\hat {\rho }}+9.56z^{3}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 8 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.17 + 1.5 z ) ρ 2 ρ ^ + 8.75 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.17+1.5z)\rho ^{2}{\hat {\rho }}+8.75z^{2}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 9 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. 10 A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. 11 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. 12 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. 13 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. 14 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 1.93 + 2.31 z ) ρ 3 ρ ^ + 7.21 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(1.93+2.31z)\rho ^{3}{\hat {\rho }}+7.21z^{2}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 15 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.24 + 1.11 z ) ρ 3 ρ ^ + 8.16 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.24+1.11z)\rho ^{3}{\hat {\rho }}+8.16z^{3}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 16 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 1.96 + 2.52 z ) ρ 2 ρ ^ + 7.11 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(1.96+2.52z)\rho ^{2}{\hat {\rho }}+7.11z^{2}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 17 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. 18 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.24 + 2.08 z ) ρ 2 ρ ^ + 8.93 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.24+2.08z)\rho ^{2}{\hat {\rho }}+8.93z^{3}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 19 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. 20 A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.37 + 2.6 z ) ρ 2 ρ ^ + 8.84 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.37+2.6z)\rho ^{2}{\hat {\rho }}+8.84z^{3}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 21 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. 22 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 1.88 + 1.29 z ) ρ 2 ρ ^ + 7.2 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(1.88+1.29z)\rho ^{2}{\hat {\rho }}+7.2z^{2}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. 23 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.44 + 2.86 z ) ρ 2 ρ ^ + 7.42 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.44+2.86z)\rho ^{2}{\hat {\rho }}+7.42z^{3}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ t o p F → ⋅ n ^ d A | {\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the top surface of the cylinder. A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.35 + 2.57 z ) ρ 3 ρ ^ + 7.45 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.35+2.57z)\rho ^{3}{\hat {\rho }}+7.45z^{3}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∫ s i d e F → ⋅ n ^ d A | {\displaystyle \left|\int _{side}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the curved side surface of the cylinder. A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,   F → = ( 2.35 + 2.57 z ) ρ 3 ρ ^ + 7.45 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.35+2.57z)\rho ^{3}{\hat {\rho }}+7.45z^{3}{\hat {z}}} Let n ^ {\displaystyle {\hat {n}}} be the outward unit normal to this cylinder and evaluate ,   | ∮ F → ⋅ n ^ d A | {\displaystyle \left|\oint {\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,} over the entire surface of the cylinder.
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