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1 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 → = ( 2.03 + 1.29 z ) ρ 2 ρ ^ + 8.35 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.03+1.29z)\rho ^{2}{\hat {\rho }}+8.35z^{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=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 → = ( 2.03 + 1.29 z ) ρ 2 ρ ^ + 8.35 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.03+1.29z)\rho ^{2}{\hat {\rho }}+8.35z^{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 curved side surface of the cylinder.
3 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 → = ( 2.03 + 1.29 z ) ρ 2 ρ ^ + 8.35 z 3 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.03+1.29z)\rho ^{2}{\hat {\rho }}+8.35z^{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.
1 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 → = ( 1.74 + 1.27 z ) ρ 3 ρ ^ + 9.08 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(1.74+1.27z)\rho ^{3}{\hat {\rho }}+9.08z^{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.
2 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 → = ( 1.74 + 1.27 z ) ρ 3 ρ ^ + 9.08 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(1.74+1.27z)\rho ^{3}{\hat {\rho }}+9.08z^{2}{\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.
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 → = ( 1.74 + 1.27 z ) ρ 3 ρ ^ + 9.08 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(1.74+1.27z)\rho ^{3}{\hat {\rho }}+9.08z^{2}{\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.
1 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 → = ( 2.48 + 2.38 z ) ρ 3 ρ ^ + 8.41 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.48+2.38z)\rho ^{3}{\hat {\rho }}+8.41z^{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.
2 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 → = ( 2.48 + 2.38 z ) ρ 3 ρ ^ + 8.41 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.48+2.38z)\rho ^{3}{\hat {\rho }}+8.41z^{2}{\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.
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 → = ( 2.48 + 2.38 z ) ρ 3 ρ ^ + 8.41 z 2 z ^ {\displaystyle {\vec {\mathfrak {F}}}=(2.48+2.38z)\rho ^{3}{\hat {\rho }}+8.41z^{2}{\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.
