Physics equations/19-Electric Potential and Electric Field/Q:SurfaceIntegralsCalculus

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,
 ${\displaystyle {\vec {\mathfrak {F}}}=(2.03+1.29z)\rho ^{2}{\hat {\rho }}+8.35z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
 ${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

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,
 ${\displaystyle {\vec {\mathfrak {F}}}=(2.03+1.29z)\rho ^{2}{\hat {\rho }}+8.35z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
 ${\displaystyle \left|\int _{side}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over curved side surface of the cylinder.

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,
 ${\displaystyle {\vec {\mathfrak {F}}}=(2.03+1.29z)\rho ^{2}{\hat {\rho }}+8.35z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
 ${\displaystyle \left|\oint {\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the entire surface of the cylinder.

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,
 ${\displaystyle {\vec {\mathfrak {F}}}=(1.74+1.27z)\rho ^{3}{\hat {\rho }}+9.08z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
 ${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

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,
 ${\displaystyle {\vec {\mathfrak {F}}}=(1.74+1.27z)\rho ^{3}{\hat {\rho }}+9.08z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
 ${\displaystyle \left|\int _{side}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the curved side surface of the cylinder.

A cylinder of radius, r=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,
 ${\displaystyle {\vec {\mathfrak {F}}}=(1.74+1.27z)\rho ^{3}{\hat {\rho }}+9.08z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
 ${\displaystyle \left|\oint {\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the entire surface of the cylinder.

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,
 ${\displaystyle {\vec {\mathfrak {F}}}=(2.48+2.38z)\rho ^{3}{\hat {\rho }}+8.41z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
 ${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

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,
 ${\displaystyle {\vec {\mathfrak {F}}}=(2.48+2.38z)\rho ^{3}{\hat {\rho }}+8.41z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
 ${\displaystyle \left|\int _{side}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the curved side surface of the cylinder.

A cylinder of radius, r=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,
 ${\displaystyle {\vec {\mathfrak {F}}}=(2.48+2.38z)\rho ^{3}{\hat {\rho }}+8.41z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
 ${\displaystyle \left|\oint {\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the entire surface of the cylinder.