# Properties and Applications of Surface Integrals

Scalar functions: $$f\left( {x,y,z} \right),$$ $$f\left( {x,y} \right)$$
Position vectors: $$\mathbf{r}\left( {u,v} \right),$$ $$\mathbf{r}\left( {x,y,z} \right)$$
Unit vectors: $$\mathbf{i},$$ $$\mathbf{j},$$ $$\mathbf{k}$$
Surface: $$S$$
Vector field: $$\mathbf{F}\left( {P,Q,R} \right)$$
Divergence of a vector field: $$\text {div }\mathbf{F} = \nabla \cdot \mathbf{F}$$
Curl of a vector field: $$\text {rot }\mathbf{F} = \nabla \times \mathbf{F}$$
Vector element of a surface: $$d\mathbf{S}$$
Normal to a surface: $$\mathbf{n}$$
Surface area: $$A$$
Mass of a surface: $$m$$
Density of a surface: $$\mu\left( {x,y,z} \right)$$
Coordinates of the center of mass: $${x_C},$$ $${y_C},$$ $${z_C}$$
First moments: $${M_{xy}},$$ $${M_{yz}},$$ $${M_{xz}}$$
Moments of inertia: $${I_{xy}},$$ $${I_{yz}},$$ $${I_{xz}},$$ $${I_x},$$ $${I_y},$$ $${I_z}$$
Volume of a solid: $$V$$
Force: $$\vec{F}$$
Gravitational constant: $$G$$
Fluid velocity: $$\mathbf{v}\left( \mathbf{r} \right)$$
Fluid or mass density: $$\rho$$
Pressure: $$\mathbf{p}\left( \mathbf{r} \right)$$
Mass flux, electric flux: $$\Phi$$
Surface charge: $$Q$$
Charge density: $$\sigma\left( {x,y} \right)$$
Magnitude of the electric field: $$\mathbf{E}$$
1. Surface integral of a scalar function
Let a surface $$S$$ be given by the position vector $$\mathbf{r}\left( {u,v} \right) =$$ $$x\left( {u,v} \right)\mathbf{i}$$ $$+\; y\left( {u,v} \right)\mathbf{j}$$ $$+\; z\left( {u,v} \right)\mathbf{k},$$ where the coordinates $$\left( {u,v} \right)$$ range over some domain $$D\left( {u,v} \right)$$ of the $$uv$$-plane. The surface integral of the scalar function $$f\left( {x,y,z} \right)$$ over the surface $$S$$ is defined to be
$${\large\iint\limits_S\normalsize} {f\left( {x,y,z} \right)dS} =$$ $${\large\iint\limits_{D\left( {u,v} \right)}\normalsize} {f\left( {x\left( {u,v} \right),y\left( {u,v} \right),z\left( {u,v} \right)} \right) }$$ $${ \left| {{\large\frac{{\partial \mathbf{r}}}{{\partial u}}\normalsize} \times {\large\frac{{\partial \mathbf{r}}}{{\partial v}}\normalsize}} \right|dudv} ,$$
where the partial derivatives $$\partial \mathbf{r}/\partial u$$ and $$\partial \mathbf{r}/\partial v$$ are expressed by the formulas
$${\large\frac{{\partial \mathbf{r}}}{{\partial u}}\normalsize} = {\large\frac{{\partial x}}{{\partial u}}\normalsize}\left( {u,v} \right)\mathbf{i}$$ $$+\;{\large\frac{{\partial y}}{{\partial u}}\normalsize}\left( {u,v} \right)\mathbf{j}$$ $$+\;{\large\frac{{\partial z}}{{\partial u}}\normalsize}\left( {u,v} \right)\mathbf{k},$$
$${\large\frac{{\partial \mathbf{r}}}{{\partial v}}\normalsize} = {\large\frac{{\partial x}}{{\partial v}}\normalsize}\left( {u,v} \right)\mathbf{i}$$ $$+\;{\large\frac{{\partial y}}{{\partial v}}\normalsize}\left( {u,v} \right)\mathbf{j}$$ $$+\;{\large\frac{{\partial z}}{{\partial v}}\normalsize}\left( {u,v} \right)\mathbf{k},$$
and $${{\large\frac{{\partial \mathbf{r}}}{{\partial u}}\normalsize} \times {\large\frac{{\partial \mathbf{r}}}{{\partial v}}\normalsize}}$$ is the cross product.
2. If the surface $$S$$ is given by the explicit equation $$z = z\left( {x,y} \right)$$ where $$z\left( {x,y} \right)$$ is a differentiable function in the domain $$D\left( {x,y} \right),$$ then the surface integral of the scalar function is written as
$${\large\iint\limits_S\normalsize} {f\left( {x,y,z} \right)dS} =$$ $${\large\iint\limits_{D\left( {u,v} \right)}\normalsize} {f\left( {x,y,z\left( {x,y} \right)} \right) }$$ $${\sqrt {1 + {{\left( {\large\frac{{\partial z}}{{\partial x}}\normalsize} \right)}^2} + {{\left( {\large\frac{{\partial z}}{{\partial y}}\normalsize} \right)}^2}} dxdy} .$$
3. Surface integral of a vector field $$\mathbf{F}$$ over an oriented surface $$S$$
– If the surface $$S$$ is oriented outward, then
$${\large\iint\limits_S\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot d\mathbf{S}} =$$ $${\large\iint\limits_S\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot \mathbf{n}dS} =$$ $${\large\iint\limits_{D\left( {u,v} \right)}\normalsize} {\mathbf{F}\left( {x\left( {u,v} \right),y\left( {u,v} \right),z\left( {u,v} \right)} \right) \cdot }$$ $${ \left[ {\large\frac{{\partial \mathbf{r}}}{{\partial u}}\normalsize \times \large\frac{{\partial \mathbf{r}}}{{\partial v}}\normalsize} \right]dudv} ;$$
– If the surface $$S$$ is oriented inward, then
$${\large\iint\limits_S\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot d\mathbf{S}} =$$ $${\large\iint\limits_S\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot \mathbf{n}dS} =$$ $${\large\iint\limits_{D\left( {u,v} \right)}\normalsize} {\mathbf{F}\left( {x\left( {u,v} \right),y\left( {u,v} \right),z\left( {u,v} \right)} \right) \cdot }$$ $${ \left[ {\large\frac{{\partial \mathbf{r}}}{{\partial v}}\normalsize \times \large\frac{{\partial \mathbf{r}}}{{\partial u}}\normalsize} \right]dudv}.$$
Here $$d\mathbf{S} = \mathbf{n}dS$$ is called the vector element of the surface. Dot means the scalar product of the vectors. The partial derivatives $$\partial \mathbf{r}/\partial u$$ and $$\partial \mathbf{r}/\partial v$$ are determined by the formulas
$${\large\frac{{\partial \mathbf{r}}}{{\partial u}}\normalsize} = {\large\frac{{\partial x}}{{\partial u}}\normalsize}\left( {u,v} \right)\mathbf{i}$$ $$+\; {\large\frac{{\partial y}}{{\partial u}}\normalsize}\left( {u,v} \right)\mathbf{j}$$ $$+\;{\large\frac{{\partial z}}{{\partial u}}\normalsize}\left( {u,v} \right)\mathbf{k},$$
$${\large\frac{{\partial \mathbf{r}}}{{\partial v}}\normalsize} = {\large\frac{{\partial x}}{{\partial v}}\normalsize}\left( {u,v} \right)\mathbf{i}$$ $$+\; {\large\frac{{\partial y}}{{\partial v}}\normalsize}\left( {u,v} \right)\mathbf{j}$$ $$+\;{\large\frac{{\partial z}}{{\partial v}}\normalsize}\left( {u,v} \right)\mathbf{k}.$$
4. If the surface $$S$$ is defined by the explicit equation $$z = z\left( {x,y} \right)$$ where $$z\left( {x,y} \right)$$ is a differentiable function in the domain $$D\left( {x,y} \right),$$ then the surface integral of the vector field is written as follows:
– If the surface $$S$$ is oriented upward (the $$k$$th component of the normal vector is positive), then
$${\large\iint\limits_S\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot d\mathbf{S}} =$$ $${\large\iint\limits_S\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot \mathbf{n}dS} =$$ $${\large\iint\limits_{D\left( {x,y} \right)}\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot }$$ $${\left( { – {\large\frac{{\partial z}}{{\partial x}}\normalsize} \mathbf{i} – {\large\frac{{\partial z}}{{\partial y}}\normalsize} \mathbf{j} + \mathbf{k}} \right)dxdy} ;$$
– If the surface $$S$$ is oriented downward (the $$k$$th component of the normal vector is negative), then
$${\large\iint\limits_S\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot d\mathbf{S}} =$$ $${\large\iint\limits_S\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot \mathbf{n}dS} =$$ $${\large\iint\limits_{D\left( {x,y} \right)}\normalsize} {\mathbf{F}\left( {x,y,z} \right) \cdot }$$ $${\left( {{\large\frac{{\partial z}}{{\partial x}}\normalsize} \mathbf{i} + {\large\frac{{\partial z}}{{\partial y}}\normalsize} \mathbf{j} – \mathbf{k}} \right)dxdy} .$$
5. Surface integral of a vector field in coordinate form
$${\large\iint\limits_S\normalsize} {\left( {\mathbf{F} \cdot \mathbf{n}} \right)dS} =$$ $${\large\iint\limits_S\normalsize} {Pdydz + Qdzdx + Rdxdy} =$$ $${\large\iint\limits_S\normalsize} {\big( {P\cos \alpha + Q\cos \beta }}$$ $$+\,{{ R\cos \gamma } \big)dS} ,$$
where $$P\left( {x,y,z} \right)$$, $$Q\left( {x,y,z} \right)$$, $$R\left( {x,y,z} \right)$$ are the components of the vector field $$\mathbf{F}$$ and $$\cos \alpha,$$ $$\cos \beta,$$ $$\cos \gamma$$ are the direction cosines of the outer normal $$\mathbf{n}$$ to the surface $$S$$.
6. Surface integral of a vector field in parametric form
If the surface $$S$$ is given in parametric form by the vector $$\mathbf{r}\left( {x\left( {u,v} \right),y\left( {u,v} \right),z\left( {u,v} \right)} \right),$$ then the surface integral is written as
$${\large\iint\limits_S\normalsize} {\left( {\mathbf{F} \cdot \mathbf{n}} \right)dS} =$$ $${\large\iint\limits_S\normalsize} {Pdydz + Qdzdx + Rdxdy} =$$ $${\large\iint\limits_{D\left( {u,v} \right)}\normalsize} {\left| {\begin{array}{*{20}{c}} P & Q & R\\ {\large\frac{{\partial x}}{{\partial u}}\normalsize} & {\large\frac{{\partial y}}{{\partial u}}\normalsize} & {\large\frac{{\partial z}}{{\partial u}}\normalsize}\\ {\large\frac{{\partial x}}{{\partial v}}\normalsize} & {\large\frac{{\partial y}}{{\partial v}}\normalsize} & {\large\frac{{\partial z}}{{\partial v}}\normalsize} \end{array}} \right|} dudv,$$
where the coordinates$$\left( {u,v} \right)$$ range over some domain $$D\left( {u,v} \right)$$ in the $$uv$$-plane.
7. Divergence theorem
$${\large\iint\limits_S\normalsize} {\mathbf{F} \cdot d\mathbf{S}} =$$ $${\large\iiint\limits_G\normalsize} {\left( {\nabla \cdot \mathbf{F}} \right)dV} ,$$
where $$\mathbf{F}\left( {x,y,z} \right) =$$ $$\big( {P\left( {x,y,z} \right),Q\left( {x,y,z} \right)}\big.,$$ $$\big.{R\left( {x,y,z} \right)} \big)$$ is a vector field whose components $$P$$, $$Q$$, $$R$$ have continuous partial derivatives and
$$\nabla \cdot \mathbf{F} =$$ $${\large\frac{{\partial P}}{{\partial x}}\normalsize} + {\large\frac{{\partial Q}}{{\partial y}}\normalsize} + {\large\frac{{\partial R}}{{\partial z}}\normalsize}$$
denotes the divergence of $$\mathbf{F}$$. The surface integral in the divergence theorem is taken over a closed surface.
8. Divergence theorem in coordinate form
$${\large\iint\limits_S\normalsize} {Pdydz + Qdzdx + Rdxdy} =$$ $${\large\iiint\limits_G\normalsize} {\left( {{\large\frac{{\partial P}}{{\partial x}}\normalsize} + {\large\frac{{\partial Q}}{{\partial y}}\normalsize} + {\large\frac{{\partial R}}{{\partial z}}\normalsize}} \right)dxdydz}$$
9. Stoke’s theorem
$${\large\oint\limits_C\normalsize} {\mathbf{F} \cdot d\mathbf{r}} =$$ $$\iint\limits_S {\left( {\nabla \times \mathbf{F}} \right) \cdot d\mathbf{S}} ,$$
where $$\mathbf{F}\left( {x,y,z} \right) =$$ $$\big( {P\left( {x,y,z} \right),Q\left( {x,y,z} \right)}\big.,$$ $$\big.{R\left( {x,y,z} \right)} \big)$$ is a vector field whose components $$P$$, $$Q$$, $$R$$ have continuous partial derivatives and
$$\nabla \times \mathbf{F} =$$ $$\left| {\begin{array}{*{20}{c}} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ {\large\frac{\partial }{{\partial x}}\normalsize} & {\large\frac{\partial }{{\partial y}}\normalsize} & {\large\frac{\partial }{{\partial z}}\normalsize}\\ P & Q & R \end{array}} \right| =$$ $$\left( {{\large\frac{{\partial R}}{{\partial y}}\normalsize} – {\large\frac{{\partial Q}}{{\partial z}}\normalsize}} \right)\mathbf{i}$$ $$+\; \left( {{\large\frac{{\partial P}}{{\partial z}}\normalsize} – {\large\frac{{\partial R}}{{\partial x}}\normalsize}} \right)\mathbf{j}$$ $$+\;\left( {{\large\frac{{\partial Q}}{{\partial x}}\normalsize} – {\large\frac{{\partial P}}{{\partial y}}\normalsize}} \right)\mathbf{k}$$
denotes the curl of the vector field $$\mathbf{F}$$. The line integral in the left side of the Stoke’s formula is taken over a closed curve.
10. Stoke’s theorem in coordinate form
$${\large\oint\limits_C\normalsize} {Pdx + Qdy + Rdz} =$$ $${\large\iint\limits_S\normalsize} {\left( {{\large\frac{{\partial R}}{{\partial y}}\normalsize} – {\large\frac{{\partial Q}}{{\partial z}}\normalsize}} \right)dydz }$$ $$+\;{ \left( {{\large\frac{{\partial P}}{{\partial z}}\normalsize} – {\large\frac{{\partial R}}{{\partial x}}\normalsize}} \right)dzdx }$$ $$+\;{ \left( {{\large\frac{{\partial Q}}{{\partial x}}\normalsize} – {\large\frac{{\partial P}}{{\partial y}}\normalsize}} \right)dxdy}$$
11. Surface area
$$A = {\large\iint\limits_S\normalsize} {dS}$$
12. If the surface $$S$$ is parametrized by the vector $$\mathbf{r}\left( {u,v} \right) =$$ $$x\left( {u,v} \right)\mathbf{i}$$ $$+\; y\left( {u,v} \right)\mathbf{j}$$ $$+\; z\left( {u,v} \right)\mathbf{k},$$ then the surface area is expressed by the formula
$$A =$$ $${\large\iint\limits_{D\left( {u,v} \right)}\normalsize} {\left| {{\large\frac{{\partial r}}{{\partial u}}\normalsize} \times {\large\frac{{\partial r}}{{\partial v}}\normalsize}} \right|dudv} ,$$
where $$D\left( {u,v} \right)$$ is the domain where the surface $$\mathbf{r}\left( {u,v} \right)$$ is defined.
13. If the surface $$S$$ is given explicitly by the function $$z\left( {x,y} \right),$$ then the surface area is
$$A =$$ $${\large\iint\limits_{D\left( {x,y} \right)}\normalsize} {\sqrt {1 + {{\left( {\large\frac{{\partial z}}{{\partial x}}\normalsize} \right)}^2} + {{\left( {\large\frac{{\partial z}}{{\partial y}}\normalsize} \right)}^2}} }$$ $${dxdy} ,$$
where $$D\left( {x,y} \right)$$ is the projection of the surface $$S$$ onto the $$xy$$-plane.
14. Mass of a surface
$$m = {\large\iint\limits_S\normalsize} {\mu \left( {x,y,z} \right)dS} ,$$
where $${\mu \left( {x,y,z} \right)}$$ is the mass per unit area (density function).
15. Center of mass of a surface
$${x_C} = {\large\frac{{{M_{yz}}}}{m}\normalsize},\;$$ $${y_C} = {\large\frac{{{M_{xz}}}}{m}\normalsize},\;$$ $${z_C} = {\large\frac{{{M_{xy}}}}{m}\normalsize},$$ where
$${M_{yz}} = {\large\iint\limits_S\normalsize} {x\mu \left( {x,y,z} \right)dS},\;$$
$${M_{xz}} = {\large\iint\limits_S\normalsize} {y\mu \left( {x,y,z} \right)dS},\;$$
$${M_{xy}} = {\large\iint\limits_S\normalsize} {z\mu \left( {x,y,z} \right)dS}$$
are the first moments about the coordinate planes $$x = 0,$$ $$y = 0,$$ $$z = 0,$$ respectively, and $${\mu \left( {x,y,z} \right)}$$ is the density function.
16. Moments of inertia of a surface about the coordinate planes
The moments of inertia about the $$xy$$-plane (or $$z = 0$$), $$yz$$-plane ($$x = 0$$) and $$xz$$-plane ($$y = 0$$) are given, respectively, by the formulas
$${I_{xy}} = {\large\iint\limits_S\normalsize} {{z^2}\mu \left( {x,y,z} \right)dS},\;$$ $${I_{yz}} = {\large\iint\limits_S\normalsize} {{x^2}\mu \left( {x,y,z} \right)dS},\;$$ $${I_{xz}} = {\large\iint\limits_S\normalsize} {{y^2}\mu \left( {x,y,z} \right)dS}.$$
17. Moments of inertia of a surface about the coordinate axes
The moments of inertia about the $$x$$-axis, $$y$$-axis and $$z$$-axis are computed, respectively, by the formulas
$${I_x} = {\large\int\limits_S\normalsize} {\left( {{y^2} + {z^2}} \right) }{\mu \left( {x,y,z} \right)dS},\;$$ $${I_y} = {\large\int\limits_S\normalsize} {\left( {{x^2} + {z^2}} \right) }{\mu \left( {x,y,z} \right)dS},\;$$ $${I_z} = {\large\int\limits_S\normalsize} {\left( {{x^2} + {y^2}} \right) }{\mu \left( {x,y,z} \right)dS}.$$
18. Volume of a solid bounded by a closed surface
$$V = {\large\frac{1}{3}\normalsize}\Big| {{\large\iint\limits_S\normalsize} {xdydz + ydxdz }}$$ $$+\,{{ zdxdy} } \Big|$$
19. Gravitational force between a surface and a point mass
$$\vec{F} =$$ $$Gm{\large\iint\limits_S\normalsize} {\mu \left( {x,y,z} \right){\large\frac{\mathbf{r}}{{{r^3}}}\normalsize} dS} ,$$
where $$m$$ is a mass at a point $$\left( {x_0},{y_0},{z_0} \right)$$ outside the surface, $${\mu \left( {x,y,z} \right)}$$ is the surface density, $$G$$ is the gravitational constant, and $$\mathbf{r} =$$ $$\left( {x – {x_0},y – {y_0},z – {z_0}} \right)$$.
20. Pressure force
$$\vec{F} = {\large\iint\limits_S\normalsize} {p\left( \mathbf{r} \right)d\mathbf{S}} ,$$
where the pressure $$p\left( \mathbf{r} \right)$$ acts on the surface $$S$$ given by the position vector $$\mathbf{r}$$.
21. Fluid flux across the surface S
$$\Phi = {\large\iint\limits_S\normalsize} {\mathbf{v}\left( \mathbf{r} \right) \cdot d\mathbf{S}} ,$$
where $$\mathbf{v}\left( \mathbf{r} \right)$$ is the fluid velocity.
22. Mass flux across the surface S
$$\Phi = {\large\iint\limits_S\normalsize} {\rho\mathbf{v}\left( \mathbf{r} \right) \cdot d\mathbf{S}} ,$$
where $$\mathbf{F} = \rho \mathbf{v}$$ is the vector field, $$\rho$$ is the density.
23. Surface charge
$$Q = {\large\iint\limits_S\normalsize} {\sigma \left( {x,y} \right)dS} ,$$
where $${\sigma \left( {x,y} \right)}$$ is the surface charge density.
24. Gauss’ law
The electric flux Φ through any closed surface is proportional to the charge Q enclosed by the surface:
$$\Phi = {\large\iint\limits_S\normalsize} {\mathbf{E} \cdot d\mathbf{S}} = {\large\frac{Q}{{{\varepsilon _0}}}\normalsize},$$
where $$\mathbf{E}$$ is the electric field strength, $$\Phi$$ is the electric flux, $${\varepsilon_0} = 8,85$$ $$\times\, 10^{-12} \text{F/m}$$ is permittivity of free space.