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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.