Details and Options

• Surface integrals are also known as flux integrals.
• Scalar surface integrals integrate scalar functions over a hypersurface. They are typically used to compute things like area, mass and charge for a surface.
• Vector surface integrals are used to compute the flux of a vector function through a surface in the direction of its normal. Typical vector functions include a fluid velocity field, electric field and magnetic field.
• The scalar surface integral of a function f over a surface is given by:

• where is the measure of a parametric surface element.
• The scalar surface integral of f over a hypersurface is given by:



• The scalar surface integral is independent of the parametrization and orientation of the surface. Any dimensional RegionQ object in can be use for the surface.
• The vector surface integral of a vector function over a surface is given by:

• where is the projection of the vector function onto the normal direction so only the component in the normal direction gets integrated.
• The vector surface integral of over a a hypersurface is given by:



• The vector surface integral is independent of the parametrization, but depends on the orientation.
• The orientation for a hypersurface is given by a normal vector field over the surface.

• For a parametric hypersurface ParametricRegion[{r₁[u₁,…,u_n-1],…,r_n[u₁,…,u_n-1]},…], the normal vector field is taken to be Cross[∂_u1r[u],…,∂_un-1r[u]].
• The RegionQ objects in Wolfram Language are not oriented. However for the convenience of this function, you can assume the following rules for getting oriented hypersurfaces.
• For solid (of dimension ) and bounded RegionQ objects ℛ, take the surface to be the region boundary (RegionBoundary[ℛ]) and the normal orientation to be pointed outward.
• Special solids in with their assumed boundary surface (edge) normal orientation include:
• Triangleoutward normal Rectangleoutward normal Polygonoutward normal Diskoutward normal Ellipsoidoutward normal Annulusoutward normal
• Special solids in with their assumed boundary surface (face) normal orientation include:
• Tetrahedronoutward normal Cuboidoutward normal Polyhedronoutward normal Balloutward normal Ellipsoidoutward normal Cylinderoutward normal Coneoutward normal
• Special solids in with their assumed surface (facet) and normal orientation:
• Simplexoutward normal Cuboidoutward normal Balloutward normal Ellipsoidoutward normal
• The following options can be given:
• Assumptions $Assumptionsassumptions to make about parameters GenerateConditions Automaticwhether to generate answers that involve conditions on parameters WorkingPrecision Automaticthe precision used in internal computations
• SurfaceIntegrate uses a combination of symbolic and numerical methods when the input involves inexact quantities.

Examples

Basic Examples  (6)

Surface integral of a scalar function over a spherical surface:

Surface integral of a vector field over a spherical surface:

Surface integral of a scalar field over a parametric surface:

Surface integral of a vector field over a parametric surface:

Surface integral of a scalar field over a surface:

Visualize the scalar field on the surface:

Surface integral of a vector field over a surface:

Visualize the scalar field on the surface:

Scope  (32)

Basic Uses  (5)

Surface integral of a scalar field over a sphere in three dimensions:

Surface integral of a vector field in three dimensions:

SurfaceIntegrate works with many special surfaces:

Surface integral over a parametric surface:

SurfaceIntegrate works in dimensions different from three:

Scalar Functions  (5)

Surface integral of a scalar field over a three-dimensional surface:

Plot of the surface:

Surface integral:

Surface integral of a scalar field:

The surface integral:

Surface integral of a scalar field in three dimensions over a sphere:

Surface integral of a scalar field over the surface of a pyramid:

Surface integral:

Surface integral of a scalar field over a parametric surface in three dimensions:

The surface and its plot:

Vector Functions  (5)

Surface integral of a vector field in three dimensions over a sphere:

Visualize the vector field on the surface:

Surface integral:

Surface integral of a vector field in three dimensions over a triangle:

Surface integral:

Surface integral of a vector field over a parametric surface in three dimensions:

Surface integral of a vector field over the boundary of an ellipsoid:

Surface integral of a vector field in three dimensions over the boundary of a cone:

Visualization of the vector field on the surface:

Surface integral:

Special Surfaces  (10)

Surface integral of a vector field over a sphere of radius :

Surface integral of a vector field over the boundary of a cube of side centered at the origin:

Surface integral of a vector field over the boundary of a tetrahedron:

Surface integral of a vector field over a triangle:

Surface integral of a vector field over an ellipsoid:

Surface integral of a vector field over the boundary of a cone:

Surface integral of a vector field over the boundary of a cylinder:

Surface integral of a vector field over the boundary of a parallelepiped:

Surface integral of a vector field over the boundary of a prism:

Surface integral over a polygon in three dimensions:

The orientation of the polygon depends on the order in which the points are given:

Parametric Surfaces  (4)

Surface integral of a vector field over a parametric surface:

Surface integral of a vector field over a parametrized dome-like surface:

Surface integral over a parametrized cylinder:

Surface integral of a vector field over a parametrized hyperboloid:

Hypersurfaces  (3)

Surface integral over a 1D hypersurface in 2D:

Surface integral over a 3D hypersurface in 4D:

Volume of a five-dimensional sphere, computed using a surface integral:

Options  (4)

Assumptions  (1)

Assumptions can be specified for symbolic parameters:

With Assumptions, a result valid under the given assumptions is given:

GenerateConditions  (1)

SurfaceIntegrate can work with symbolic parameters:

Generate conditions on the parameters:

WorkingPrecision  (2)

If a WorkingPrecision is specified, a numerical result is given:

The result has finite precision if the integrand has a finite precision:

Applications  (18)

College Calculus  (5)

Surface integral over the boundary of a cube of side 2 centered at the origin:

Surface integral over a paraboloid:

Surface integral over the side of a cylinder:

Surface integral over a hemispherical shell of radius :

Surface integral over the boundary of a cube:

Areas  (3)

Area of a sphere:

Area of an ellipsoid:

Area of a triangle:

Volumes  (3)

Volume of an ellipsoid computed using a surface integral:

Volume of a icosahedron computed using a surface integral:

Volume of a cube of side computed using a surface integral:

Flux  (3)

Flux of the electric field generated by a point charge at the origin over a sphere surrounding it:

Flux of the uniform magnetic field of an infinite solenoid with windings per unit length traversed by a current over a disk orthogonal to it:

Electric field due to an infinite charged wire of linear charge density :

Flux across a cylinder of height and radius having the axis on the charged wire:

Centroids  (2)

Mass of a hemispherical shell of unit density and radius :

coordinate of the center of mass:

coordinate of the center of mass:

coordinate of the center of mass:

Moments of inertia of a thin cut cone:

About the axis:

About the axis:

About the axis:

Classical Theorems  (2)

Compute the Curl of a vector field :

The surface integral of over an open surface is:

This is the same as the line integral of over the boundary of the surface:

Compute the surface integral of a vector field over a closed surface:

This is the same as the integral of Div[f] over the interior of the surface:

Properties & Relations  (5)

Apply N[SurfaceIntegrate[...]] to obtain a numerical solution if the symbolic calculation fails:

Find the center of mass of a thin triangular surface of unit mass per unit area:

Find the total mass:

Find the component of the center of mass:

Find the component of the center of mass:

Find the component of the center of mass:

The center of mass can also be obtained using RegionCentroid:

Find the moment of inertia around the axis of a thin cylindrical shell of unit area density:

The answer can also be computed with MomentOfInertia:

Find the area of a tetrahedron:

The answer can also be computed with Area:

Find the volume of a icosahedron:

The answer can also be computed with Volume:

Neat Examples  (9)

Volume of a pseudosphere computed using a surface integral:

Plot of a finite part of the pseudosphere:

Volume of a drop-shaped solid using a surface integral:

Volume of a Dupin cyclide:

Flux of a vector field across a part of a Bohemian dome:

Surface integral of a vector field over a portion of a conocuneus of Wallis:

Surface integral of a vector field over a funnel-shaped surface:

Area of a Gaudi surface:

Compute numerically the area of Guimard's surface:

Surface integral of a vector field over a neiloid: