SurfaceIntegrate
SurfaceIntegrate[f,{x,y,…}∈surface]
computes the scalar surface integral of the function f[x,y,…] over the surface.
SurfaceIntegrate[{p,q,…},{x,y,…}∈surface]
computes the vector surface integral of the vector field {p[x,y,…],q[x,y,…],…}.
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,…,un-1],…,rn[u1,…,un-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:
-
Triangle outward normal Rectangle outward normal Polygon outward normal Disk outward normal Ellipsoid outward normal Annulus outward normal - Special solids in with their assumed boundary surface (face) normal orientation include:
-
Tetrahedron outward normal Cuboid outward normal Polyhedron outward normal Ball outward normal Ellipsoid outward normal Cylinder outward normal Cone outward normal - Special solids in with their assumed surface (facet) and normal orientation:
-
Simplex outward normal Cuboid outward normal Ball outward normal Ellipsoid outward normal - The following options can be given:
-
Assumptions $Assumptions assumptions to make about parameters GenerateConditions Automatic whether to generate answers that involve conditions on parameters WorkingPrecision Automatic the precision used in internal computations - SurfaceIntegrate uses a combination of symbolic and numerical methods when the input involves inexact quantities.
Examples
open allclose allBasic 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:
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:
Surface integral of a scalar field:
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 of a scalar field over a parametric surface in three dimensions:
Vector Functions (5)
Surface integral of a vector field in three dimensions over a sphere:
Visualize the vector field on the surface:
Surface integral of a vector field in three dimensions over a triangle:
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:
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)
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:
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)
Volumes (3)
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)
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 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:
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:
Text
Wolfram Research (2023), SurfaceIntegrate, Wolfram Language function, https://reference.wolfram.com/language/ref/SurfaceIntegrate.html.
CMS
Wolfram Language. 2023. "SurfaceIntegrate." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SurfaceIntegrate.html.
APA
Wolfram Language. (2023). SurfaceIntegrate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SurfaceIntegrate.html