
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration.

Surface integrals have applications in physics, particularly in the classical theories of electromagnetism and fluid mechanics.

Surface integrals of scalar fields
Assume that f is a scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be r(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by
where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of r(s, t), and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere, where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced). The surface integral can also be expressed in the equivalent form
where g is the determinant of the first fundamental form of the surface mapping r(s, t).
For example, if we want to find the surface area of the graph of some scalar function, say z = f(x, y), we have
where r = (x, y, z) = (x, y, f(x, y)). So that , and
. So,
which is the standard formula for the area of a surface described this way. One can recognize the vector in the second-last line above as the normal vector to the surface.
Because of the presence of the cross product, the above formulas only work for surfaces embedded in three-dimensional space.
This can be seen as integrating a Riemannian volume form on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface.
Surface integrals of vector fields
Consider a vector field v on a surface S, that is, for each r = (x, y, z) in S, v(r) is a vector.
The integral of v on S was defined in the previous section. Suppose now that it is desired to integrate only the normal component of the vector field over the surface, the result being a scalar, usually called the flux passing through the surface. For example, imagine that we have a fluid flowing through S, such that v(r) determines the velocity of the fluid at r. The flux is defined as the quantity of fluid flowing through S per unit time.
This illustration implies that if the vector field is tangent to S at each point, then the flux is zero because the fluid just flows in parallel to S, and neither in nor out. This also implies that if v does not just flow along S, that is, if v has both a tangential and a normal component, then only the normal component contributes to the flux. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above. In other words, we have to integrate v with respect to the vector surface element , which is the vector normal to S at the given point, whose magnitude is
We find the formula
The cross product on the right-hand side of this expression is a (not necessarily unital) surface normal determined by the parametrisation.
This formula defines the integral on the left (note the dot and the vector notation for the surface element).
We may also interpret this as a special case of integrating 2-forms, where we identify the vector field with a 1-form, and then integrate its Hodge dual over the surface. This is equivalent to integrating over the immersed surface, where
is the induced volume form on the surface, obtained by interior multiplication of the Riemannian metric of the ambient space with the outward normal of the surface.
Surface integrals of differential 2-forms
Let
be a differential 2-form defined on a surface S, and let
be an orientation preserving parametrization of S with in D. Changing coordinates from
to
, the differential forms transform as
So transforms to
, where
denotes the determinant of the Jacobian of the transition function from
to
. The transformation of the other forms are similar.
Then, the surface integral of f on S is given by
where
is the surface element normal to S.
Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as components ,
and
.
Theorems involving surface integrals
Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, magnetic flux, and its generalization, Stokes' theorem.
Dependence on parametrization
Let us notice that we defined the surface integral by using a parametrization of the surface S. We know that a given surface might have several parametrizations. For example, if we move the locations of the North Pole and the South Pole on a sphere, the latitude and longitude change for all the points on the sphere. A natural question is then whether the definition of the surface integral depends on the chosen parametrization. For integrals of scalar fields, the answer to this question is simple; the value of the surface integral will be the same no matter what parametrization one uses.
For integrals of vector fields, things are more complicated because the surface normal is involved. It can be proven that given two parametrizations of the same surface, whose surface normals point in the same direction, one obtains the same value for the surface integral with both parametrizations. If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. It follows that given a surface, we do not need to stick to any unique parametrization, but, when integrating vector fields, we do need to decide in advance in which direction the normal will point and then choose any parametrization consistent with that direction.
Another issue is that sometimes surfaces do not have parametrizations which cover the whole surface. The obvious solution is then to split that surface into several pieces, calculate the surface integral on each piece, and then add them all up. This is indeed how things work, but when integrating vector fields, one needs to again be careful how to choose the normal-pointing vector for each piece of the surface, so that when the pieces are put back together, the results are consistent. For the cylinder, this means that if we decide that for the side region the normal will point out of the body, then for the top and bottom circular parts, the normal must point out of the body too.
Last, there are surfaces which do not admit a surface normal at each point with consistent results (for example, the Möbius strip). If such a surface is split into pieces, on each piece a parametrization and corresponding surface normal is chosen, and the pieces are put back together, we will find that the normal vectors coming from different pieces cannot be reconciled. This means that at some junction between two pieces we will have normal vectors pointing in opposite directions. Such a surface is called non-orientable, and on this kind of surface, one cannot talk about integrating vector fields.
See also
- Area element
- Divergence theorem
- Stokes' theorem
- Line integral
- Line element
- Volume element
- Volume integral
- Cartesian coordinate system
- Volume and surface area elements in spherical coordinate systems
- Volume and surface area elements in cylindrical coordinate systems
- Holstein–Herring method
References
- Edwards, C. H. (1994). Advanced Calculus of Several Variables. Mineola, NY: Dover. p. 335. ISBN 0-486-68336-2.
- Hazewinkel, Michiel (2001) [1994]. "Surface Integral". Encyclopedia of Mathematics. EMS Press.
External links
- Weisstein, Eric W. "Surface Integral". MathWorld.
In mathematics particularly multivariable calculus a surface integral is a generalization of multiple integrals to integration over surfaces It can be thought of as the double integral analogue of the line integral Given a surface one may integrate over this surface a scalar field that is a function of position which returns a scalar as a value or a vector field that is a function which returns a vector as value If a region R is not flat then it is called a surface as shown in the illustration The definition of the surface integral relies on splitting the surface into small surface elements Surface integrals have applications in physics particularly in the classical theories of electromagnetism and fluid mechanics An illustration of a single surface element These elements are made infinitesimally small by the limiting process so as to approximate the surface Surface integrals of scalar fieldsAssume that f is a scalar vector or tensor field defined on a surface S To find an explicit formula for the surface integral of f over S we need to parameterize S by defining a system of curvilinear coordinates on S like the latitude and longitude on a sphere Let such a parameterization be r s t where s t varies in some region T in the plane Then the surface integral is given by SfdS Tf r s t r s r t dsdt displaystyle iint S f mathrm d S iint T f mathbf r s t left partial mathbf r over partial s times partial mathbf r over partial t right mathrm d s mathrm d t where the expression between bars on the right hand side is the magnitude of the cross product of the partial derivatives of r s t and is known as the surface element which would for example yield a smaller value near the poles of a sphere where the lines of longitude converge more dramatically and latitudinal coordinates are more compactly spaced The surface integral can also be expressed in the equivalent form SfdS Tf r s t gdsdt displaystyle iint S f mathrm d S iint T f mathbf r s t sqrt g mathrm d s mathrm d t where g is the determinant of the first fundamental form of the surface mapping r s t For example if we want to find the surface area of the graph of some scalar function say z f x y we have A SdS T r x r y dxdy displaystyle A iint S mathrm d S iint T left partial mathbf r over partial x times partial mathbf r over partial y right mathrm d x mathrm d y where r x y z x y f x y So that r x 1 0 fx x y displaystyle partial mathbf r over partial x 1 0 f x x y and r y 0 1 fy x y displaystyle partial mathbf r over partial y 0 1 f y x y So A T 1 0 f x 0 1 f y dxdy T f x f y 1 dxdy T f x 2 f y 2 1dxdy displaystyle begin aligned A amp iint T left left 1 0 partial f over partial x right times left 0 1 partial f over partial y right right mathrm d x mathrm d y amp iint T left left partial f over partial x partial f over partial y 1 right right mathrm d x mathrm d y amp iint T sqrt left partial f over partial x right 2 left partial f over partial y right 2 1 mathrm d x mathrm d y end aligned which is the standard formula for the area of a surface described this way One can recognize the vector in the second last line above as the normal vector to the surface Because of the presence of the cross product the above formulas only work for surfaces embedded in three dimensional space This can be seen as integrating a Riemannian volume form on the parameterized surface where the metric tensor is given by the first fundamental form of the surface Surface integrals of vector fieldsA curved surface S displaystyle S with a vector field F displaystyle mathbf F passing through it The red arrows vectors represent the magnitude and direction of the field at various points on the surfaceSurface divided into small patches dS dudv displaystyle dS du dv by a parameterization of the surface u x v x displaystyle u mathbf x v mathbf x The flux through each patch is equal to the normal perpendicular component of the field Fn x F x cos 8 displaystyle F n mathbf x F mathbf x cos theta at the patch s location x displaystyle mathbf x multiplied by the area dS displaystyle dS The normal component is equal to the dot product of F x displaystyle mathbf F mathbf x with the unit normal vector n x displaystyle mathbf n mathbf x blue arrows The total flux through the surface is found by adding up F ndS displaystyle mathbf F cdot mathbf n dS for each patch In the limit as the patches become infinitesimally small this is the surface integral SF ndS textstyle int S mathbf F cdot n dS Consider a vector field v on a surface S that is for each r x y z in S v r is a vector The integral of v on S was defined in the previous section Suppose now that it is desired to integrate only the normal component of the vector field over the surface the result being a scalar usually called the flux passing through the surface For example imagine that we have a fluid flowing through S such that v r determines the velocity of the fluid at r The flux is defined as the quantity of fluid flowing through S per unit time This illustration implies that if the vector field is tangent to S at each point then the flux is zero because the fluid just flows in parallel to S and neither in nor out This also implies that if v does not just flow along S that is if v has both a tangential and a normal component then only the normal component contributes to the flux Based on this reasoning to find the flux we need to take the dot product of v with the unit surface normal n to S at each point which will give us a scalar field and integrate the obtained field as above In other words we have to integrate v with respect to the vector surface element ds nds displaystyle mathrm d mathbf s mathbf n mathrm d s which is the vector normal to S at the given point whose magnitude is ds ds displaystyle mathrm d s mathrm d mathbf s We find the formula Sv ds S v n ds T v r s t r s r t r s r t r s r t dsdt Tv r s t r s r t dsdt displaystyle begin aligned iint S mathbf v cdot mathrm d mathbf s amp iint S left mathbf v cdot mathbf n right mathrm d s amp iint T left mathbf v mathbf r s t cdot frac partial mathbf r partial s times frac partial mathbf r partial t over left frac partial mathbf r partial s times frac partial mathbf r partial t right right left frac partial mathbf r partial s times frac partial mathbf r partial t right mathrm d s mathrm d t amp iint T mathbf v mathbf r s t cdot left frac partial mathbf r partial s times frac partial mathbf r partial t right mathrm d s mathrm d t end aligned The cross product on the right hand side of this expression is a not necessarily unital surface normal determined by the parametrisation This formula defines the integral on the left note the dot and the vector notation for the surface element We may also interpret this as a special case of integrating 2 forms where we identify the vector field with a 1 form and then integrate its Hodge dual over the surface This is equivalent to integrating v n dS displaystyle left langle mathbf v mathbf n right rangle mathrm d S over the immersed surface where dS displaystyle mathrm d S is the induced volume form on the surface obtained by interior multiplication of the Riemannian metric of the ambient space with the outward normal of the surface Surface integrals of differential 2 formsLet f dxdyfz dydzfx dzdxfy displaystyle f mathrm d x mathrm d y f z mathrm d y mathrm d z f x mathrm d z mathrm d x f y be a differential 2 form defined on a surface S and let r s t x s t y s t z s t displaystyle mathbf r s t x s t y s t z s t be an orientation preserving parametrization of S with s t displaystyle s t in D Changing coordinates from x y displaystyle x y to s t displaystyle s t the differential forms transform as dx x sds x tdt displaystyle mathrm d x frac partial x partial s mathrm d s frac partial x partial t mathrm d t dy y sds y tdt displaystyle mathrm d y frac partial y partial s mathrm d s frac partial y partial t mathrm d t So dxdy displaystyle mathrm d x mathrm d y transforms to x y s t dsdt displaystyle frac partial x y partial s t mathrm d s mathrm d t where x y s t displaystyle frac partial x y partial s t denotes the determinant of the Jacobian of the transition function from s t displaystyle s t to x y displaystyle x y The transformation of the other forms are similar Then the surface integral of f on S is given by D fz r s t x y s t fx r s t y z s t fy r s t z x s t dsdt displaystyle iint D left f z mathbf r s t frac partial x y partial s t f x mathbf r s t frac partial y z partial s t f y mathbf r s t frac partial z x partial s t right mathrm d s mathrm d t where r s r t y z s t z x s t x y s t displaystyle partial mathbf r over partial s times partial mathbf r over partial t left frac partial y z partial s t frac partial z x partial s t frac partial x y partial s t right is the surface element normal to S Let us note that the surface integral of this 2 form is the same as the surface integral of the vector field which has as components fx displaystyle f x fy displaystyle f y and fz displaystyle f z Theorems involving surface integralsVarious useful results for surface integrals can be derived using differential geometry and vector calculus such as the divergence theorem magnetic flux and its generalization Stokes theorem Dependence on parametrizationLet us notice that we defined the surface integral by using a parametrization of the surface S We know that a given surface might have several parametrizations For example if we move the locations of the North Pole and the South Pole on a sphere the latitude and longitude change for all the points on the sphere A natural question is then whether the definition of the surface integral depends on the chosen parametrization For integrals of scalar fields the answer to this question is simple the value of the surface integral will be the same no matter what parametrization one uses For integrals of vector fields things are more complicated because the surface normal is involved It can be proven that given two parametrizations of the same surface whose surface normals point in the same direction one obtains the same value for the surface integral with both parametrizations If however the normals for these parametrizations point in opposite directions the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization It follows that given a surface we do not need to stick to any unique parametrization but when integrating vector fields we do need to decide in advance in which direction the normal will point and then choose any parametrization consistent with that direction Another issue is that sometimes surfaces do not have parametrizations which cover the whole surface The obvious solution is then to split that surface into several pieces calculate the surface integral on each piece and then add them all up This is indeed how things work but when integrating vector fields one needs to again be careful how to choose the normal pointing vector for each piece of the surface so that when the pieces are put back together the results are consistent For the cylinder this means that if we decide that for the side region the normal will point out of the body then for the top and bottom circular parts the normal must point out of the body too Last there are surfaces which do not admit a surface normal at each point with consistent results for example the Mobius strip If such a surface is split into pieces on each piece a parametrization and corresponding surface normal is chosen and the pieces are put back together we will find that the normal vectors coming from different pieces cannot be reconciled This means that at some junction between two pieces we will have normal vectors pointing in opposite directions Such a surface is called non orientable and on this kind of surface one cannot talk about integrating vector fields See alsoArea element Divergence theorem Stokes theorem Line integral Line element Volume element Volume integral Cartesian coordinate system Volume and surface area elements in spherical coordinate systems Volume and surface area elements in cylindrical coordinate systems Holstein Herring methodReferencesEdwards C H 1994 Advanced Calculus of Several Variables Mineola NY Dover p 335 ISBN 0 486 68336 2 Hazewinkel Michiel 2001 1994 Surface Integral Encyclopedia of Mathematics EMS Press External linksWeisstein Eric W Surface Integral MathWorld