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Surface integral
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a surface integral is a definite integral taken over a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 (which may be a curved set in space
Space

Space is the boundless, three-dimensional extent in which Physical body and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physics usually consider it, with time, to be part of the boundless four-dimensional continuum known as spacetime....
); it can be thought of as the double integral analog of the line integral
Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use....
. Given a surface, one may integrate over it scalar field
Scalar field

In mathematics and physics, a scalar field associates a scalar value, which can be either scalar in definition, or scalar , to every point in space....
s (that is, function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
s which return number
Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
s as values), and vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
s (that is, functions which return vectors as values).

Surface integrals have applications in physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, particularly with the classical theory
Classical theory

Classical theory has at least two distinct meanings in Physics:#In the context of quantum mechanics, "classical theory" refers to theory of physics that do not use the Quantization paradigm, particularly Newtonian mechanics ....
 of electromagnetism
Electromagnetism

Electromagnetism is the physics of the electromagnetic field, a field which exerts a force on Elementary particles with the property of electric charge and which is reciprocally affected by the presence and motion of such particles....
.

ider a surface S on which a scalar field f is defined.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a surface integral is a definite integral taken over a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 (which may be a curved set in space
Space

Space is the boundless, three-dimensional extent in which Physical body and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physics usually consider it, with time, to be part of the boundless four-dimensional continuum known as spacetime....
); it can be thought of as the double integral analog of the line integral
Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use....
. Given a surface, one may integrate over it scalar field
Scalar field

In mathematics and physics, a scalar field associates a scalar value, which can be either scalar in definition, or scalar , to every point in space....
s (that is, function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
s which return number
Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
s as values), and vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
s (that is, functions which return vectors as values).

Surface integrals have applications in physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, particularly with the classical theory
Classical theory

Classical theory has at least two distinct meanings in Physics:#In the context of quantum mechanics, "classical theory" refers to theory of physics that do not use the Quantization paradigm, particularly Newtonian mechanics ....
 of electromagnetism
Electromagnetism

Electromagnetism is the physics of the electromagnetic field, a field which exerts a force on Elementary particles with the property of electric charge and which is reciprocally affected by the presence and motion of such particles....
.

Surface integrals of scalar fields

Consider a surface S on which a scalar field f is defined. If we think of S as made of some material, and for each x in S the number f(x) is the density
Density

The density of a material is defined as its mass per unit volume. The symbol of density is ....
 of material at x, then the surface integral of f over S is the mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
 per unit thickness of S. (This only holds as long as the surface is an infinitesimally thin shell.) One approach to calculate the surface integral is then to split the surface in many very small pieces, assume that on each piece the density is approximately constant, find the mass per unit thickness of each piece by multiplying the density of the piece by its area, and then sum up the resulting numbers to find the total mass per unit thickness of S.

To find an explicit formula for the surface integral, we need to parametrize
Coordinate system

In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalar to each Point in an n-dimensional space....
 S by considering on S a system of curvilinear coordinates
Curvilinear coordinates

Curvilinear coordinates are a coordinate system for the Euclidean space based on some transformation that converts the standard Cartesian coordinate system to a coordinate system with the same number of coordinates in which the coordinate lines are curved....
, like the latitude and longitude
Geographic coordinate system

A geographic coordinate system enables every location on the Earth to be specified in three coordinates, using mainly a Spherical coordinates#Spherical coordinates....
 on a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
. Let such a parametrization be x(s, t), where (s, t) varies in some region T in the plane
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
. Then, the surface integral is given by

where the expression between bars on the right-hand side is the magnitude
Magnitude (mathematics)

The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
 of the cross product
Cross product

In mathematics, the cross product is a binary operation on two vector s in a three-dimensional Euclidean space that results in another vector which is orthogonal to the plane containing the two input vectors....
 of the partial derivative
Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant ....
s of x(s, t).

For example, if we want to find the surface area of some general functional shape, say , we have

where . So, , and . So,

which is the familiar formula we get for the surface area of a general functional shape. One can recognize the vector in the second line above as the normal vector
Surface normal

A surface normal, or simply normal, to a Flatness is a vector which is perpendicular to that surface. A normal to a non-flat surface at a Point P on the surface is a vector perpendicular to the Tangent space to that surface at P....
 to the surface.

Note that because of the presence of the cross product, the above formulas only work for surfaces embedded in three dimensional space.

Surface integrals of vector fields

Consider a vector field v on S, that is, for each x in S, v(x) is a vector. Imagine that we have a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux
Flux

In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.*In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time....
 is defined as the quantity of fluid flowing through S in unit amount of time.

This illustration implies that if the vector field is tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 to S at each point, then the flux is zero, because the fluid just flows in parallel
Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more line s or plane , or a combination of these....
 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
Dot product

In mathematics, the dot product, also known as the scalar product, is an operation which takes two vector over the real numbers R and returns a real-valued scalar quantity....
 of v with the unit surface normal
Surface normal

A surface normal, or simply normal, to a Flatness is a vector which is perpendicular to that surface. A normal to a non-flat surface at a Point P on the surface is a vector perpendicular to the Tangent space to that surface at P....
 to S at each point, which will give us a scalar field, and integrate the obtained field as above. We find the formula

The cross product on the right-hand side of this expression is a surface normal determined by the parametrization.

This formula is defined to be the integral of the vector field v on S.

Surface integrals of differential 2-forms


Let be a differential 2-form
Differential form

In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....
 defined on the surface S, and let

be an orientation preserving
Orientability

A surface S in the Euclidean space R3 is orientable if a two-dimensional figure cannot be moved around the surface and back to where it started so that it looks like its own mirror image ....
 parametrization of S with in D. Then, the surface integral of f on S is given by

where is the surface 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
Vector calculus

Vector calculus is a branch of mathematics concerned with derivative and integral of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial derivative and multiple integral....
, such as the divergence theorem
Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss?s theorem , Ostrogradsky?s theorem , or Gauss-Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface....
, and its generalization, Stokes' theorem
Stokes' theorem

In differential geometry, Stokes' theorem is a statement about the integral of differential forms which generalizes several theorems from vector calculus....
.

Advanced issues


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 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 proved 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 which direction the normal will point to and then choose any parametrization consistent with that direction.

Another issue is that sometimes surfaces do not have parametrizations which cover the whole surface; this is true for example for the surface of a cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
 (of finite height). The obvious solution is then to split that surface in 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.

Lastly, there are surfaces which do not admit a surface normal at each point with consistent results (for example, the Möbius strip
Möbius strip

The M?bius strip or M?bius band is a surface with only one side and only one boundary component. The M?bius strip has the mathematical property of being orientability....
). 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
Orientability

A surface S in the Euclidean space R3 is orientable if a two-dimensional figure cannot be moved around the surface and back to where it started so that it looks like its own mirror image ....
, and on this kind of surface one cannot talk about integrating vector fields.

See also

  • Divergence theorem
    Divergence theorem

    In vector calculus, the divergence theorem, also known as Gauss?s theorem , Ostrogradsky?s theorem , or Gauss-Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface....
  • Stokes' theorem
    Stokes' theorem

    In differential geometry, Stokes' theorem is a statement about the integral of differential forms which generalizes several theorems from vector calculus....
  • Volume integral
    Volume integral

    In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain.Volume integral is a triple integral of the constant function 1, which gives the volume of the region D, that is, the integral...
  • Volume and surface elements in different co-ordinate systems
    Volume and surface elements in different co-ordinate systems

    This page outlines the value of different volume element and surface elements in several different coordinate systems.Surface elements depend on the orientation of the surface over which the integration is performed....


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