In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. Given a surface, one can integrate over it scalar fields (that is, functions which return numbers as values), and vector fields (that is, functions which return vectors as values). Euclid, detail from The School of Athens by Raphael. ... This article deals with the concept of an integral in calculus. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematical analysis, there is a serious distinction between a double integral and an iterated integral. ... In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... Partial plot of a function f. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...

Surface integrals have applications in physics, especially in the classical theory of electromagnetism. A Superconductor demonstrating the Meissner Effect. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, which exerts a force on those particles that possess the property of electric charge, and is in turn 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 of material at x, then the surface integral of f over S is the mass per unit thickness of S (this only holds as long as the surface is an infintessimally thin shell.) One approach to calculating 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. Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... Mass is a property of a physical object that quantifies the amount of matter it contains. ...

To find an explicit formula for the surface integral, we need to parametrize S by considering on S a system of curvilinear coordinates, like the latitude and longitude on a sphere. Let such a parametrization be x(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system. ... Map of Earth showing lines of latitude (horizontally) and longitude (vertically); large version (pdf) The geographic (earth-mapping) coordinate system expresses every horizontal position on Earth by two of the three coordinates of a spherical coordinate system which is aligned with the spin axis of the Earth. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x(s, t). // Real numbers The magnitude of a real number is usually called the absolute value or modulus. ... In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ... 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 (as opposed to the total derivative, in which all variables are allowed to vary). ...

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 direction and velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in unit amount of time. In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks. ...

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 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 to S at each point, which will give us a scalar field, and integrate the obtained field as above. We find the formula This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... Perpendicular is a geometric term that may be used as a noun or adjective. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

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 defined on the surface S, and let A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

be an orientation preserving parametrization of S with (s,t) in D. Then, the surface integral of f on S is given by It has been suggested that this article or section be merged with orientable manifold. ...

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 f₁, f₂ and f₃.

Theorems involving surface integrals

Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradsky–Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ... The Stokes theorem in differential geometry is a statement about the integration 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. The obvious solution is then to split that surface in several pieces, calculate the surface integral on each piece, and then add things 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. A right circular cylinder In mathematics, a cylinder is a quadric, i. ...

Lastly, 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 surfaces one cannot talk about integrating vector fields. A Möbius strip made with a piece of paper and tape. ... It has been suggested that this article or section be merged with orientable manifold. ...

See also

• Path integral
• Volume integral

In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve. ... In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain. ...

External links

• Surface Integral -- from MathWorld