Part of a series of articles about |

Calculus |
---|

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 a scalar field (that is, a function of position which returns a scalar as a value) over the surface, 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 of scalar fields
- Surface integrals of vector fields
- Surface integrals of differential 2-forms
- Theorems involving surface integrals
- Dependence on parametrization
- See also
- References
- External links

Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.

To find an explicit formula for the surface integral over a surface *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*).^{ [1] }^{ [2] }

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.

Note that 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.

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 surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector. This applies for example in the expression of the electric field at some fixed point due to an electrically charged surface, or the gravity at some fixed point due to a sheet of material.

Alternatively, if we integrate the normal component of the vector field over the surface, the result is a scalar, usually called the flux passing through the surface. 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. 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.

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 .

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.

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.

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 surface, one cannot talk about integrating vector fields.

In physics the **Lorentz force** is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity **v** in an electric field **E** and a magnetic field **B** experiences a force of

In vector calculus and differential geometry, the **generalized Stokes theorem**, also called the **Stokes–Cartan theorem**, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.

**Flux** describes any effect that appears to pass or travel through a surface or substance. A flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.

In mathematics, **curvature** is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

In vector calculus and physics, a **vector field** is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used 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 or gravitational force, as it changes from one point to another point.

In mathematics, **Cauchy's integral formula**, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.

In vector calculus, the **divergence theorem**, also known as **Gauss's theorem** or **Ostrogradsky's theorem**, is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

In geometry, a **normal** is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the **normal line** to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one or its length may represent the curvature of the object ; its algebraic sign may indicate sides.

In vector calculus, **Green's theorem** relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem.

An **electromagnetic four-potential** is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.

**Magnetic vector potential**, **A**, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field: . Together with the electric potential *φ*, the magnetic vector potential can be used to specify the electric field **E** as well. Therefore, many equations of electromagnetism can be written either in terms of the fields **E** and **B**, or equivalently in terms of the potentials *φ* and **A**. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

The following are important identities involving derivatives and integrals in vector calculus.

A **parametric surface** is a surface in the Euclidean space which is defined by a parametric equation with two parameters . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

The **gradient theorem**, also known as the **fundamental theorem of calculus for line integrals**, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the fundamental theorem of calculus to any curve in a plane or space rather than just the real line.

In differential calculus, there is no single uniform **notation for differentiation**. Instead, several different notations for the derivative of a function or variable have been proposed by different mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation are listed below.

**Two-dimensional space** is a geometric setting in which two values are required to determine the position of an element. The set ℝ^{2} of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension.

In mathematics, a **line integral** is an integral where the function to be integrated is evaluated along a curve. The terms *path integral*, *curve integral*, and *curvilinear integral* are also used; *contour integral* is used as well, although that is typically reserved for line integrals in the complex plane.

**Stokes' theorem**, also known as **Kelvin–Stokes theorem** after Lord Kelvin and George Stokes, the **fundamental theorem for curls** or simply the **curl theorem**, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical Stokes' theorem can be stated in one sentence: The line integral of a vector field over a loop is equal to the *flux of its curl* through the enclosed surface.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

- ↑ Edwards, C. H. (1994).
*Advanced Calculus of Several Variables*. Mineola, NY: Dover. p. 335. ISBN 0-486-68336-2. - ↑ Hazewinkel, Michiel (2001).
*Encyclopedia of Mathematics*. Springer. pp. Surface Integral. ISBN 978-1-55608-010-4.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.