The multiple integral is a type of definite integral Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral extended to functions The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain of more than one real variable A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use. The concepts of constants and variables are fundamental to all modern mathematics, science, engineering, and computer programming, for example, ƒ(x, y) or ƒ(x, y, z).

Contents 1 Introduction 2 Mathematical definition 2.1 Properties 2.2 Particular cases 3 Methods of integration 3.1 Integrating constant functions 3.2 Use of symmetry 3.3 Normal domains on R2 3.3.1 x-axis 3.3.2 y-axis 3.3.3 Example 3.4 Normal domains on R3 3.5 Change of variables 3.5.1 Polar coordinates 3.5.2 Cylindrical coordinates 3.5.3 Spherical coordinates 4 Examples 4.1 Double integral 4.2 Volumes 4.3 Computing a volume 5 Multiple improper integral 6 Multiple integrals and iterated integrals 7 Some practical applications 8 See also 9 References 10 External links

Introduction

Just as the definite integral of a positive function of one variable represents the area Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The surface area of a 3-dimensional solid is the total area of the exposed surface, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential geometry of of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume Volume is how much three-dimensional space a substance or shape occupies or contains, often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of of the region between the surface defined by the function (on the three dimensional Cartesian plane A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length where z = ƒ(x, y)) and the plane which contains its domain In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 (ignoring complex numbers in both cases). In a representation of a function in a. (Note that the same volume can be obtained via the triple integral—the integral of a function in three variables—of the constant function ƒ(x, y, z) = 1 over the above-mentioned region between the surface and the plane.) If there are more variables, a multiple integral will yield hypervolumes of multi-dimensional functions.

Multiple integration of a function in n variables: f(x₁, x₂, ..., x_n) over a domain D is most commonly represented by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments in proper order (the integral with respect to the rightmost argument is computed last). The domain of integration is either represented symbolically for every argument over each integral sign, or is abbreviated by a variable at the rightmost integral sign:

Since the concept of an antiderivative In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i.e., F ′ = f. The process of solving for antiderivatives is called antidifferentiation and its opposite function is called differentiation, which is the process of finding a derivative. Antiderivatives are related to is only defined for functions of a single real variable, the usual definition of the indefinite integral In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i.e., F ′ = f. The process of solving for antiderivatives is called antidifferentiation and its opposite function is called differentiation, which is the process of finding a derivative. Antiderivatives are related to does not immediately extend to the multiple integral.

Mathematical definition

Let n be an integer greater than 1. Consider a so-called half-open n-dimensional rectangle (from here on simply called rectangle). For a plane In mathematics, a plane is any flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line (one-dimension) and a space (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of, n = 2, and the multiple integral is just a double integral

Divide each interval (a_i, b_i) into a finite number of non-overlapping subintervals, with each subinterval closed at the left end, and open at the right end. Denote such a subinterval by I_i.

Then the family of subrectangles of the form

is a partition of T that is, the subrectangles C are non-overlapping and their union is T. The diameter of a subrectangle C is, by definition, the largest of the lengths of the intervals whose Cartesian product is Cand the diameter of a given partition of T is defined as the largest of the diameters of the subrectangles in the partition.

Let f : T → R be a function defined on a rectangle T. Consider a partition of T

defined as above, where m is a positive integer.

A Riemann sum is a sum of the form

where for each k the point P_k is in C_k and m(C_k) is the product of the lengths of the intervals whose Cartesian product is C_k.

The function f is said to be Riemann integrable if the limit In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus and are used to define continuity, derivatives and integrals

exists, where the limit is taken over all possible partitions of T of diameter at most δ.

If f is Riemann integrable, S is called the Riemann integral of f over T and is denoted

The Riemann integral of a function defined over an arbitrary bounded n-dimensional set can be defined by extending that function to a function defined over a half-open rectangle whose values are zero outside the domain of the original function. Then the integral of the original function over the original domain is defined to be the integral of the extended function over its rectangular domain, if it exists.

In what follows the Riemann integral in n dimensions will be called multiple integral.

Properties

Multiple integrals have many of the same properties of integrals of functions of one variable (linearity, additivity, monotonicity, etc.). Moreover, just as in one variable, one can use the multiple integral to find the average of a function over a given set. Given a set D ⊆ Rⁿ and an integrable function f over D, the average value of f over its domain is given by

where m(D) is the measure In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, volume, etcetera. A particularly important example is the Lebesgue measure on a of D.

Particular cases

In the case of T ⊆ R², the integral

is the double integral of f on T, and if T ⊆ R³ the integral

is the triple integral of f on T.

Notice that, by convention, the double integral has two integral signs, and the triple integral has three; this is a notational convention which is convenient when computing a multiple integral as an iterated integral, as shown later in the article.

Methods of integration

The resolution of problems with multiple integrals consists, in most of cases, of finding a way to reduce the multiple integral to an iterated integral, a series of integrals of one variable, each being directly solvable. Sometimes, it is possible to obtain the result of the integration by direct examination without any calculations.

Integrating constant functions

When the integrand is a constant function c, the integral is equal to the product of c and the measure of the domain of integration. If c = 1 and the domain is a subregion of R², the integral gives the area of the region, while if the domain is a subregion of R³, the integral gives the volume of the region.

• For example:

and

in which case

,

since by definition.

Use of symmetry

When the domain of integration is symmetric about the origin with respect to at least one of the variables of integration and the integrand is odd with respect to this variable, the integral is equal to zero, as the integrals over the two halves of the domain have the same absolute value but opposite signs. When the integrand is even with respect to this variable, the integral is equal to twice the integral over one half of the domain, as the integrals over the two halves of the domain are equal.

• Example (1):

Consider the function f(x,y) = 2sinx − 3y³ + 5 integrated over the domain , a disc with radius In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter 1 centered at the origin with the boundary included.

Using the linearity property, the integral can be decomposed into three pieces:

2 sin  x and 3y³ are both odd functions and moreover it is evident that the T disc has a symmetry for the x and even the y axis; therefore the only contribution to the final result of the integrals is that of the constant function 5 because the other two pieces are null.

• Example (2):

Consider the function f(x, y, z) = x exp(y² + z²) and as integration region the sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The with radius 2 centered at the origin of the axes T = x² + y² + z² ≤ 4. The "ball" is symmetric about all three axes, but it is sufficient to integrate with respect to x-axis to show that the integral is 0, because the function is an odd function of that variable.

Normal domains on R²

See also: Order of integration (calculus) In calculus, interchange of the order of integration is a methodology that transforms iterated integrals of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot

This method is applicable to any domain D for which:

• the projection of D onto either the x-axis or the y-axis is bounded by the two values, a and b
• any line perpendicular to this axis that passes between these two values intersects the domain in an interval whose endpoints given by the graphs of two functions, α and β.

x-axis

If the domain D is normal with respect to the x-axis, and is a continuous function In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive though imprecise idea of continuity is; then α(x) and β(x) (defined on the interval [a, b]) are the two functions that determine D. Then:

y-axis

If D is normal with respect to the y-axis and is a continuous function; then α(y) and β(y) (defined on the interval [a, b]) are the two functions that determine D. Then:

Example

Example: double integral over the normal region D

Consider this region: (please see the graphic in the example). Calculate

This domain is normal with respect to both the x- and y-axes. To apply the formulas you have to find the functions that determine D and the intervals over which these are defined.

In this case the two functions are:

while the interval is given by the intersections of the functions with x = 0, so the interval is [a, b] = [0, 1] (normality has been chosen with respect to the x-axis for a better visual understanding).

It is now possible to apply the formulas:

(at first the second integral is calculated considering x as a constant). The remaining operations consist of applying the basic techniques of integration:

If we choose normality with respect to the y-axis we could calculate

and obtain the same value.

Example of domain in R³ that is normal with respect to the xy-plane.

Normal domains on R³

The extension of these formulae to triple integrals should be apparent:

if T is a domain that is normal with respect to the xy-plane and determined by the functions α (x,y) and β(x,y), then

(this definition is the same for the other five normality cases on R³).

Change of variables

See also: Integration by substitution#Substitution for multiple variables In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians. It is the counterpart to the chain rule of differentiation

The limits of integration are often not easily interchangeable (without normality or with complex formulae to integrate). One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae. To do so, the function must be adapted to the new coordinates.

Example (1-a):

The function is ;

if one adopts this substitution therefore

one obtains the new function .

• Similarly for the domain because it is delimited by the original variables that were transformed before (x and y in example).
• the differentials dx and dy transform via the absolute value of the determinant of the Jacobian matrix containing the partial derivatives of the transformations regarding the new variable (consider, as an example, the differential transformation in polar coordinates).

There exist three main "kinds" of changes of variable (one in R², two in R³); however, more general substitutions can be made using the same principle.

Polar coordinates

See also: Polar coordinate system In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction Transformation from cartesian to polar coordinates.

In R² if the domain has a circular "symmetry" and the function has some "particular" characteristics you can apply the transformation to polar coordinates (see the example in the picture) which means that the generic points P(x,y) in Cartesian coordinates switch to their respective points in polar coordinates. That allows one to change the "shape" of the domain and simplify the operations.

The fundamental relation to make the transformation is the following:

Example (2-a):

The function is

and applying the transformation one obtains

Example (2-b):

The function is

In this case one has:

using the Pythagorean trigonometric identity (very useful to simplify this operation).

The transformation of the domain is made by defining the radius' crown length and the amplitude of the described angle to define the ρ, φ intervals starting from x, y.

Example of a domain transformation from cartesian to polar.

Example (2-c):

The domain is , that is a circumference of radius 2; it's evident that the covered angle is the circle angle, so φ varies from 0 to 2π, while the crown radius varies from 0 to 2 (the crown with the inside radius null is just a circle).

Example (2-d):

The domain is , that is the circular crown in the positive y half-plane (please see the picture in the example); note that φ describes a plane angle while ρ varies from 2 to 3. Therefore the transformed domain will be the following rectangle In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD:

The Jacobian determinant of that transformation is the following:

which has been obtained by inserting the partial derivatives of x = ρ cos(φ), y = ρ sin(φ) in the first column respect to ρ and in the second respect to φ, so the dx dy differentials in this transformation becomes ρ dρ dφ.

Once the function is transformed and the domain evaluated, it is possible to define the formula for the change of variables in polar coordinates:

Please note that φ is valid in the [0, 2π] interval while ρ, which is a measure of a length, can only have positive values.

Example (2-e):

The function is ƒ(x, y) = x and as the domain the same in 2-d example.

From the previous analysis of D we know the intervals of ρ (from 2 to 3) and of φ (from 0 to π). Now let's change the function:

finally let's apply the integration formula:

Once the intervals are known, you have

Cylindrical coordinates

Cylindrical coordinates.

In R³ the integration on domains with a circular base can be made by the passage in cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative; the transformation of the function is made by the following relation:

The domain transformation can be graphically attained, because only the shape of the base varies, while the height follows the shape of the starting region.

Example (3-a):

The region is (that is the "tube" whose base is the circular crown of the 2-d example and whose height is 5); if the transformation is applied, this region is obtained: (that is the parallelepiped whose base is similar to the rectangle in 2-d example and whose height is 5).

Because the z component is unvaried during the transformation, the dx dy dz differentials vary as in the passage in polar coordinates: therefore, they become ρ dρ dφ dz.

Finally, it is possible to apply the final formula to cylindrical coordinates:

This method is convenient in case of cylindrical or conical domains or in regions where is easy to individuate the z interval and even transform the circular base and the function.

Example (3-b):

The function is and as integration domain this cylinder A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since: .

The transformation of D in cylindrical coordinates is the following:

while the function becomes

Finally you can apply the integration's formula:

developing the formula you have

Spherical coordinates

Spherical coordinates.

In R³ some domains have a spherical symmetry, so it's possible to specify the coordinates of every point of the integration region by two angles and one distance. It's possible to use therefore the passage in spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane; the function is transformed by this relation:

Note that points on z axis do not have a precise characterization in spherical coordinates, so φ can vary between 0 to π .

The better integration domain for this passage is obviously the sphere.

Example (4-a):

The domain is (sphere with radius 4 and center in the origin); applying the transformation you get this region:

The Jacobian determinant of this transformation is the following:

The dx dy dz differentials therefore are transformed to ρ² sin(φ) dρ dθ dφ.

Finally you obtain the final integration formula:

It's better to use this method in case of spherical domains and in case of functions that can be easily simplified, by the first fundamental relation of trigonometry, extended in R³ (please see example 4-b); in other cases it can be better to use cylindrical coordinates (please see example 4-c).

Note that the extra ρ² and sinφ come from the Jacobian.

Note that in the following examples the roles of φ and θ have been reversed.

Example (4-b):

D is the same region of the 4-a example and is the function to integrate.

Its transformation is very easy:

while we know the intervals of the transformed region T from D:

Let's therefore apply the integration's formula:

and, developing, we get

Example (4-c):

The domain D is the ball with center in the origin and radius 3a () and is the function to integrate.

Looking at the domain, it seems convenient to adopt the passage in spherical coordinates, in fact, the intervals of the variables that delimit the new T region are obviously:

However, applying the transformation, we get

.

Applying the formula for integration we would obtain:

which is very hard to solve. This problem will be solved by using the passage in cylindrical coordinates. The new T intervals are

the z interval has been obtained by dividing the ball in two hemispheres simply by solving the inequality from the formula of D (and then directly transforming x² + y² in ρ²). The new function is simply ρ². Applying the integration formula

.

Then we get

Now let's apply the transformation

(the new intervals become ). We get

because , we get

after inverting the integration's bounds and multiplying the terms between parenthesis, it is possible to decompose the integral in two parts that can be directly solved:

Thanks to the passage in cylindrical coordinates it was possible to reduce the triple integral to an easier one-variable integral.

See also the differential volume entry in nabla in cylindrical and spherical coordinates.

Examples

Double integral

Let us assume that we wish to integrate a multivariable function f over a region A.

From this we formulate the double integral

The inner integral is performed first, integrating with respect to x and taking y as a constant, as it is not the variable of integration. The result of this integral, which is a function depending only on y, is then integrated with respect to y.

We then integrate the result with respect to y.

Volumes

The volume of the parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just like a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts (i.e., parallelepiped, parallelogram, cube, and square). In this context of affine geometry, in which angles of sides 4 × 6 × 5 may be obtained in two ways:

• By calculating the double integral of the function f(x, y) = 5 over the region D in the xy-plane which is the base of the parallelepiped.

• By calculating the triple integral of the constant function 1 over the parallelepiped itself

Computing a volume

Using the methods previously described, it is possible to calculate the volumes of some common solids.

• Cylinder A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since: The volume of a cylinder with height h and circular base of radius R can be calculated by integrating the constant function h over the circular base, using polar coordinates.

This is in agreement with the formula

.

• Sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The: The volume of a sphere with radius R can be calculated by integrating the constant function 1 over the sphere, using spherical coordinates.

• Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids. The tetrahedron is the only convex polyhedron that has four faces (triangular pyramid A pyramid is a structure where the outer surfaces are triangular and converge at a point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three triangular surfaces (at least four faces including the base). The square pyramid, with square base and four triangular outer surfaces, is a or 3-simplex In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron. A single point may be): The volume of a tetrahedron with its apex at the origin and edges of length l along the x, y and z axes can be calculated by integrating the constant function 1 over the tetrahedron.

This is in agreement with the formula

.

Example of an improper domain.

Multiple improper integral

In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the double improper integral or the triple improper integral.

Multiple integrals and iterated integrals

See also: Order of integration (calculus) In calculus, interchange of the order of integration is a methodology that transforms iterated integrals of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot

Fubini's theorem In mathematical analysis, Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to change the order of integration states that if

that is, if the integral is absolutely convergent, then the multiple integral will give the same result as the iterated integral,

In particular this will occur if |f(x,y)| is a bounded function In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a number M>0 such that and A and B are bounded sets In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely a set which is not bounded is called unbounded.

If the integral is not absolutely convergent, care is needed not to confuse the concepts of multiple integral and iterated integral, especially since the same notation is often used for either concept. The notation

means, in some cases, an iterated integral rather than a true double integral. In an iterated integral, the outer integral

is the integral with respect to x of the following function of x:

A double integral, on the other hand, is defined with respect to area in the xy-plane. If the double integral exists, then it is equal to each of the two iterated integrals (either "dy dx" or "dx dy") and one often computes it by computing either of the iterated integrals. But sometimes the two iterated integrals exist when the double integral does not, and in some such cases the two iterated integrals are different numbers, i.e., one has

This is an instance of rearrangement of a conditionally convergent integral.

The notation

may be used if one wishes to be emphatic about intending a double integral rather than an iterated integral.

Some practical applications

These integrals are used in many applications in physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves. The examples below also show some variations in the notation.

In mechanics Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. The discipline has its roots in several ancient civilizations (see History of classical mechanics and Timeline of classical mechanics). During the early modern, the moment of inertia Moment of inertia, also called mass moment of inertia, rotational inertia, or the angular mass, is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass, the inertia of a rigid rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass is calculated as the volume integral (triple integral) of the density The density of a material is defined as its mass per unit volume. The symbol of density is ρ . In some countries (for instance, in the United States), density is also defined as its weight per unit volume weighed with the square of the distance from the axis:

The gravitational potential In classical mechanics, the gravitational potential at a location represents the work per unit mass as an object moves to that location from a reference location. It is analogous to the electric potential with mass playing the role of charge. By convention, the gravitational potential is defined as zero infinitely far away from any mass. As a associated with a mass distribution given by a mass measure In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a dm on three-dimensional Euclidean space In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity R³ is

If there is a continuous function ρ(x) representing the density of the distribution at x, so that dm(x) = ρ(x)d³x, where d³x is the Euclidean volume element, then the gravitational potential is

In electromagnetism Electromagnetism is one of the four fundamental interactions of nature, along with strong interaction, weak interaction and gravitation. It is the force that causes the interaction between electrically charged particles; the areas in which this happens are called electromagnetic fields, also known as B fields in physics classes, Maxwell's equations Maxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave. Individually, the equations are known as Gauss's law, Gauss's law for magnetism, Faraday's law of can be written using multiple integrals to calculate the total magnetic and electric fields. In the following example, the electric field In physics, an electric field is a property that describes the space that surrounds electrically charged particles or that which is in the presence of a time-varying magnetic field. This electric field exerts a force on other electrically charged objects. The concept of an electric field was introduced by Michael Faraday produced by a distribution of charges Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields. The interaction between a moving charge and an electromagnetic field is the source of the electromagnetic force, which is one of the given by the volume charge density The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume respectively. It is measured in coulombs per metre , square metre (C/m²), or cubic metre (C/m³), respectively. Since there are positive as well as negative charges, the charge density can take on negative values. Like any density it can is obtained by a triple integral of a vector function:

This can also be written as an integral with respect to a signed measure In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values representing the charge distribution.

See also

• Main analysis Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and theorems that relate multiple integrals:
  • Divergence theorem In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem (Mikhail Vasilievich Ostrogradsky), or Gauss–Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface
  • Stokes' theorem In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which generalizes several theorems from vector calculus. William Thomson first discovered the result and communicated it to George Stokes in July 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to
  • Green's theorem In physics and mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is the two-dimensional special case of the more general Stokes' theorem, and is named after British mathematician George Green

References

• Robert A. Adams - Calculus: A Complete Course (5th Edition) ISBN 0201791315.
• R.K.Jain and S.R.K Iyengar- Advanced Engineering Mathematics (Third edition) 2009, Narosa Publishing House ISBN 9788173197307

External links

• Weisstein, Eric W. Eric W. Weisstein is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He currently works for Wolfram Research, Inc, "Multiple Integral" from MathWorld.
• L.D. Kudryavtsev (2001), "Multiple integral", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/M/m065370.htm
• Mathematical Assistant on Web online evaluation of double integrals in Cartesian coordinates and polar coordinates (includes intermediate steps in the solution, powered by Maxima (software))

Categories: Integral calculus | Multivariable calculus

 

 

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