Moin Ul Haq Babar Ali Naveed Ramzan

The process, which is the reverse of differentiation, is called integration. The process of integration is sometimes referred to as finding the anti-derivative of the specified function.

The function to be integrated is called the integrand. When a function has been

integrated the result is referred to as the integral.

Introduction In calculus, the integral of a function of more

than one variable is multiple integral. As the integral of a function of one variable over an interval results in an area, the double integral of a function of two variables calculated over a region results in a volume. Functions of three variables have triple integrals, and so on. Like the single integral, such constructions are useful in calculating the net change in a function that results from changes in its input values.

Multiple integration of a function in n variables f(x1, x2, ..., xn) 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:

Integrals of a function of two variables over a region in ℝ2 are called double integrals.

The double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three dimensional Cartesian plane where z = ƒ(x, y)) and the plane which contains its domain.

We have   The inner integral is

Note that we treat y as a constant as we integrate with respect to x. The outer integral is

 

In R2 if the domain has a circular "symmetry" and the function has some "particular" characteristics you can apply the transformation to polar coordinates

The fundamental relation to make the transformation is the following

D is the portion of the region between the circles of radius 2 and radius 5 centered at the origin that lies in the first quadrant.

 First let’s get D in terms of polar coordinates.  The circle of radius 2 is given by r=2 and the circle of radius 5 is given by r=5.  We want the region between them so we will have the following inequality for r.

Also, since we only want the portion that is in the first quadrant we get the following range of  ’s.

Now that we’ve got these we can do the integral.

Don’t forget to do the conversions and to add in the extra r.  Now, let’s simplify and make use of the double angle formula for sine to make the integral a little easier.

Integrals of a function of three variables over a region in ℝ3 are called triple integrals. The 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 region between the surface and the plane

Example:

Solution: Just to make the point that order doesn’t matter let’s

use a different order from that listed above.  We’ll do the integral in the following order.

In R3 the integration on domains with a circular base can be made by the passage in cylindrical coordinates; the transformation of the function is made by the following relation:

So it becomes:

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; the function is transformed by this relation:

A line integral is an integral where the function to be integrated is evaluated along a curve. The value of the line integral is the sum of values of the field at all points on the curve.

Evaluate  where C is the curve shown below.

  Solution So, first we need to parameterize each of the curves.

Now let’s do the line integral over each of these curves.                                                                                                                                               

Finally, the line integral that we were asked to compute is

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. Let C be a positively oriented, piecewise smooth, simple closed curve in the plane and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then

For positive orientation, an arrow pointing in the counterclockwise direction may be drawn in the small circle in the integral symbol.

The outward flux of a field F = M i + N j across a simple closed curve C equals the double integral of div F over the region R enclosed by C.

Note: The divergence of a fluid's velocity field measures the rate

at which fluid is being piped into or out of the region at any given point. The curl measures the fluid's rate of rotation at each point.

Stokes' theorem include the concept of curl and it simplifies and generalizes several theorems from vector calculus. It says that the circulation of a vector field around the boundary of an oriented surface in space in the direction counter clockwise with respect to the surface’s unit normal vector field n equals the integral of the normal component of the curl of the field over the surface.

∮ f.dr=∬∇×F.n d∑ This is the mathematical formula of calculating stokes

theorem.

Divergence theorem is similar to stokes theorem but main difference is that it contain dot product instead of cross product of nebla sign and function and it include triple integration.

the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region

Example: Use the divergence theorem to evaluate  where   and the surface consists of the three surfaces,  on the top, ,  on the sides and  on the bottom. SolutionLet’s start this off with a sketch of the surface. 

The region E for the triple integral is then the region enclosed by these surfaces.  Note that cylindrical coordinates would be a perfect coordinate system for this region.  If we do that here are the limits for the ranges.                                                                                           

We’ll also need the divergence of the vector field so let’s get that

The integral is then

When talking about application of multiple integral it has lot of application also in telecomm but here we discussed its some application in telecomm engineering.

The main thing is in electromagnetic is Maxwell equation which is based on this stokes theorem and divergence theorem. When we calculate stoke theorem then definitely double integrals is used.

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.

In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields.

Gauss's law electric

0 Gauss's law in magnetism

Faraday's law

Ampere-Maxwell lawI

oS

S

B

Eo o o

qd

ε

d

dd

dtd

d μ ε μdt

E A

B A

E s

B s

Gauss’s law (electrical): The total electric flux

through any closed surface equals the net charge inside that surface divided by o

This relates an electric field to the charge distribution that creates it

Gauss’s law (magnetism): The total magnetic flux

through any closed surface is zero

This says the number of field lines that enter a closed volume must equal the number that leave that volume

This implies the magnetic field lines cannot begin or end at any point

oS

qd

ε E A

0S

d B A

Faraday’s law of Induction: This describes the creation of an electric

field by a changing magnetic flux The law states that the emf, which is the

line integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path

One consequence is the current induced in a conducting loop placed in a time-varying B

The Ampere-Maxwell law is a generalization of Ampere’s law

It describes the creation of a magnetic field by an electric field and electric currents

The line integral of the magnetic field around any closed path is the given sum

Bdd

dt

E s

I Eo o o

dd μ ε μ

dt

B s

Picture a shows first half cycle. When current reverses in picture b, the fields reverse. See the first disturbance moving outward. These are the electromagnetic waves.

Notice that the electric and magnetic fields are at right angles to one another! They are also perpendicular to the direction of motion of the wave.

B

C

dE d

dt

Eo oC

dB d

dt

Applying Ampere to radiation

Helmholtz equation is also a optics equation whose solution is an double integral

 

It has application in electromagnetic mode which is depend upon width of the fiber optics so here also we use multiple integrals because this modes comes from Maxwell equation and to solve these modes we need to evaluate multiple integrals.

Modes are the possible solutions of the Helmholtz equation for waves, which is obtained by combining Maxwell's equations and the boundary conditions. These modes define the way the wave travels through space, i.e. how the wave is distributed in space

Its one application is in quantum well laser whose wavelength is small as compare to ordinary laser.

Multiple integration is also important in Karhunen-Loève theorem. It is used here for expansion of this theorem. This is probability/stochastic theorem and probability is used in digital communication. Specially it is applied in receiver’s detection portion.

Quantum Well Laser