Submitted to: Sir Asad Ijaz

Department : BSCS “A”

GROUP MEMBERS:

Eisha (20-ARID-1678)

Amshal Ejaz (20-ARID-1669)

Laiba Irfan (20-ARID-1696)

Maryam Zainab (20-ARID-1697)

Fatima Bilal (20-ARID-1683)

Ayesha sadeeqa (20-ARID-1675)

Arooj khalid (20-ARID-1672)

Multi-variable Calculus

Line integral

Fundamental theorem on line integrals

Exact differential form

Green’s theorem in the plane

oDivergence(flux density)

oK-component of curl(circulation density)

oGreen theorem (flux divergence or normal form)

oGreen theorem(circulation curl and tangential form)

Using Green’s theorem to evaluate line integrals

Proof Of Green’s Theorem

Summary

Content

Definition:

In Calculus, a line integral is an integral in which the

function to be integrated is evaluated along a curve.

A line integral is also called the path integral or a curve

integral or a curvilinear integral.

Line Integral

Let F = Mi + Nj + Pk be a vector field whose components are

continuous throughout an open connected region D in space. Then

there exists a differentiable function ƒ such that

if and only if for all points A and B in D the value of is independent

of the path joining A to B in D.

If the integral is independent of the path from A to B, its value is

Fundamental Theorem Of Line Integrals

Theorem 2: K-component Of

Curl(Circulation Density)

Definition:

The k-component of the curl (circulation density) of a

vector field F = Mi + Nj at the point (x, y) is the scalar

Example:

Definition:

The outward flux of a field F = Mi + Nj across a simple

closed curve C equals the double integral of div F over

the region R enclosed by C.

Theorem 3: Green Theorem (Flux

Divergence Or Normal Form)

Definition:

The counterclockwise circulation of a field F = Mi + Nj

around a simple closed curve C in the plane equals the

double integral of (curl F). k over the region R enclosed

by C.

Theorem 4: Green Theorem(circulation

Curl And Tangential Form)

So guys here is a quick summary of all those topics that we have

discussed…

Line integral..

As discussed before an integral is the function to be integrated and

is evaluated along a curve. The fundamental theorem of line

integral is as but when we talk

about the integral is independent of path A and B the value will be

as and hopefully you are clear with

the example as my partner has told it very clearly.

Summary:

Green theorem in the plane sound interesting.

What do you think when you heard green

theorem in the plane?? The first question

that comes to us is why is it called

Green theorem.

Well lets talk about it is named after George

Green, who stated a similar result in an 1828 paper titled An Essay on

the Application of Mathematical Analysis to the Theories of

Electricity and Magnetism. In 1846, Augustin-Louis Cauchy published

a paper stating Green’s theorem as the penultimate sentence. This is in

fact the first printed version of Green’s theorem in the form appearing

in modern textbooks. Bernhard Riemann gave the first proof of

Green’s theorem in his doctoral dissertation on the theory of functions

of a complex variable.

Green Theorem In The Plane

Types Of Theorem

There are four basic theorem of green theorem

Divergence ( Flux density)

It is a vector field F= Mi + Nj

K- Component of Curl (circulation density)

In a vector field F = Mi + Nj

Green Theorem (Flux Divergence Or Normal Form)

Outward flux of a field F = Mi + Nj

Green Theorem(circulation Curl And Tangential Form)

The counterclockwise circulation of a field F = Mi + Nj around a

simple closed curve C in the plane equals the double integral of

(curl F). k over the region R enclosed by C.

So this was a quick overview/ summary of the following topics we

have discussed today.

CONT…