This document contains a group project submission on multi-variable calculus topics to Sir Asad Ijaz. It lists the group members and covers line integrals, the fundamental theorem of line integrals, exact differential forms, and Green's theorem in the plane. Green's theorem relates the line integral around a closed curve to a double integral over the interior region and has two forms - the flux-divergence form and the circulation-curl form. The document provides definitions, examples, and exercises for understanding these concepts.

2. 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

3.  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

4. 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

5.  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

6. Example:

7.  Any expression M(x,y,z)dx + N(x,y,z)dy + P(x,y,z)dz
is a differential form. A differential form is exact on a domain D
in space if
for some scalar function ƒ throughout D.
Exact Differential Form

8. Example

9. CONT….

10. Exercise Questions

11. CONT….

12. Green’s Theorem In The
Plane

13. Definition:
The divergence of a vector field F = Mi + Nj at the point
(x, y) is
Example:
Theorem 1:Divergence(flux Density)

14. 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:

15. 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)

16. 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)

17. Example Of Theorem 3 And 4

18. CONT….

19. EXAMPLE:
Using Green’s Theorem To Evaluate Line
Integrals

20. Proof Of Green’s Theorem
Statement:

21. CONT….

22. CONT….

23. CONT….

24. Exercise Question:

25. CONT…

26. CONT…

27. CONT…

28. 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:

29. 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

30. 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

31.  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…