1. MAT
Integration
Integration

2. 1. Line Integrals
The line integral of the scalar field Ф over the curve c is denoted by
Note :
When the scalar field Ф is identically 1
gives the length of the curve.
E.g.
Evaluate the line integrals of
The scalar field
over the following lines.
(2, 1, 0 )
(2, 1, 2 )

3. Line integral of a vector field
Definition
Line integral of a vector field
We denote the integral of the vector field A over the curve c by
E.g.
Evaluate the integral of the unit tangent drawn to the unit circle.

4. Flux – i%djh
Suppose c is a simple closed curve in the space. The outward flux of the vector field A over c is denoted by
, and defined as
Here, s-arc length.
normal
tangent
parallel to
parallel to
Outward unit normal is

5. Let
c ir, ixjD; jl%h msrsjid msg;g A ys i%djh

6. Circulation – ixirKh Circulation is defined by
c ir, ixjD; jl%h msrsjid msg;g A ys ixirKh

7. Green’s Theorem in 2-dimension - oaúudkfha .%skaf.a m%fushh
Suppose c is a simple closed curve in a plane, and S is the surface enclosed by c. The outward flux of a vector field A over c is same as the surface integral of div A over S.
c ir, ixjD; jl%h msrsjid msg;g A ffoYsl lafIa;%fhys i%djh iy S msrsjid wmid A ys mDIaG wkql,h iudk fjs .
since
div
So Green’s Theorem can be written as

8. Circulation Flux form of Green’s Theorem.
.%skaf.a m%fushfha i%dj wdldrh
Flux form of Green’s Theorem.
Circulation
.%skaf.a m%fushfha ixirK wdldrh
Circulation form of Green’s Theorem.

9. E.g. Solution : Evaluate . Here c ic the boundary of the triangle Let
Integral becomes

10. E.g. Verify the flux form of the Green’s Theorem, by taking and
Flux form (i%dj wdldrh) of the Green’s Theorem is

11. iu.ska
i;Hdmkh lsrSu
L.H.S.=
jD;a;h mrdï;slj

12. R.H.S.

13. With Polar cdts
jus mi yd ol=Kq mi iudk ksid .%skaf.a m%fushfha i%dj wdldrh i;Hdmkh fjs.