the yoga upanishads - t. r. sreenivasa ayyangar [english].pdf
Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of...
Transcript of Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of...
![Page 1: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/1.jpg)
1
1
Sreenivasa Institute of Technology and Management Studies
Engineering Mathematics-III
II B.TECH. I SEMESTER
Course Code: 18SAH211 Regulation:R18
![Page 2: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/2.jpg)
UNIT-I
NUMERICAL INTEGRATION
Motivation:
Most such integrals cannot be evaluated explicitly. Many others it is often faster to integrate them numerically rather than evaluating them exactly using a complicated antiderivative of f(x)
Example:
The solution of this integral equation with Matlab is 1/2*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2)*x)
we cannot find this solution analytically by techniques in calculus.
2
dxx )sin( 2
![Page 3: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/3.jpg)
3
3
![Page 4: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/4.jpg)
4
![Page 5: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/5.jpg)
5
Graphical Representation of
Integral
Integral = area
under the curve
Use of a grid to
approximate an integral
![Page 6: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/6.jpg)
6
Use of strips to
approximate an
integral
![Page 7: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/7.jpg)
7
Numerical Integration
Net force
against a
skyscraper
Cross-sectional area
and volume flowrate
in a river
Survey of land
area of an
irregular lot
![Page 8: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/8.jpg)
Methods of Numerical Integration
Trapezoidal Rule’s
1/3 Simpson’s method
3/8 Simpson’s method
Applied in two dimensional domain
8
![Page 9: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/9.jpg)
9
![Page 10: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/10.jpg)
10
![Page 11: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/11.jpg)
11
![Page 12: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/12.jpg)
12
Simpson’s 1/3- Rule is given by
Note: While applying the Simpson’s 1/3 rule, the number
of sub-intervals (n) should be taken as multiple of 2.
![Page 13: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/13.jpg)
13
13
Simpson’s 3/8- Rule is given by
Note: While applying the Simpson’s 3/8 rule, the number
of sub-intervals (n) should be taken as multiple of 3.
![Page 14: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/14.jpg)
14
![Page 15: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/15.jpg)
15
![Page 16: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/16.jpg)
16
![Page 17: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/17.jpg)
17
![Page 18: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/18.jpg)
18
![Page 19: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/19.jpg)
19
19
Numerical solution of
ordinary differential
equations
![Page 20: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/20.jpg)
20
Introduction
![Page 21: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/21.jpg)
21
Taylor’s Series Method
![Page 22: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/22.jpg)
22
![Page 23: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/23.jpg)
23
![Page 24: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/24.jpg)
24
Picard’s Method
![Page 25: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/25.jpg)
25
![Page 26: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/26.jpg)
26
![Page 27: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/27.jpg)
27
![Page 28: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/28.jpg)
28
![Page 29: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/29.jpg)
29
Euler’s Method
![Page 30: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/30.jpg)
30
![Page 31: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/31.jpg)
31
![Page 32: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/32.jpg)
32
Runge-Kutta Formula
![Page 33: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/33.jpg)
33
UNIT-II
Multiple Integrals
Double Integration
![Page 34: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/34.jpg)
34
Evaluation of Double Integration
![Page 35: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/35.jpg)
35
![Page 36: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/36.jpg)
36
![Page 37: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/37.jpg)
37
![Page 38: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/38.jpg)
38
![Page 39: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/39.jpg)
39
![Page 40: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/40.jpg)
40
Triple Integration
![Page 41: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/41.jpg)
41
![Page 42: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/42.jpg)
42
![Page 43: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/43.jpg)
43
UNIT-III
Partial Differential Equations
![Page 44: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/44.jpg)
44
![Page 45: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/45.jpg)
45
![Page 46: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/46.jpg)
46
![Page 47: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/47.jpg)
47
![Page 48: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/48.jpg)
48
![Page 49: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/49.jpg)
49
UNIT-IV
Vector Differentiation
Elementary Vector Analysis
Definition (Scalar and vector)
Vector is a directed quantity, one with both
magnitude and direction.
For instance acceleration, velocity, force
Scalar is a quantity that has magnitude but not
direction.
For instance mass, volume, distance
![Page 50: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/50.jpg)
50
We represent a vector as an arrow from the
origin O to a point A.
The length of the arrow is the magnitude of
the vector written as or .
O
A
or O
A
OAa
aOA
![Page 51: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/51.jpg)
51
Basic Vector System
Unit vectors , ,
•Perpendicular to each other
•In the positive directions
of the axes
•have magnitude (length) 1
![Page 52: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/52.jpg)
52
Magnitude of vectors
Let P = (x, y, z). Vector is defined by
with magnitude (length)
OP = = + +p x i y j z k
= [ ]x, y, z
OP = p
OP = = + +p x y z2 2 2
![Page 53: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/53.jpg)
53
Calculation of Vectors
Vector Equation
Two vectors are equal if and only if the
corresponding components are equals
332211
321321
, ,
Then
. and Let
babababa
kbjbibbkajaiaa
====
++=++=
![Page 54: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/54.jpg)
54
Addition and Subtraction of Vectors
Multiplication of Vectors by Scalars
kbajbaibaba )()()( 332211 ++=
kbjbibb )()()(
thenscalar, a is If
321
++=
![Page 55: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/55.jpg)
55
55
Example 1
Given 5 3 and 4 3 2 . Findp i j k q i j k= + - = - +
a p q) +
) b p q-
) 2 10 d q p-
c p) Magnitude of vector
![Page 56: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/56.jpg)
56
Vector Products
1 2 3 1 2 3~ ~ ~ ~ ~ ~~ ~
If and , a a i a j a k b b i b j b k= + + = + +
1 1 2 2 3 3~ ~a b a b a b a b = + +
1) Scalar Product (Dot product)
2) Vector Product (Cross product)
~ ~~
1 2 3~ ~
1 2 3
2 3 3 2 1 3 3 1 1 2 2 1~ ~~
i j k
a b a a a
b b b
a b a b i a b a b j a b a b k
=
= - - - + -
~~~~ and between angle theis ,cos||||.or bababa =
![Page 57: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/57.jpg)
57
The area of parallelogram
The volume of tetrahedrone
The volume of parallelepiped
a
ba bxA =
a b
c
321
321
321
6
1
ccc
bbb
aaa
=6
1=V a . b cx
a b
c321
321
321
ccc
bbb
aaa
==V a . b cx
![Page 58: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/58.jpg)
58
Differentiation of Two Vectors
If both and are vectors, then
)(~
uA )(~
uB
58
~
~~
~~~
~
~~
~~~
~~
~~
~
~
)()
..).()
)()
)()
Bdu
Ad
du
BdABA
du
dd
Bdu
Ad
du
BdABA
du
dc
du
Bd
du
AdBA
du
db
du
AdcAc
du
da
+=
+=
+=+
=
![Page 59: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/59.jpg)
Del Operator Or Nabla (Symbol )
• Operator is called vector differential operator,
defined as
59
.~~~
+
+
= k
zj
yi
x
![Page 60: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/60.jpg)
60
Grad (Gradient of Scalar Functions)
• If x,y,z is a scalar function of three
variables and is differentiable, the gradient
of is defined as
. grad~~~k
zj
yi
x
+
+
==
function vector a is *
functionscalar a is *
![Page 61: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/61.jpg)
61
Example 1
zxyyzx
xyzzx
zyxyz
zxyyzx
zxyyzx
222
232
223
2232
2232
23z
2y
2x
hence ,Given
(1,3,2).Pat grad determine , If
+=
+=
+=
+=
=+=
Solution
61
![Page 62: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/62.jpg)
62
.723284
.))2()3)(1(2)2)(3()1(3(
))2)(3)(1(2)2()1(())2()3()2)(3)(1(2(
have we(1,3,2),PAt
.)23(
)2()2(
zyx
Therefore,
~~~
~
222
~
232
~
223
~
222
~
232
~
223
~~~
kji
k
ji
kzxyyzx
jxyzzxizyxyz
kji
++=
++
+++=
=
++
+++=
+
+
=
![Page 63: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/63.jpg)
63
Exercise 2
(1,2,3).Ppoint at grad determine
, If 323
=
+=
zxyyzx
63
Solution
.110111126(1,2,3),PAt
Grad
z
y
x
then,Given
~~~
323
kji
zxyyzx
++==
==
=
=
=
+=
63
![Page 64: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/64.jpg)
64
Grad Properties
If A and B are two scalars, then
)()()()2
)()1
ABBAAB
BABA
+=
+=+
64
![Page 65: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/65.jpg)
65
Divergence of a Vector
..
).(
.
as defined
is of divergence the, If
~~
~~~~~~
~~
~~~~~
z
a
y
a
x
aAAdiv
kajaiakz
jy
ix
AAdiv
AkajaiaA
zyx
zyx
zyx
+
+
==
++
+
+
=
=
++=
65
![Page 66: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/66.jpg)
66
Example 1
.13
)3)(2(2)3)(1()2)(1(2
(1,2,3),point At
.22
.
(1,2,3).point at determine
, If
~
~~
~
~
2
~~
2
~
=
+-=
+-=
+
+
==
+-=
Adiv
yzxzxy
z
a
y
a
x
aAAdiv
Adiv
kyzjxyziyxA
zyx
Answer
![Page 67: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/67.jpg)
67
Exercise 2
.114
(3,2,1),point At
.
(3,2,1).point at determine
, If
~
~~
~
~
3
~
2
~
23
~
=
=
=
+
+
==
-+=
Adiv
z
a
y
a
x
aAAdiv
Adiv
kyzjzxyiyxA
zyxAnswer
![Page 68: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/68.jpg)
68
Remarks
.called is vector ,0 If
function.scalar a is but function, vector a is
~~
~~
vectorsolenoidAAdiv
AdivA
=
![Page 69: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/69.jpg)
69
Curl of a Vector
.
)(
by defined is of curl the,If
~~~
~~
~~~~~~
~~
~~~~~
zyx
zyx
zyx
aaa
zyx
kji
AAcurl
kajaiakz
jy
ix
AAcurl
AkajaiaA
==
++
+
+
=
=
++=
![Page 70: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/70.jpg)
70
Example 1
.)2,3,1(at determine
,)()( If
~
~
2
~
22
~
224
~
-
-++-=
Acurl
kyzxjyxizxyA
Solution
.)42()22(
)()(
)()(
)()(
~
3
~
2
~
2
~
22422
~
2242
~
222
222224
~~~
~~
kyxjzxxyzizx
kzxyy
yxx
jzxyz
yzxx
iyxz
yzxy
yzxyxzxy
zyx
kji
AAcurl
-++---=
-
-+
+
-
--
-
+
--
=
-+-
==
70
![Page 71: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/71.jpg)
71
Exercise 2
.10682
))3(4)1(2(
))2()1(2)2)(3)(1(2()2()1(
(1,3,-2),At
~~~
~
3
~
2
~
2
~
kji
k
jiAcurl
--=
-+
-+-----=
.)3,2,1(point at determine
,)()( If
~
~
22
~
22
~
223
~
Acurl
kyzxjzxizyxyA -++-=
![Page 72: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/72.jpg)
72
Answer
.261215 (1,2,3),At
.)232(
)22()2(
~~~~
~
22
~
22
~
22
~
kjiAcurl
kyzxyx
jzyxyzizzxAcurl
++-=
+-+
+----=
Remark
function. vector a also is
andfunction vector a is
~
~
Acurl
A
![Page 73: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/73.jpg)
73
UNIT-V
Vector Integration
Polar Coordinate for Plane (r, θ)
ddrrdS
ry
rx
=
=
=
sin
cos
x
ds
y
d
![Page 74: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/74.jpg)
74
Polar Coordinate for Cylinder (, , z)
dzdddV
dzddS
zz
y
x
=
=
=
=
=
sin
cos
x
y
z
dv
z
ds
![Page 75: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/75.jpg)
75
Polar Coordinate for Sphere (r, ,
dddrrdV
ddrdS
rz
ry
rx
sin
sin
cos
sinsin
cossin
2
2
=
=
=
=
=
y
x
r
z
![Page 76: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/76.jpg)
76
Example 1 (Volume Integral)
.9 and
4,0by bounded space a is and
22 where Calculate
22
~~~~~
=+
==
++=
yx
zzV
kyjziFdVFV
x
z
y
4 -
3 3
![Page 77: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/77.jpg)
77
Solution
Since it is about a cylinder, it is easier if we use
cylindrical polar coordinates, where
.40,20,30 where
,,sin,cos
====
z
dzdddVzzyx
![Page 78: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/78.jpg)
78
Line Integral
Ordinary integral f (x) dx, we integrate along
the x-axis. But for line integral, the integration
is along a curve.
f (s) ds = f (x, y, z) ds
A
O
B
~~rdr+
~r
![Page 79: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/79.jpg)
79
Scalar Field, V Integral
If there exists a scalar field V along a curve C,
then the line integral of V along C is defined by
.where~~~~
~
kdzjdyidxrd
rdVc
++=
![Page 80: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/80.jpg)
80
Example 1
(3,2,1).B to(0,0,0)A from
along findthen
,,2,3
by given is curve a and z If
~
32
2
==
===
=
CrdV
uzuyux
CxyV
c
![Page 81: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/81.jpg)
81
Solution
.1
,1,22,33 (3,2,1),BAt
.0
,0,02,03 (0,0,0),AAt
.343
And,
.12)()2)(3(
zGiven
32
32
~
2
~~
~~~~
8322
2
=
====
=
====
++=
++=
==
=
u
uuu
u
uuu
kduujduuidu
kdzjdyidxrd
uuuu
xyV
![Page 82: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/82.jpg)
82
.11
36
5
244
11
36
5
244
364836
)343)(12(
~~~
~
1
0
11
~
1
0
10
~
1
0
9
1
0 ~
101
0~
91
0~
8
1
0 ~
2
~~
8
~
kji
kujuiu
kduujduuiduu
kduujuduiduurdVu
u
B
A
++=
+
+=
++=
++=
=
=
![Page 83: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/83.jpg)
83
83
Exercise 2
.11
768144
5
384
(4,3,2).B to(0,0,0)A from
curve thealong calculate
,2,3,4
by given is curve theand If
~~~~
~
23
22
kjirdV
C rdV
uzuyux
CyzxV
B
A
c
++=
==
===
=
Answer
![Page 84: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/84.jpg)
84
Vector Field, Integral
Let a vector field
and
The scalar product is written as
.
)).((. ~~~~~~~~
dzFdyFdxF
kdzjdyidxkFjFiFrdF
zyx
zyx
++=
++++=
~F
~~~~kFjFiFF zyx ++=
.~~~~kdzjdyidxrd ++=
~~. rdF
![Page 85: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/85.jpg)
85
. .
bygiven is Bpoint another A topoint a from
curve thealong of integral line then the
, curve thealong is field vector a If
~~
~
~
++=c
zc
yc
xc
dzFdyFdxFrdF
CF
CF
![Page 86: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/86.jpg)
86
Example 1
.y2y
if,2,4curve thealong
(4,2,1)B to(0,0,0)A from .Calculate
~~~
2
~
32
~~
kzjxzixF
tztytx
rdFc
-+=
===
==
![Page 87: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/87.jpg)
87
Solution
.344
And
.4432
)()2(2)()4()2()4(
2yGiven
~
2
~~
~~~~
~
5
~
4
~
4
~
32
~
3
~
22
~~~
2
~
kdttjdttidt
kdzjdyidxrd
ktjtit
kttjttitt
kyzjxzixF
++=
++=
-+=
-+=
-+=
![Page 88: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/88.jpg)
88
.)1216128(
1216128
)3)(4()4)(4()4)(32(
)344)(4432(.
Then
754
754
2544
~
2
~~~
5
~
4
~
4
~~
dtttt
dttdttdtt
dttttdttdtt
kdttjdttidtktjtitrdF
-+=
-+=
-++=
++-+=
.1
,1,22,44 (4,2,1),Bat and,
.0
,0,02,04 (0,0,0),AAt
32
32
=
====
=
====
t
ttt
t
ttt
![Page 89: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/89.jpg)
89
.30
2326
2
3
3
8
5
128
2
3
3
8
5
128
)1216128(.
1
0
865
1
0
754
~~
=
-+=
-+=
-+= =
=
ttt
dttttrdFt
t
B
A
![Page 90: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/90.jpg)
90
Surface Integral
Scalar Field, V Integral
If scalar field V exists on surface S, surface
integral V of S is defined by
=S S
dSnVSVd~~
where
S
Sn
=
~
![Page 91: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/91.jpg)
91
Example 1
Scalar field V = x y z defeated on the surface
S : x2 + y2 = 4 between z = 0 and z = 3 in the
first octant.
Evaluate SSVd~
Solution
Given S : x2 + y2 = 4 , so grad S is
~~~~~22 jyixk
z
Sj
y
Si
x
SS +=
+
+
=
![Page 92: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/92.jpg)
92
Also,
4422)2()2( 2222 ==+=+= yxyxS
Therefore,
)(2
1
4
22
~~
~~
~jyix
jyix
S
Sn +=
+
=
=
Then,
+
=
S SdSjyixxyzdSnV )(
2
1
~~~
+= dSjzxyiyzx )(2
1
~
2
~
2
![Page 93: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/93.jpg)
93
Surface S : x2 + y2 = 4 is bounded by z = 0 and z = 3
that is a cylinder with z-axis as a cylinder axes and
radius,
So, we will use polar coordinate of cylinder to find
the surface integral.
.24 ==
x
z
y
2
2
3
O
![Page 94: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/94.jpg)
94
Polar Coordinate for Cylinder
cos 2cos
sin 2sin
ρ
x
y
z z
dS d dz
= =
= =
=
=
where 2
0
30 z(1st octant) and
![Page 95: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/95.jpg)
95
Using polar coordinate of cylinder,
cossin8)()sin2)(cos2(
sincos8)sin2()cos2(
222
222
zzzxy
zzyzx
==
==
From
=+=S
SS
SVddSjzxyiyzxdSnV~~
2
~
2
~)(
2
1
![Page 96: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/96.jpg)
96
96
= =+=
Sz
dzdjzizSVd 2
0
3
0 ~
2
~
2
~)2)(cossin8sincos8(
2
1
3
2 2 2 22
0 ~ ~ 0
2 22
0 ~ ~
1 18 cos sin sin cos
2 2
9 98 cos sin sin cos
2 2
z i z j d
i j d
= +
= +
2 22
0 ~ ~
3 3 2
~ ~0
~ ~
98 cos sin sin cos
2
cos sin sin cos36
3( sin ) 3(cos )
12( )
i j d
i j
i j
= +
= +
-
= +
Therefore,
![Page 97: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/97.jpg)
97
Green’s Theorem
If c is a closed curve in counter-clockwise on
plane-xy, and given two functions P(x, y) and
Q(x, y),
where S is the area of c.
+=
-
cSdyQdxPdydx
y
P
x
Q)(
![Page 98: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/98.jpg)
98
Example 1
2 2
2 2
Prove Green's Theorem for
[( ) ( 2 ) ]
which has been evaluated by boundary that defined as
0, 0 4 in the first quarter.
cx y dx x y dy
x y and x y
+ + +
= = + =
y
2
x 2
C3
C2
C1 O
x2 + y2 = 22
Solution
![Page 99: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/99.jpg)
99
1 1
2 2
2 2
1 2 3
1
2 2
22
0
2
3
0
Given [( ) ( 2 ) ] where
and 2 . We defined curve
as , .
i) For : 0, 0 0 2
( ) ( ) ( 2 )
1 8.
3 3
c
c c
x y dx x y dy
P x y Q x y c
c c and c
c y dy and x
Pdx Qdy x y dx x y dy
x dx
x
+ + +
= + = +
= =
+ = + + +
=
= =
![Page 100: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/100.jpg)
100
2 2
2ii) For : 4 , in the first quarter from (2,0) to (0,2).
This curve actually a part of a circle.
Therefore, it's more easier if we integrate by using polar
coordinate of plane,
2cos , 2sin , 0
c x y
x y
+ =
= =2
2sin , 2cos .dx d dy d
= - =
![Page 101: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/101.jpg)
101
.448
sin42sin2cos8
)cossin82cos22sin8(
)cossin8cos4sin8(
)]cos2))(sin2(2cos2((
)sin2)()sin2()cos2(([
)2()()(
2
2
2
2
22
0
2
0
0
2
22
0
22
-=++-=
+++=
+++-=
++-=
++
-+=
+++=+
d
d
d
d
dyyxdxyxQdyPdxcc
![Page 102: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/102.jpg)
102
3 3
3
2 2
0
2
02
2
iii) : 0, 0, 0 2
( ) ( ) ( 2 )
2
4.
8 16( ) ( 4) 4 .
3 3
c c
c
For c x dx y
Pdx Qdy x y dx x y dy
y dy
y
Pdx Qdy
= =
+ = + + +
=
=
= -
+ = + - - = -
![Page 103: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/103.jpg)
103
b) Now, we evaluate
where 1 2 .
Again,because this is a part of the circle,
we shall integrate by using polar coordinate of plane,
cos , sin
where
S
Q Pdxdy
x y
Q Pand y
x y
x r y r
-
= =
= =
0 r 2, 0 .2
and dxdy dS r dr d
= =
![Page 104: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/104.jpg)
104
.3
16
cos3
162
sin3
162
sin3
2
2
1
)sin21(
)21(
2
2
2
2
0
0
2
00
32
0
2
0
-=
+=
-=
-=
-=
-=
-
=
=
= =
d
drr
ddrrr
dydxydydxy
P
x
Q
r
SS
![Page 105: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/105.jpg)
105
Therefore,
( )
16.
3
Green's Theorem has been proved.
C S
Q PPdx Qdy dx dy
x y
LHS RHS
+ = -
= -
=
![Page 106: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/106.jpg)
106
Divergence Theorem (Gauss’ Theorem)
If S is a closed surface including region V
in vector field
..~~~ =
SVSdFdVFdiv
~F
~
yx zff f
div Fx y z
= + +
![Page 107: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/107.jpg)
107
Example 1
2
~ ~ ~~
2 2
Prove Gauss' Theorem for vector field,
2 in the region bounded by
planes 0, 4, 0, 0 4
in the first octant.
F x i j z k
z z x y and x y
= + +
= = = = + =
![Page 108: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/108.jpg)
108
Solution
x
z
y
2
2
4
O
S3
S4
S2
S1
S5
![Page 109: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/109.jpg)
109
1
2
3
4
2 2
5
~
~ ~
For this problem, the region of integration is bounded
by 5 planes :
: 0
: 4
: 0
: 0
: 4
To prove Gauss' Theorem, we evaluate both
. ,
The answer should be the same.
V
S
S z
S z
S y
S x
S x y
div F dV
and F d S
=
=
=
=
+ =
![Page 110: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/110.jpg)
110
2
~ ~ ~ ~~
2
~
~
1) We evaluate . Given 2 .
So,
( ) (2) ( )
1 2 .
Also, (1 2 ) .
The region is a part of the cylinder. So, we integrate by using
polar c
V
V V
div F dV F x i j z k
div F x zx y z
z
div F dV z dV
= + +
= + +
= +
= +
oordinate of cylinder ,
; sin ;
where 0 2, 0 , 0 4.2
x = cos y z z
dV d d dz
z
= =
=
![Page 111: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/111.jpg)
111
2
2
2
2
2
2
2 4
0 0 0
22 4
00 0
2
0 0
2 2
00
0
0
~
Therefore,
(1 2 ) (1 2 )
[ ]
(20 )
[10 ]
(40)
40
20 .
20 .
V z
V
z dV z dzd d
z z d d
d d
d
d
div F dV
= = =
= =
= =
=
=
+ = +
= +
=
=
=
=
=
=
![Page 112: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/112.jpg)
112
1
~ ~~ ~
1~ ~
~ ~ ~~
~ ~ ~ ~~
~ ~
2) Now, we evaluate . . .
i) : 0, ,
2 0
. ( 2 ).( ) 0
. 0.
S S
S
F d S F n dS
S z n k dS rdrd
F x i j k
F n x i j k
F n dS
=
= = - =
= + +
= + - =
=
![Page 113: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/113.jpg)
113
2
2
2~ ~
2
~ ~ ~ ~ ~~ ~
~ ~ ~ ~ ~~
2
2
0 0~ ~
ii) : 4, ,
2 (4) 2 16
. ( 2 16 ).( ) 16.
Therefore for , 0 r 2, 02
. 16
16 .
S r
S z n k dS rdrd
F x i j k x i j k
F n x i j k k
S
F n dS rdrd
= =
= = =
= + + = + +
= + + =
=
=
=
![Page 114: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/114.jpg)
114
3
3~ ~
2
~ ~ ~~
2
~ ~ ~ ~~ ~
3
2 4
0 0~ ~
iii) : 0, ,
2
. ( 2 ).( )
2.
Therefore for S , 0 2, 0 4
. ( 2)
16.
S x z
S y n j dS dxdz
F x i j z k
F n x i j z k j
x z
F n dS dzdx= =
= = - =
= + +
= + + -
= -
= -
=
= -
![Page 115: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/115.jpg)
115
4
4~ ~
2 2
~ ~ ~ ~~ ~
2
~ ~ ~ ~~
~ ~
iv) : 0, ,
0 2 2
. (2 ).( ) 0.
. 0.S
S x n i dS dydz
F i j z k j z k
F n j z k i
F n dS
= = - =
= + + = +
= + - =
=
![Page 116: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/116.jpg)
116
2 2
5
5 5~ ~
~5 ~
~5
~ ~
5
v) : 4,
2 2 4
2 2
4
1( ).
2
By using polar coordinate of cylinder :
cos , sin ,
where for :
2, 0 , 0 4, 22
S x y dS d dz
S x i y j and S
x i y jS
nS
x i y j
x y z z
S
z dS d dz
+ = =
= + =
+
= =
= +
= = =
= =
![Page 117: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/117.jpg)
117
.416
)2)(sin)(cos2(.
).sin(cos2
2.kerana ;sin2cos2
)sin()cos(2
1
2
1
2
1
2
1).2(.
5
2
0
4
0
2
~~
2
2
2
2
~~~
2
~~~~
+=
=
+=
+=
=+=
+=
+=
+++=
= =
S zdzddSnF
yx
jyixkzjixnF
![Page 118: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/118.jpg)
118
1 2 3 4 5~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~
~ ~
Finally,
. . . . . .
0 16 16 0 16 4
20 .
. 20 .
Gauss' Theorem has been proved.
S S S S S S
S
F d S F d S F d S F d S F d S F d S
F d S
LHS RHS
= + + + +
= + - + + +
=
=
=
![Page 119: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/119.jpg)
119
Stokes’ Theorem
If is a vector field on an open surface S and
boundary of surface S is a closed curve c,
therefore
=S c
rdFSdFcurl~~~~
~F
~ ~~
~ ~
x y z
i j k
curl F Fx y z
f f f
= =
![Page 120: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/120.jpg)
120
Example 1
Surface S is the combination of
2 2
~ ~ ~~
i) part of the cylinder 9 0
and 4 0.
ii) half of the circle with radius 3 at 4, and
iii) 0
, prove Stokes' Theorem
for this case.
a x y between z
z for y
a z
plane y
If F z i xy j xz k
+ = =
=
=
=
= + +
![Page 121: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/121.jpg)
121
Solution
2 2
1
2
3
We can divide surface S as
S : x y 9 0 z 4 y 0
S : z 4, half of the circle with radius 3
S : y 0
for and+ =
=
=
z
y x
3
4
O
S3
C2
S2
C1
S1
3
![Page 122: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/122.jpg)
122
We can also mark the pieces of curve C as
C1 : Perimeter of a half circle with radius 3.
C2 : Straight line from (-3,0,0) to (3,0,0).
Let say, we choose to evaluate first.
Given
~ ~Scurl F d S
~~~~kxzjxyizF ++=
![Page 123: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/123.jpg)
123
So,
~~
~
~~
~~~
~
)1(
)()(
)()()()(
kyjz
kzy
xyx
jxzx
zz
ixyz
xzy
xzxyzzyx
kji
Fcurl
+-=
-
+
-
+
-
=
=
![Page 124: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/124.jpg)
124
By integrating each part of the surface,
2 2
1
1~ ~
2 2
1
2 2
( ) : 9,
2 2
(2 ) (2 )
2 6
i For surface S x y
S x i y j
and S x y
x y
+ =
= +
= +
= + =
![Page 125: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/125.jpg)
125
)(3
1
6
22
~~
~~
1
1
~jyix
jyix
S
Sn +=
+
=
=
and
).1(3
1
3
1
3
1)1(
~~~~~~
zy
jyixkyjznFcurl
-=
+
+-=
Then ,
![Page 126: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/126.jpg)
126
By using polar coordinate of cylinder ( because
is a part of the cylinder), 9: 22
1 =+ yxS
cos , sin ,
3, 0 0 4.
x y z z
dS d dz
where
dan z
= = =
=
=
![Page 127: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/127.jpg)
127
Therefore,
~ ~
1(1 )
3
1sin 1
3
sin (1 ) ; 3
curl F n y z
z
z because
= -
= -
= - =
Also, dzddS 3=
![Page 128: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/128.jpg)
128
1 1~ ~~ ~
4
0 0
4
00
4
0
3 sin (1 )
3 (1 ) cos
3 (1 )(1 ( 1))
24
S S
z
curl F d S curl F n dS
z d dz
z dz
z dz
= =
=
= -
= - -
= - - -
= -
![Page 129: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/129.jpg)
129
(ii) For surface , normal vector unit to the
surface is
By using polar coordinate of plane ,
4:2 =zS
.~~kn =
ddrrdSdanzry === 4,sin
0 r 3 and 0 .where
![Page 130: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/130.jpg)
130
2 2
~ ~ ~ ~~
~ ~~ ~
3
0 0
32
0 0
(1 )
sin
( sin )( )
sin
18
S S
r
r
curl F n z j y k k
y r
curl F d S curl F n dS
r rdrd
r d dr
= =
= =
= - +
= =
=
=
=
=
![Page 131: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/131.jpg)
131
(iii) For surface S3 : y = 0, normal vector unit
to the surface is
dS = dxdz
The integration limits :
.~~jn -=
3 3 0 4x and z-
So,
1
)())1((~~~~~
-=
-+-=
z
jkyjznFcurl
![Page 132: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/132.jpg)
132
3 3
1 2 3
~ ~~ ~
3 4
3 0
~ ~ ~ ~~ ~ ~ ~
Then,
. .
( 1)
24.
. . . .
24 18 24
18.
S S
x z
S S S S
curl F d S curl F n dS
z dzdx
curl F d S curl F d S curl F d S curl F d S
=- =
=
= -
=
=
= + +
= - + +
=
![Page 133: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/133.jpg)
133
~ ~
1
1
Now, we evaluate . for each pieces of the curve C.
i) is a half of the circle.
Therefore, integration for will be more easier if we use
polar coordinate for plane with radius
CF d r
C
C
3, that is
3cos , 3sin dan z 0
where 0 .
r
x y
=
= = =
![Page 134: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/134.jpg)
134
~ ~ ~~
~
~
~ ~~
~ ~
(3cos )(3sin )
9sin cos
and
3sin 3cos .
F z i xy j xz k
j
j
dr dx i dy j dz k
d i d j
= + +
=
=
= + +
= - +
![Page 135: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/135.jpg)
135
1
2
~ ~
2
0~ ~
3
0
From here,
. 27sin cos .
. 27sin cos
9cos
18.
C
F d r d
F d r d
=
=
= -
=
![Page 136: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/136.jpg)
136
2
2
~ ~ ~~
~
~ ~
ii) Curve is a straight line defined as
, 0 z 0, where 3 3.
Therefore,
0.
. 0.C
C
x t y and t
F z i xy j xz k
F d r
= = = -
= + +
=
=
![Page 137: Sreenivasa Institute of Technology and Management Studies€¦ · 1 1 Sreenivasa Institute of Technology and Management Studies Engineering Mathematics-III II B.TECH. I SEMESTER Course](https://reader033.fdocuments.us/reader033/viewer/2022053101/60611fa4ac44140777570744/html5/thumbnails/137.jpg)
137
1 2~ ~ ~ ~ ~ ~
~ ~ ~~
. . .
18 0
18.
We already show that
. .
Stokes' Theorem has been proved.
C C C
S C
F d r F d r F d r
curl F d S F d r
= +
= +
=
=