Vector calculus 1st 2
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JETGI 1
Vector Algebra
Mr. HIMANSHU DIWAKARAssistant professor
ECED
Mr. Himanshu Diwakar
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JETGI 2Mr. Himanshu Diwakar
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JETGI 3
VECTOR CALCULUS
Introduction
Scalars And Vectors
Gradient Of A Scalar
Divergence Of A Vector
Divergence Theorem
Curl Of A Vector
Stokes’s Theorem
Laplacian Of A Scalar
Mr. Himanshu Diwakar
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JETGI 4
Introduction• Electromagnetics(EM):-
Interaction between electric charges at rest and
in motion.
• It entails the analysis, synthesis, physical interpretation, and application of electric and magnetic fields.
Mr. Himanshu Diwakar
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JETGI 5
Scalars and vectors
• A scalar is a quantity that has only magnitudeEx:- time, mass, distance, temperature, entropy etc.
• A vector is a quantity that has both magnitude and direction• Velocity, force, displacement and electric field intensity.
Mr. Himanshu Diwakar
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JETGI 6
Unit vectors
Mr. Himanshu Diwakar
x
z
y
az
ay
ax
Unit vectorsaz ,ay ,az
Similarly a A vector in Cartesian co-ordinate
A=Ax.ax+Ay.ay+Az.az
So Unit vector of A
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JETGI 7
Position and distance vector
Mr. Himanshu Diwakar
x
z
y
az
ay
ax
P (3, 4, 5)
O (0, 0, 0)
The distance vector is the displacement from one point to another.
A= 3.ax+ 4.ay+ 5.az
OP distanceOP=
= =7.071
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JETGI 8
Vector multiplication
• Scalar (or dot) product = A.B
• Vector (or cross) product = AB
• Scalar triple product :
• Vector triple product :
Mr. Himanshu Diwakar
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JETGI 9
Angels
Mr. Himanshu Diwakar
If A and B are vectors then the angle between these vectors can be find
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JETGI 10
Differential Length (Cartesian Coordinates )
• Differential elements in length, area, and volume are useful in vector calculus. They are defined in the Cartesian, cylindrical, and spherical coordinate systems.
1. Differential displacement is given by
2. Differential normal area is given by
3. Differential volume is given by
Mr. Himanshu Diwakar
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JETGI 11Mr. Himanshu Diwakar
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JETGI 12
Cylindrical Coordinates• Notice from Figure that in cylindrical coordinates, differential
elements can be found as follows:1. Differential displacement is given by
2. Differential normal area is given by
3. Differential volume is given by
Mr. Himanshu Diwakar
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JETGI 13Mr. Himanshu Diwakar
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JETGI 14Mr. Himanshu Diwakar
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JETGI 15
Spherical Coordinates
From Figure, we notice that in spherical coordinates,1. The differential displacement is
Mr. Himanshu Diwakar
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JETGI 16
2. The differential normal area is
Mr. Himanshu Diwakar
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JETGI 17
3. The differential volume is
Mr. Himanshu Diwakar
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JETGI 18
Que:- Consider the object shown in Figure and CalculateThe distance BC, The distance CD, The surface area ABCD,
The surface area ABO, The surface area A OFD, The volume ABDCFO
Mr. Himanshu Diwakar
C(0, 5, 0)
B(0, 5, 0)
D(5, 0, 10)
A(5, 0, 0)
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JETGI 19
Line, Surface And Volume Integrals
• The familiar concept of integration will now be extended to cases when the integrand involves a vector. By a line we mean the path along a curve in space. We shall use terms such as line, curve, and contour interchangeably.• The line integral is the integral of the tangential component of A
along curve L. Or simplyAnd For closed surface
Mr. Himanshu Diwakar
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JETGI 20
Example:- Given that , calculate the circulation of F around the (closed) path shown in Figure
Ans:
Mr. Himanshu Diwakar
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JETGI 21
Dell operator
• The del operator, written , is the vector differential operator. In Cartesian coordinates,
• This vector differential operator, otherwise known as the gradient operator, is not a vector in itself, but when it operates on a scalar function, for example, a vector ensues. The operator is useful in defining
Mr. Himanshu Diwakar
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JETGI 22
1. The gradient of a scalar V, written, as 2. The divergence of a vector A, written as (• A)3. The curl of a vector A, written as ( X A)4. The Laplacian of a scalar V, written as
Each of these will be denned in detail in the subsequent sections.
Mr. Himanshu Diwakar
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JETGI 23Mr. Himanshu Diwakar
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JETGI 24Mr. Himanshu Diwakar
So the solution for above equations is
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JETGI 25
CLASSIFICATION OF VECTOR FIELDS
• A vector field is uniquely characterized by its divergence and curl. Neither the divergence nor curl of a vector field is sufficient to completely describe the field.
• All vector fields can be classified in terms of their vanishing or non vanishing divergence or curl as follows:
Mr. Himanshu Diwakar
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JETGI 26
Typical fields with vanishing and non vanishing divergence or curl.
Mr. Himanshu Diwakar
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JETGI 27
•A vector field A is said to be solenoidal (or divergenceless) if .
•A vector field A is said to be irrotational (or potential) if .
Mr. Himanshu Diwakar
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JETGI 28
The differential distances are the components of the differential distance vector :
dzdydx ,,
zyx dzdydxd aaaL Ld
However, from differential calculus, the differential temperature:
dzzTdy
yTdx
xTTTdT
12
GRADIENT OF A SCALAR
Mr. Himanshu Diwakar
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JETGI 29
But,
z
y
x
ddz
ddyddx
aL
aLaL
So, previous equation can be rewritten as:
Laaa
LaLaLa
dzT
yT
xT
dzTd
yTd
xTdT
zyx
zyx
GRADIENT OF A SCALAR (Cont’d)
Mr. Himanshu Diwakar
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JETGI 30
The vector inside square brackets defines the change of temperature corresponding to a vector change in position .This vector is called Gradient of Scalar T.
LddT
GRADIENT OF A SCALAR (Cont’d)
For Cartesian coordinate:
zyx zV
yV
xVV aaa
Mr. Himanshu Diwakar
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JETGI 31
GRADIENT OF A SCALAR (Cont’d)
For Circular cylindrical coordinate:
zzVVVV aaa
1
For Spherical coordinate:
aaa
V
rV
rrVV r sin
11
Mr. Himanshu Diwakar
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JETGI 32
EXAMPLE 1
Find gradient of these scalars:
yxeV z cosh2sin
2cos2zU
cossin10 2rW
(a)
(b)
(c)
Mr. Himanshu Diwakar
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JETGI 33
SOLUTION TO EXAMPLE 1
(a) Use gradient for Cartesian coordinate:
zz
yz
xz
zyx
yxe
yxeyxe
zV
yV
xVV
a
aa
aaa
cosh2sin
sinh2sincosh2cos2
Mr. Himanshu Diwakar
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JETGI 34
SOLUTION TO EXAMPLE 1 (Cont’d)
(b) Use gradient for Circular cylindrical coordinate:
z
z
zzzUUUU
a
aa
aaa
2cos
2sin22cos2
1
2
Mr. Himanshu Diwakar
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JETGI 35
SOLUTION TO EXAMPLE 1 (Cont’d)
(c) Use gradient for Spherical coordinate:
a aa
aaa
sinsin10cos2sin10cossin10
sin11
2
r
rW
rW
rrWW
Mr. Himanshu Diwakar
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JETGI 36
DIVERGENCE OF A VECTOR
Illustration of the divergence of a vector field at point P:
Positive Divergence
Negative Divergence
Zero Divergence
Mr. Himanshu Diwakar
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JETGI 37
DIVERGENCE OF A VECTOR (Cont’d)
The divergence of A at a given point P is the outward flux per unit volume:
v
dSdiv s
v
AA A lim
0
Mr. Himanshu Diwakar
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JETGI 38
DIVERGENCE OF A VECTOR (Cont’d)
What is ?? s
dSA Vector field A at closed surface S
Mr. Himanshu Diwakar
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JETGI 39
Where,dSdS
bottomtoprightleftbackfronts
AA
And, v is volume enclosed by surface S
DIVERGENCE OF A VECTOR (Cont’d)
Mr. Himanshu Diwakar
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JETGI 40
For Cartesian coordinate:
zA
yA
xA zyx
A
For Circular cylindrical coordinate:
z
AAA z
11A
DIVERGENCE OF A VECTOR (Cont’d)
Mr. Himanshu Diwakar
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JETGI 41
For Spherical coordinate:
A
rA
rAr
rr r sin1sin
sin11 2
2A
DIVERGENCE OF A VECTOR (Cont’d)
Mr. Himanshu Diwakar
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JETGI 42
EXAMPLE 11
Find divergence of these vectors:
zx xzyzxP aa 2
zzzQ aaa cossin 2
aaa coscossincos12 r
rW r
(a)
(b)
(c)
Mr. Himanshu Diwakar
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JETGI 43
(a) Use divergence for Cartesian coordinate:
SOLUTION TO EXAMPLE 11
xxyz
xzzy
yzxx
zP
yP
xP zyx
2
02
P
Mr. Himanshu Diwakar
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JETGI 44
(b) Use divergence for Circular cylindrical coordinate:
cossin2
cos1sin1
11
22
Q
zz
z
zQQ
Q z
SOLUTION TO EXAMPLE 11 (Cont’d)
Mr. Himanshu Diwakar
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JETGI 45
SOLUTION TO EXAMPLE 11 (Cont’d)
(c) Use divergence for Spherical coordinate:
coscos2
cossin1
cossinsin1cos1
sin1sin
sin11
22
22
W
r
rrrr
Wr
Wr
Wrrr r
Mr. Himanshu Diwakar
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JETGI 46
It states that the total outward flux of a vector field A at the closed surface S is the same as volume integral of divergence of A.
VV
dVdS AA
DIVERGENCE THEOREM
Mr. Himanshu Diwakar
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JETGI 47
EXAMPLE 12
A vector field exists in the region between two concentric cylindrical surfaces defined by ρ = 1 and ρ = 2, with both cylinders extending between z = 0 and z = 5. Verify the divergence theorem by evaluating:
aD 3
S
dsD
V
DdV
(a)
(b)
Mr. Himanshu Diwakar
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JETGI 48
SOLUTION TO EXAMPLE 12
(a) For two concentric cylinder, the left side:
topbottomouterinnerS
d DDDDSD
Where,
10)(
)(
2
0
5
01
4
2
0
5
0 13
z
zinner
dzd
dzdD
aa
aa
Mr. Himanshu Diwakar
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JETGI 49
160)(
)(
2
0
5
0 24
2
0
5
02
3
z
zouter
dzd
dzdD
aa
aa
2
1
2
05
3
2
1
2
00
3
0)(
0)(
zztop
zzbottom
ddD
ddD
aa
aa
SOLUTION TO EXAMPLE 12 Cont’d)
Mr. Himanshu Diwakar
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JETGI 50
Therefore
150
0016010
SDS
d
SOLUTION TO EXAMPLE 12 Cont’d)
Mr. Himanshu Diwakar
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JETGI 51
SOLUTION TO EXAMPLE 12 Cont’d)
(b) For the right side of Divergence Theorem, evaluate divergence of D
23 41
D
So,
150
4
5
0
2
0
2
14
5
0
2
0
2
1
2
zr
zdzdddVD
Mr. Himanshu Diwakar
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JETGI 52
CURL OF A VECTOR
The curl of vector A is an axial (rotational) vector whose magnitude is the maximum circulation of A per unit area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum.
Mr. Himanshu Diwakar
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JETGI 53
maxlim0
aA
A A ns
s s
dlCurl
Where,
CURL OF A VECTOR (Cont’d)
dldldacdbcabs
AA
Mr. Himanshu Diwakar
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JETGI 54
CURL OF A VECTOR (Cont’d)
The curl of the vector field is concerned with rotation of the vector field. Rotation can be used to measure the uniformity of the field, the more non uniform the field, the larger value of curl.
Mr. Himanshu Diwakar
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JETGI 55
For Cartesian coordinate:
CURL OF A VECTOR (Cont’d)
zyx
zyx
AAAzyx
aaa
A
zxy
yxz
xyz
yA
xA
zA
xA
zA
yA aaaA
Mr. Himanshu Diwakar
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JETGI 56
z
z
AAAz
aaa
A 1
z
zz
AA
zAA
zAA
a
aaA
1
1
For Circular cylindrical coordinate:
CURL OF A VECTOR (Cont’d)
Mr. Himanshu Diwakar
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JETGI 57
CURL OF A VECTOR (Cont’d)
For Spherical coordinate:
ArrAArrr
r
sinsin1
2
aaa
A
a
aaA
r
rr
Ar
rAr
rrAA
rAA
r
)(1
sin11sin
sin1
Mr. Himanshu Diwakar
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JETGI 58
EXAMPLE 13
zx xzyzxP aa 2
zzzQ aaa cossin 2
aaa coscossincos12 r
rW r
(a)
(b)
(c)
Find curl of these vectors:
Mr. Himanshu Diwakar
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JETGI 59
SOLUTION TO EXAMPLE 13
(a) Use curl for Cartesian coordinate:
zy
zyx
zxy
yxz
xyz
zxzyx
zxzyx
yP
xP
zP
xP
zP
yP
aa
aaa
aaaP
22
22 000
Mr. Himanshu Diwakar
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JETGI 60
(b) Use curl for Circular cylindrical coordinate
z
z
zzz
zz
z
z
yQ
xQQ
zQ
zQQ
aa
a
aa
aaaQ
cos3sin1
cos31
00sin
11
3
2
2
SOLUTION TO EXAMPLE 13 (Cont’d)
Mr. Himanshu Diwakar
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JETGI 61
(c) Use curl for Spherical coordinate:
a
aa
a
aaW
22
2
cos)cossin(1
coscos
sin11cossinsincos
sin1
)(1
sin11sin
sin1
rr
rr
rrr
rr
r
Wr
rWr
rrWW
rWW
r
r
r
rr
SOLUTION TO EXAMPLE 13 (Cont’d)
Mr. Himanshu Diwakar
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JETGI 62
SOLUTION TO EXAMPLE 13 (Cont’d)
a
aa
a
aa
sin1cos2
cossinsin
2cos
sincossin21
cos01sinsin2cossin1
3
2
r
rr
rr
r
rr
r
r
r
Mr. Himanshu Diwakar
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JETGI 63
STOKE’S THEOREM
The circulation of a vector field A around a closed path L is equal to the surface integral of the curl of A over the open surface S bounded by L that A and curl of A are continuous on S.
SL
dSdl AA
Mr. Himanshu Diwakar
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JETGI 64
STOKE’S THEOREM (Cont’d)
Mr. Himanshu Diwakar
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JETGI 65
EXAMPLE 14
By using Stoke’s Theorem, evaluate for
dlA
aaA sincos
Mr. Himanshu Diwakar
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JETGI 66
EXAMPLE 14 (Cont’d)
Mr. Himanshu Diwakar
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JETGI 67
SOLUTION TO EXAMPLE 14
Stoke’s Theorem,
SL
dSdl AA
where, and
zddd aS Evaluate right side to get left side,
zaA
sin11
Mr. Himanshu Diwakar
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JETGI 68
SOLUTION TO EXAMPLE 14 (Cont’d)
941.4
sin110
0
60
30
5
2
aA zS
dddS
Mr. Himanshu Diwakar
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JETGI 69
EXAMPLE 15
Verify Stoke’s theorem for the vector field for given figure by evaluating: aaB sincos
(a) over the semicircular contour.
LB d
(b) over the surface of semicircular contour.
SB d
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JETGI 70
SOLUTION TO EXAMPLE 15
(a) To find LB d
321 LLL
dddd LBLBLBLB
Where,
dd
dzddd z
sincos
sincos
aaaaaLB
Mr. Himanshu Diwakar
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JETGI 71
So
2021
sincos
2
0
2
0
0
00,0
2
01
LBzzL
ddd
4cos20
sincos
0
0,200
2
22
LBzzL
ddd
SOLUTION TO EXAMPLE 15 (Cont’d)
Mr. Himanshu Diwakar
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JETGI 72
2021
sincos
0
2
2
00,0
0
23
r
zzLddd
LB
SOLUTION TO EXAMPLE 15 (Cont’d)
Therefore the closed integral,
8242 LB d
Mr. Himanshu Diwakar
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JETGI 73
SOLUTION TO EXAMPLE 15 (Cont’d)
(b) To find SB d
z
z
z
zz
a
aaa
a
aa
aaB
11sin
sinsin100
cossin1
0cossin01
sincos
Mr. Himanshu Diwakar
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JETGI 74
SOLUTION TO EXAMPLE 15 (Cont’d)
Therefore
821cos
1sin
11sin
0
2
0
2
0
2
0
0
2
0
aaSB
dd
ddd zz
Mr. Himanshu Diwakar
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JETGI 75
LAPLACIAN OF A SCALAR
The Laplacian of a scalar field, V written as: V2
Where, Laplacian V is:
zyxzyx zV
yV
xV
zyx
VV
aaaaaa
2
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JETGI 76
For Cartesian coordinate:
2
2
2
2
2
22
zV
yV
xVV
For Circular cylindrical coordinate:
2
22
22 11
zVVVV
LAPLACIAN OF A SCALAR (Cont’d)
Mr. Himanshu Diwakar
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JETGI 77
LAPLACIAN OF A SCALAR (Cont’d)
For Spherical coordinate:
2
2
22
22
22
sin1
sinsin11
Vr
Vrr
Vrrr
V
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JETGI 78
EXAMPLE 16
Find Laplacian of these scalars:
yxeV z cosh2sin 2cos2zU
cossin10 2rW
(a)
(b)
(c)
Try yourself !!
Mr. Himanshu Diwakar
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JETGI 79
SOLUTION TO EXAMPLE 16
yxeV z cosh2sin22
02 U
2cos21cos102 r
W
(a)
(b)
(c)
Mr. Himanshu Diwakar
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JETGI 80
THANK YOU
Mr. Himanshu Diwakar