vector calculus
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TRANSCRIPT
![Page 1: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/1.jpg)
VECTOR CALCULUS
1.10 GRADIENT OF A SCALAR
1.11 DIVERGENCE OF A VECTOR
1.12 DIVERGENCE THEOREM
1.13 CURL OF A VECTOR
1.14 STOKES’S THEOREM
1.15 LAPLACIAN OF A SCALAR
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1.10 GRADIENT OF A SCALAR
Suppose is the temperature at ,
and is the temperature at
as shown.
zyxT ,,1 zyxP ,,1
2P dzzdyydxxT ,,2
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The differential distances are the components of the differential distance vector :
dzdydx ,,
zyx dzdydxd aaaL Ld
However, from differential calculus, the differential temperature:
dzz
Tdy
y
Tdx
x
TTTdT
12
GRADIENT OF A SCALAR (Cont’d)
![Page 4: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/4.jpg)
But,
z
y
x
ddz
ddy
ddx
aL
aL
aL
So, previous equation can be rewritten as:
Laaa
LaLaLa
dz
T
y
T
x
T
dz
Td
y
Td
x
TdT
zyx
zyx
GRADIENT OF A SCALAR (Cont’d)
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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 z
V
y
V
x
VV aaa
![Page 6: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/6.jpg)
GRADIENT OF A SCALAR (Cont’d)
For Circular cylindrical coordinate:
zz
VVVV aaa
1
For Spherical coordinate:
aaa
V
r
V
rr
VV r sin
11
![Page 7: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/7.jpg)
EXAMPLE 10
Find gradient of these scalars:
yxeV z cosh2sin
2cos2zU
cossin10 2rW
(a)
(b)
(c)
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SOLUTION TO EXAMPLE 10
(a) Use gradient for Cartesian coordinate:
zz
yz
xz
zyx
yxe
yxeyxe
z
V
y
V
x
VV
a
aa
aaa
cosh2sin
sinh2sincosh2cos2
![Page 9: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/9.jpg)
SOLUTION TO EXAMPLE 10 (Cont’d)
(b) Use gradient for Circular cylindrical
coordinate:
z
z
zz
z
UUUU
a
aa
aaa
2cos
2sin22cos2
1
2
![Page 10: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/10.jpg)
SOLUTION TO EXAMPLE 10 (Cont’d)
(c) Use gradient for Spherical coordinate:
a
aa
aaa
sinsin10
cos2sin10cossin10
sin
11
2
r
rW
r
W
rr
WW
![Page 11: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/11.jpg)
1.11 DIVERGENCE OF A VECTOR
Illustration of the divergence of a vector field at point P:
Positive Divergence
Negative Divergence
Zero Divergence
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DIVERGENCE OF A VECTOR (Cont’d)
The divergence of A at a given point P is the outward flux per unit volume:
v
dS
div s
v
A
A A lim0
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DIVERGENCE OF A VECTOR (Cont’d)
What is ?? s
dSA Vector field A at closed surface S
![Page 14: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/14.jpg)
Where,
dSdSbottomtoprightleftbackfronts
AA
And, v is volume enclosed by surface S
DIVERGENCE OF A VECTOR (Cont’d)
![Page 15: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/15.jpg)
For Cartesian coordinate:
z
A
y
A
x
A zyx
A
For Circular cylindrical coordinate:
z
AAA z
11A
DIVERGENCE OF A VECTOR (Cont’d)
![Page 16: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/16.jpg)
For Spherical coordinate:
A
r
A
rAr
rrr sin
1sin
sin
11 22
A
DIVERGENCE OF A VECTOR (Cont’d)
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EXAMPLE 11
Find divergence of these vectors:
zx xzyzxP aa 2
zzzQ aaa cossin 2
aaa coscossincos12
rr
W r
(a)
(b)
(c)
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18
(a) Use divergence for Cartesian coordinate:
SOLUTION TO EXAMPLE 11
xxyz
xzzy
yzxx
z
P
y
P
x
P zyx
2
02
P
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(b) Use divergence for Circular cylindrical
coordinate:
cossin2
cos1
sin1
11
22
Q
zz
z
z
QQQ z
SOLUTION TO EXAMPLE 11 (Cont’d)
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SOLUTION TO EXAMPLE 11 (Cont’d)
(c) Use divergence for Spherical coordinate:
coscos2
cossin
1
cossinsin
1cos
1
sin
1sin
sin
11
22
22
W
r
rrrr
W
r
W
rWr
rrr
![Page 21: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/21.jpg)
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
1.12 DIVERGENCE THEOREM
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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)
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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
01
3
z
zinner
dzd
dzdD
aa
aa
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160)(
)(
2
0
5
02
4
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)
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Therefore
150
0016010
SDS
d
SOLUTION TO EXAMPLE 12 Cont’d)
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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
1
4
5
0
2
0
2
1
2
zr
z
dzdddVD
![Page 27: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/27.jpg)
1.13 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.
![Page 28: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/28.jpg)
maxlim0
a
A
A A ns
s s
dl
Curl
Where,
CURL OF A VECTOR (Cont’d)
dldldacdbcabs
AA
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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.
![Page 30: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/30.jpg)
For Cartesian coordinate:
CURL OF A VECTOR (Cont’d)
zyx
zyx
AAAzyx
aaa
A
zxy
yxz
xyz
y
A
x
A
z
A
x
A
z
A
y
AaaaA
![Page 31: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/31.jpg)
z
z
AAAz
aaa
A1
z
zz
AA
z
AA
z
AA
a
aaA
1
1
For Circular cylindrical coordinate:
CURL OF A VECTOR (Cont’d)
![Page 32: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/32.jpg)
CURL OF A VECTOR (Cont’d)
For Spherical coordinate:
ArrAArrr
r
sinsin
12
aaa
A
a
aaA
r
rr
A
r
rA
r
r
rAA
r
AA
r
)(1
sin
11sin
sin
1
![Page 33: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/33.jpg)
EXAMPLE 13
zx xzyzxP aa 2
zzzQ aaa cossin 2
aaa coscossincos12
rr
W r
(a)
(b)
(c)
Find curl of these vectors:
![Page 34: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/34.jpg)
SOLUTION TO EXAMPLE 13
(a) Use curl for Cartesian coordinate:
zy
zyx
zxy
yxz
xyz
zxzyx
zxzyx
y
P
x
P
z
P
x
P
z
P
y
P
aa
aaa
aaaP
22
22 000
![Page 35: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/35.jpg)
(b) Use curl for Circular cylindrical coordinate
z
z
zzz
zz
z
z
y
Q
x
z
Q
z
aa
a
aa
aaaQ
cos3sin1
cos31
00sin
11
3
2
2
SOLUTION TO EXAMPLE 13 (Cont’d)
![Page 36: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/36.jpg)
(c) Use curl for Spherical coordinate:
a
aa
a
aaW
22
2
cos)cossin(1
coscos
sin
11cossinsincos
sin
1
)(1
sin
11sin
sin
1
rr
r
r
r
rrr
r
r
W
r
rW
r
r
rWW
r
WW
r
r
r
rr
SOLUTION TO EXAMPLE 13 (Cont’d)
![Page 37: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/37.jpg)
SOLUTION TO EXAMPLE 13 (Cont’d)
a
aa
a
aa
sin1
cos2
cossin
sin
2cos
sincossin2
1
cos01
sinsin2cossin
1
3
2
r
rr
rr
r
rr
r
r
r
![Page 38: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/38.jpg)
1.14 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
![Page 39: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/39.jpg)
STOKE’S THEOREM (Cont’d)
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EXAMPLE 14
By using Stoke’s Theorem, evaluate
for dlA
aaA sincos
![Page 41: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/41.jpg)
EXAMPLE 14 (Cont’d)
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SOLUTION TO EXAMPLE 14
Stoke’s Theorem,
SL
dSdl AA
where, and
zddd aS Evaluate right side to get left side,
zaA
sin11
![Page 43: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/43.jpg)
SOLUTION TO EXAMPLE 14 (Cont’d)
941.4
sin11
0
0
60
30
5
2
aA zS
dddS
![Page 44: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/44.jpg)
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
![Page 45: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/45.jpg)
SOLUTION TO EXAMPLE 15
(a) To find LB d
321 LLL
dddd LBLBLBLB
Where,
dd
dzddd z
sincos
sincos
aaaaaLB
![Page 46: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/46.jpg)
So
202
1
sincos
2
0
2
0
0
00,0
2
01
LB
zzL
ddd
4cos20
sincos
0
0,200
2
22
LB
zzL
ddd
SOLUTION TO EXAMPLE 15 (Cont’d)
![Page 47: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/47.jpg)
202
1
sincos
0
2
2
00,0
0
23
r
zzL
ddd
LB
SOLUTION TO EXAMPLE 15 (Cont’d)
Therefore the closed integral,
8242 LB d
![Page 48: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/48.jpg)
SOLUTION TO EXAMPLE 15 (Cont’d)
(b) To find SB d
z
z
z
zz
a
aaa
a
aa
aaB
11sin
sinsin1
00
cossin1
0cossin01
sincos
![Page 49: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/49.jpg)
SOLUTION TO EXAMPLE 15 (Cont’d)
Therefore
82
1cos
1sin
11sin
0
2
0
2
0
2
0
0
2
0
aaSB
dd
ddd zz
![Page 50: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/50.jpg)
1.15 LAPLACIAN OF A SCALAR
The Laplacian of a scalar field, V written as: V2
Where, Laplacian V is:
zyxzyx z
V
y
V
x
V
zyx
VV
aaaaaa
2
![Page 51: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/51.jpg)
For Cartesian coordinate:
2
2
2
2
2
22
z
V
y
V
x
VV
For Circular cylindrical coordinate:
2
22
22 11
z
VVVV
LAPLACIAN OF A SCALAR (Cont’d)
![Page 52: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/52.jpg)
LAPLACIAN OF A SCALAR (Cont’d)
For Spherical coordinate:
2
2
22
22
22
sin
1
sinsin
11
V
r
V
rr
Vr
rrV
![Page 53: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/53.jpg)
EXAMPLE 16
Find Laplacian of these scalars:
yxeV z cosh2sin 2cos2zU
cossin10 2rW
(a)
(b)
(c)
You should try this!!
![Page 54: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/54.jpg)
SOLUTION TO EXAMPLE 16
yxeV z cosh2sin22
02 U
2cos21
cos102 r
W
(a)
(b)
(c)
![Page 55: Vector calculus](https://reader033.vdocuments.us/reader033/viewer/2022061213/54967444b47959507a8b466f/html5/thumbnails/55.jpg)
Yea, unto God belong all things in the heavens and on earth, and enough is
God to carry through all affairs
Quran:4:132
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