2. Coordinate Systems
• Cartesian or Rectangular Coordinate System
• Cylindrical Coordinate System
• Spherical Coordinate System
Choice of the system is based on the symmetry of the
problem.
3. Cartesian Or Rectangular Coordinates
P (x, y, z)
x
y
z
P(x,y,z)
z
y
x
A vector A in Cartesian coordinates can be written as
),,( zyx AAA or zzyyxx aAaAaA
where ax,ay and az are unit vectors along x, y and z-directions.
4. Cylindrical Coordinates
P (ρ, Φ, z)
x= ρ cos Φ, y=ρ sin Φ, z=z
z
Φ
z
ρ
x
y
P(ρ, Φ, z)
z
20
0
A vector A in Cylindrical coordinates can be written as
),,( zAAA or
zzaAaAaA
where aρ,aΦ and az are unit vectors along ρ, Φ and z-directions.
zz
x
y
yx
,tan, 122
5. The relationships between (ax,ay, az) and (aρ,aΦ, az)are
zz
y
x
aa
aaa
aaa
cossin
sincos
zz
yx
yx
aa
aaa
aaa
cossin
sincos
or
zzyxyx aAaAAaAAA )cossin()sincos(
Then the relationships between (Ax,Ay, Az) and (Aρ, AΦ, Az)are
8. Spherical Coordinates
P (r, θ, Φ)
x=r sin θ cos Φ, y=r sin θ sin Φ, Z=r cos θ
20
0
0
r
A vector A in Spherical coordinates can be written as
),,( AAAr or aAaAaA rr
where ar, aθ, and aΦ are unit vectors along r, θ, and Φ-directions.
θ
Φ
r
z
y
x
P(r, θ, Φ)
x
y
z
yx
zyxr 1
22
1222
tan,tan,
9. The relationships between (ax,ay, az) and (ar,aθ,aΦ)are
aaa
aaaa
aaaa
rz
ry
rx
sincos
cossincossinsin
sincoscoscossin
yx
zyx
zyxr
aaa
aaaa
aaaa
cossin
sinsincoscoscos
cossinsincossin
or
Then the relationships between (Ax,Ay, Az) and (Ar, Aθ,and AΦ)are
aAA
aAAA
aAAAA
yx
zyx
rzyx
)cossin(
)sinsincoscoscos(
)cossinsincossin(
12. Cartesian Coordinates
P(x, y, z)
Spherical Coordinates
P(r, θ, Φ)
Cylindrical Coordinates
P(ρ, Φ, z)
x
y
z
P(x,y,z)
Φ
z
r
x
y
z
P(ρ, Φ, z)
θ
Φ
r
z
y
x
P(r, θ, Φ)
13. Differential Length, Area and Volume
Differential displacement
zyx dzadyadxadl
Differential area
zyx dxdyadxdzadydzadS
Differential Volume
dxdydzdV
Cartesian Coordinates
17. Differential Length, Area and Volume
Differential displacement
adrarddradl r sin
Differential area
ardrdadrdraddrdS r sinsin2
Differential Volume
ddrdrdV sin2
Spherical Coordinates
18. Line, Surface and Volume Integrals
Line Integral
L
dlA.
Surface Integral
Volume Integral
S
dSA.
dvp
V
v
19. • Gradient of a scalar function is a
vector quantity.
• Divergence of a vector is a scalar
quantity.
• Curl of a vector is a vector quantity.
• The Laplacian of a scalar A
f Vector
A.
A
A2
The Del Operator
20. Del Operator
Cartesian Coordinates zyx a
z
a
y
a
x
Cylindrical Coordinates
Spherical Coordinates
za
z
aa
1
a
r
a
r
a
r
r
sin
11
22. GRADIENT OF A SCALAR
Suppose is the temperature at ,
and is the temperature at
as shown.
zyxT ,,1 zyxP ,,1
2P dzzdyydxxT ,,2
23. The differential distances are the
components of the differential distance
vector :
dzdydx ,,
zyx dzdydxd aaaL
Ld
However, from differential calculus, the
differential temperature:
dz
z
T
dy
y
T
dx
x
T
TTdT
12
GRADIENT OF A SCALAR (Cont’d)
25. 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.
Ld
dT
For Cartesian coordinate:
zyx
z
V
y
V
x
V
V aaa
GRADIENT OF A SCALAR (Cont’d)
26. For Circular cylindrical coordinate:
z
z
VVV
V aaa
1
For Spherical coordinate:
aaa
V
r
V
rr
V
V r
sin
11
GRADIENT OF A SCALAR (Cont’d)
27. EXAMPLE
Find gradient of these scalars:
yxeV z
cosh2sin
2cos2
zU
cossin10 2
rW
(a)
(b)
(c)
28. SOLUTIONTO EXAMPLE
(a) Use gradient for Cartesian coordinate:
z
z
y
z
x
z
zyx
yxe
yxeyxe
z
V
y
V
x
V
V
a
aa
aaa
cosh2sin
sinh2sincosh2cos2
29. SOLUTIONTO EXAMPLE (Cont’d)
(b) Use gradient for Circular cylindrical
coordinate:
z
z
zz
z
UUU
U
a
aa
aaa
2cos
2sin22cos2
1
2
30. (c) Use gradient for Spherical coordinate:
a
aa
aaa
sinsin10
cos2sin10cossin10
sin
11
2
r
r
W
r
W
rr
W
W
SOLUTIONTO EXAMPLE (Cont’d)
31. Illustration of the divergence of a vector
field at point P:
Positive
Divergence
Negative
Divergence
Zero
Divergence
DIVERGENCE OF AVECTOR
32. DIVERGENCE OF AVECTOR (Cont’d)
The divergence of A at a given point P
is the outward flux per unit volume:
v
dS
div s
v
A
AA lim
0
33. What is ??
s
dSA Vector field A at
closed surface S
DIVERGENCE OF AVECTOR (Cont’d)
35. For Cartesian coordinate:
z
A
y
A
x
A zyx
A
For Circular cylindrical coordinate:
z
AA
A z
11
A
DIVERGENCE OF AVECTOR (Cont’d)
36. For Spherical coordinate:
A
r
A
r
Ar
rr
r
sin
1sin
sin
11 2
2
A
DIVERGENCE OF AVECTOR (Cont’d)
37. Find divergence of these vectors:
zx xzyzxP aa 2
zzzQ aaa cossin 2
aaa coscossincos
1
2
r
r
W r
(a)
(b)
(c)
EXAMPLE
38. 39
(a) Use divergence for Cartesian
coordinate:
xxyz
xz
zy
yzx
x
z
P
y
P
x
P zyx
2
02
P
SOLUTIONTO EXAMPLE
39. (b) Use divergence for Circular cylindrical
coordinate:
cossin2
cos
1
sin
1
11
22
Q
z
z
z
z
QQ
Q z
SOLUTIONTO EXAMPLE (Cont’d)
40. (c) Use divergence for Spherical coordinate:
coscos2
cos
sin
1
cossin
sin
1
cos
1
sin
1sin
sin
11
2
2
2
2
W
r
r
rrr
W
r
W
r
Wr
rr
r
SOLUTIONTO EXAMPLE (Cont’d)
41. 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
DIVERGENCETHEOREM
42. 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)
EXAMPLE
43. (a) For two concentric cylinder, the left side:
topbottomouterinner
S
d DDDDSD
Where,
10)(
)(
2
0
5
0
1
4
2
0
5
0
1
3
z
z
inner
dzd
dzdD
aa
aa
SOLUTIONTO EXAMPLE
46. (b) For the right side of Divergence Theorem,
evaluate divergence of D
23
4
1
D
So,
150
4
5
0
2
0
2
1
4
5
0
2
0
2
1
2
z
r
z
dzdddVD
SOLUTIONTO EXAMPLE (Cont’d)
47. CURL OF AVECTOR
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.
49. 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.
CURL OF AVECTOR (Cont’d)
50. For Cartesian coordinate:
zyx
zyx
AAA
zyx
aaa
A
z
xy
y
xz
x
yz
y
A
x
A
z
A
x
A
z
A
y
A
aaaA
CURL OF AVECTOR (Cont’d)
52. For Spherical coordinate:
ArrAA
rr
r
r
sin
sin
1
2
aaa
A
a
aaA
r
r
r
A
r
rA
r
r
rAA
r
AA
r
)(1
sin
11sin
sin
1
CURL OF AVECTOR (Cont’d)
53. zx xzyzxP aa 2
zzzQ aaa cossin 2
aaa coscossincos
1
2
r
r
W r
(a)
(b)
(c)
Find curl of these vectors:
EXAMPLE
54. (a) Use curl for Cartesian coordinate:
zy
zyx
z
xy
y
xz
x
yz
zxzyx
zxzyx
y
P
x
P
z
P
x
P
z
P
y
P
aa
aaa
aaaP
22
22
000
SOLUTIONTO EXAMPLE
55. (b) Use curl for Circular cylindrical coordinate
z
z
z
zz
zz
z
z
y
Q
x
QQ
z
Q
z
QQ
aa
a
aa
aaaQ
cos3sin
1
cos3
1
00sin
11
3
2
2
SOLUTIONTO EXAMPLE (Cont’d)
56. (c) Use curl for Spherical coordinate:
a
aa
a
aaW
22
2
cos
)cossin(1
cos
cos
sin
11cossinsincos
sin
1
)(1
sin
11sin
sin
1
r
r
r
r
r
rr
r
r
r
W
r
rW
r
r
rWW
r
WW
r
r
r
r
r
SOLUTIONTO EXAMPLE (Cont’d)
57.
a
aa
a
aa
sin
1
cos2
cos
sin
sin
2cos
sin
cossin2
1
cos0
1
sinsin2cos
sin
1
3
2
r
rr
r
r
r
r
r
r
r
r
SOLUTIONTO EXAMPLE (Cont’d)
58. STOKE’STHEOREM
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
60. By using Stoke’s Theorem, evaluate
for
dlA
aaA sincos
EXAMPLE
61.
62. Stoke’s Theorem,
SL
dSdl AA
where, andzddd aS
Evaluate right side to get left side,
zaA
sin1
1
SOLUTIONTO EXAMPLE (Cont’d)
63.
941.4
sin1
1
0
0
60
30
5
2
aA z
S
dddS
SOLUTIONTO EXAMPLE (Cont’d)
64. 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
EXAMPLE
65. (a) To find LB d
321 LLL
dddd LBLBLBLB
Where,
dd
dzddd z
sincos
sincos
aaaaaLB
SOLUTIONTO EXAMPLE
70. 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
72. LAPLACIAN OF A SCALAR (Cont’d)
For Spherical coordinate:
2
2
22
2
2
2
2
sin
1
sin
sin
11
V
r
V
rr
V
r
rr
V
73. EXAMPLE
Find Laplacian of these scalars:
yxeV z
cosh2sin
2cos2
zU
cossin10 2
rW
(a)
(b)
(c)
You should try this!!