Formula sheet: Coordinate System & Vector Analysis | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


Formula Sheet for Coordinate Systems and Vector
Analysis (EMFT) – GATE
1. Basic Vector Operations
• Vector Representation: A = A
x
ˆ x+A
y
ˆ y +A
z
ˆ z (Cartesian).
• Magnitude: |A| =
q
A
2
x
+A
2
y
+A
2
z
.
• Dot Product: A·B = A
x
B
x
+A
y
B
y
+A
z
B
z
=|A||B|cos?.
• Cross Product:
A×B =







ˆ x ˆ y ˆ z
A
x
A
y
A
z
B
x
B
y
B
z







= (A
y
B
z
-A
z
B
y
)ˆ x+(A
z
B
x
-A
x
B
z
)ˆ y +(A
x
B
y
-A
y
B
x
)ˆ z
• Scalar Triple Product: A·(B×C) =







A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z







.
2. Coordinate Systems
2.1 Cartesian Coordinates
• Position Vector: r = xˆ x+yˆ y +zˆ z.
• Distance: r =
v
x
2
+y
2
+z
2
.
• Unit Vectors: ˆ x,ˆ y,ˆ z (orthogonal).
2.2 Cylindrical Coordinates
• Coordinates: (?,?,z ).
• Relation to Cartesian:
x = ?cos?, y = ?sin?, z = z
? =
q
x
2
+y
2
, ? = tan
-1

y
x

, z = z
• Unit Vectors: ˆ ?,
ˆ
?, ˆ z.
• Vector Representation: A = A
?
ˆ ?+A
? ˆ
? +A
z
ˆ z.
2.3 Spherical Coordinates
• Coordinates: (r,?,? ).
• Relation to Cartesian:
x = rsin?cos?, y = rsin?sin?, z = rcos?
r =
q
x
2
+y
2
+z
2
, ? = cos
-1
 
z
v
x
2
+y
2
+z
2
!
, ? = tan
-1

y
x

1
Page 2


Formula Sheet for Coordinate Systems and Vector
Analysis (EMFT) – GATE
1. Basic Vector Operations
• Vector Representation: A = A
x
ˆ x+A
y
ˆ y +A
z
ˆ z (Cartesian).
• Magnitude: |A| =
q
A
2
x
+A
2
y
+A
2
z
.
• Dot Product: A·B = A
x
B
x
+A
y
B
y
+A
z
B
z
=|A||B|cos?.
• Cross Product:
A×B =







ˆ x ˆ y ˆ z
A
x
A
y
A
z
B
x
B
y
B
z







= (A
y
B
z
-A
z
B
y
)ˆ x+(A
z
B
x
-A
x
B
z
)ˆ y +(A
x
B
y
-A
y
B
x
)ˆ z
• Scalar Triple Product: A·(B×C) =







A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z







.
2. Coordinate Systems
2.1 Cartesian Coordinates
• Position Vector: r = xˆ x+yˆ y +zˆ z.
• Distance: r =
v
x
2
+y
2
+z
2
.
• Unit Vectors: ˆ x,ˆ y,ˆ z (orthogonal).
2.2 Cylindrical Coordinates
• Coordinates: (?,?,z ).
• Relation to Cartesian:
x = ?cos?, y = ?sin?, z = z
? =
q
x
2
+y
2
, ? = tan
-1

y
x

, z = z
• Unit Vectors: ˆ ?,
ˆ
?, ˆ z.
• Vector Representation: A = A
?
ˆ ?+A
? ˆ
? +A
z
ˆ z.
2.3 Spherical Coordinates
• Coordinates: (r,?,? ).
• Relation to Cartesian:
x = rsin?cos?, y = rsin?sin?, z = rcos?
r =
q
x
2
+y
2
+z
2
, ? = cos
-1
 
z
v
x
2
+y
2
+z
2
!
, ? = tan
-1

y
x

1
• Unit Vectors: ˆ r,
ˆ
?,
ˆ
? .
• Vector Representation: A = A
r
ˆ r +A
?
ˆ
? +A
? ˆ
? .
3. Di?erential Elements
• Cartesian:
dl = dxˆ x+dyˆ y +dzˆ z, dV = dxdydz
• Cylindrical:
dl = d?ˆ ?+?d?
ˆ
? +dzˆ z, dV = ?d?d?dz
• Spherical:
dl = drˆ r +rd?
ˆ
? +rsin?d?
ˆ
?, dV = r
2
sin?drd?d?
4. Vector Calculus
• Gradient (?f):
– Cartesian: ?f =
?f
?x
ˆ x+
?f
?y
ˆ y +
?f
?z
ˆ z.
– Cylindrical: ?f =
?f
??
ˆ ?+
1
?
?f
?? ˆ
? +
?f
?z
ˆ z.
– Spherical: ?f =
?f
?r
ˆ r +
1
r
?f
??
ˆ
? +
1
rsin?
?f
?? ˆ
? .
• Divergence (?·A):
– Cartesian: ?·A =
?Ax
?x
+
?Ay
?y
+
?Az
?z
.
– Cylindrical: ?·A =
1
?
?(?A?)
??
+
1
?
?A
? ?? +
?Az
?z
.
– Spherical: ?·A =
1
r
2
?(r
2
Ar)
?r
+
1
rsin?
?(sin?A
?
)
??
+
1
rsin?
?A
? ?? .
• Curl (?×A):
– Cartesian:
?×A =
 
?A
z
?y
-
?A
y
?z
!
ˆ x+
 
?A
x
?z
-
?A
z
?x
!
ˆ y +
 
?A
y
?x
-
?A
x
?y
!
ˆ z
– Cylindrical:
?×A =
 
1
?
?A
z
?? -
?A
? ?z
!
ˆ ?+
 
?A
?
?z
-
?A
z
??
!
ˆ
? +
1
?
 
?(?A
? )
??
-
?A
?
?? !
ˆ z
– Spherical:
?×A =
1
rsin?
 
?(sin?A
? )
??
-
?A
?
?? !
ˆ r+
1
r
 
1
sin?
?A
r
?? -
?(rA
? )
?r
!
ˆ
?+
1
r
 
?(rA
?
)
?r
-
?A
r
??
!
ˆ
? 2
Page 3


Formula Sheet for Coordinate Systems and Vector
Analysis (EMFT) – GATE
1. Basic Vector Operations
• Vector Representation: A = A
x
ˆ x+A
y
ˆ y +A
z
ˆ z (Cartesian).
• Magnitude: |A| =
q
A
2
x
+A
2
y
+A
2
z
.
• Dot Product: A·B = A
x
B
x
+A
y
B
y
+A
z
B
z
=|A||B|cos?.
• Cross Product:
A×B =







ˆ x ˆ y ˆ z
A
x
A
y
A
z
B
x
B
y
B
z







= (A
y
B
z
-A
z
B
y
)ˆ x+(A
z
B
x
-A
x
B
z
)ˆ y +(A
x
B
y
-A
y
B
x
)ˆ z
• Scalar Triple Product: A·(B×C) =







A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z







.
2. Coordinate Systems
2.1 Cartesian Coordinates
• Position Vector: r = xˆ x+yˆ y +zˆ z.
• Distance: r =
v
x
2
+y
2
+z
2
.
• Unit Vectors: ˆ x,ˆ y,ˆ z (orthogonal).
2.2 Cylindrical Coordinates
• Coordinates: (?,?,z ).
• Relation to Cartesian:
x = ?cos?, y = ?sin?, z = z
? =
q
x
2
+y
2
, ? = tan
-1

y
x

, z = z
• Unit Vectors: ˆ ?,
ˆ
?, ˆ z.
• Vector Representation: A = A
?
ˆ ?+A
? ˆ
? +A
z
ˆ z.
2.3 Spherical Coordinates
• Coordinates: (r,?,? ).
• Relation to Cartesian:
x = rsin?cos?, y = rsin?sin?, z = rcos?
r =
q
x
2
+y
2
+z
2
, ? = cos
-1
 
z
v
x
2
+y
2
+z
2
!
, ? = tan
-1

y
x

1
• Unit Vectors: ˆ r,
ˆ
?,
ˆ
? .
• Vector Representation: A = A
r
ˆ r +A
?
ˆ
? +A
? ˆ
? .
3. Di?erential Elements
• Cartesian:
dl = dxˆ x+dyˆ y +dzˆ z, dV = dxdydz
• Cylindrical:
dl = d?ˆ ?+?d?
ˆ
? +dzˆ z, dV = ?d?d?dz
• Spherical:
dl = drˆ r +rd?
ˆ
? +rsin?d?
ˆ
?, dV = r
2
sin?drd?d?
4. Vector Calculus
• Gradient (?f):
– Cartesian: ?f =
?f
?x
ˆ x+
?f
?y
ˆ y +
?f
?z
ˆ z.
– Cylindrical: ?f =
?f
??
ˆ ?+
1
?
?f
?? ˆ
? +
?f
?z
ˆ z.
– Spherical: ?f =
?f
?r
ˆ r +
1
r
?f
??
ˆ
? +
1
rsin?
?f
?? ˆ
? .
• Divergence (?·A):
– Cartesian: ?·A =
?Ax
?x
+
?Ay
?y
+
?Az
?z
.
– Cylindrical: ?·A =
1
?
?(?A?)
??
+
1
?
?A
? ?? +
?Az
?z
.
– Spherical: ?·A =
1
r
2
?(r
2
Ar)
?r
+
1
rsin?
?(sin?A
?
)
??
+
1
rsin?
?A
? ?? .
• Curl (?×A):
– Cartesian:
?×A =
 
?A
z
?y
-
?A
y
?z
!
ˆ x+
 
?A
x
?z
-
?A
z
?x
!
ˆ y +
 
?A
y
?x
-
?A
x
?y
!
ˆ z
– Cylindrical:
?×A =
 
1
?
?A
z
?? -
?A
? ?z
!
ˆ ?+
 
?A
?
?z
-
?A
z
??
!
ˆ
? +
1
?
 
?(?A
? )
??
-
?A
?
?? !
ˆ z
– Spherical:
?×A =
1
rsin?
 
?(sin?A
? )
??
-
?A
?
?? !
ˆ r+
1
r
 
1
sin?
?A
r
?? -
?(rA
? )
?r
!
ˆ
?+
1
r
 
?(rA
?
)
?r
-
?A
r
??
!
ˆ
? 2
5. Laplacian
• Scalar Laplacian (?
2
f):
– Cartesian: ?
2
f =
?
2
f
?x
2
+
?
2
f
?y
2
+
?
2
f
?z
2
.
– Cylindrical: ?
2
f =
1
?
?
??

?
?f
??

+
1
?
2
?
2
f
?? 2
+
?
2
f
?z
2
.
– Spherical: ?
2
f =
1
r
2
?
?r

r
2?f
?r

+
1
r
2
sin?
?
??

sin?
?f
??

+
1
r
2
sin
2
?
?
2
f
?? 2
.
6. Vector Identities
• ?·(?×A) = 0
• ?×(?f) = 0
• ?·(fA) = f(?·A)+A·(?f)
• ?×(fA) = f(?×A)+(?f)×A
7. Line, Surface, and Volume Integrals
• Line Integral:
R
C
A·dl.
• Surface Integral:
RR
S
A·dS.
• Volume Integral:
RRR
V
f dV.
8. Theorems
• Divergence Theorem:
ZZZ
V
?·AdV =
ZZ
S
A·dS
• Stokes Theorem: ZZ
S
(?×A)·dS =
I
C
A·dl
9. Coordinate Transformation
• Cartesian to Cylindrical Unit Vectors:
ˆ ? = cos? ˆ x+sin? ˆ y,
ˆ
? =-sin? ˆ x+cos? ˆ y, ˆ z = ˆ z
• Cartesian to Spherical Unit Vectors:
ˆ r = sin?cos? ˆ x+sin?sin? ˆ y +cos?ˆ z
ˆ
? = cos?cos? ˆ x+cos?sin? ˆ y-sin?ˆ z
ˆ
? =-sin? ˆ x+cos? ˆ y
3
Page 4


Formula Sheet for Coordinate Systems and Vector
Analysis (EMFT) – GATE
1. Basic Vector Operations
• Vector Representation: A = A
x
ˆ x+A
y
ˆ y +A
z
ˆ z (Cartesian).
• Magnitude: |A| =
q
A
2
x
+A
2
y
+A
2
z
.
• Dot Product: A·B = A
x
B
x
+A
y
B
y
+A
z
B
z
=|A||B|cos?.
• Cross Product:
A×B =







ˆ x ˆ y ˆ z
A
x
A
y
A
z
B
x
B
y
B
z







= (A
y
B
z
-A
z
B
y
)ˆ x+(A
z
B
x
-A
x
B
z
)ˆ y +(A
x
B
y
-A
y
B
x
)ˆ z
• Scalar Triple Product: A·(B×C) =







A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z







.
2. Coordinate Systems
2.1 Cartesian Coordinates
• Position Vector: r = xˆ x+yˆ y +zˆ z.
• Distance: r =
v
x
2
+y
2
+z
2
.
• Unit Vectors: ˆ x,ˆ y,ˆ z (orthogonal).
2.2 Cylindrical Coordinates
• Coordinates: (?,?,z ).
• Relation to Cartesian:
x = ?cos?, y = ?sin?, z = z
? =
q
x
2
+y
2
, ? = tan
-1

y
x

, z = z
• Unit Vectors: ˆ ?,
ˆ
?, ˆ z.
• Vector Representation: A = A
?
ˆ ?+A
? ˆ
? +A
z
ˆ z.
2.3 Spherical Coordinates
• Coordinates: (r,?,? ).
• Relation to Cartesian:
x = rsin?cos?, y = rsin?sin?, z = rcos?
r =
q
x
2
+y
2
+z
2
, ? = cos
-1
 
z
v
x
2
+y
2
+z
2
!
, ? = tan
-1

y
x

1
• Unit Vectors: ˆ r,
ˆ
?,
ˆ
? .
• Vector Representation: A = A
r
ˆ r +A
?
ˆ
? +A
? ˆ
? .
3. Di?erential Elements
• Cartesian:
dl = dxˆ x+dyˆ y +dzˆ z, dV = dxdydz
• Cylindrical:
dl = d?ˆ ?+?d?
ˆ
? +dzˆ z, dV = ?d?d?dz
• Spherical:
dl = drˆ r +rd?
ˆ
? +rsin?d?
ˆ
?, dV = r
2
sin?drd?d?
4. Vector Calculus
• Gradient (?f):
– Cartesian: ?f =
?f
?x
ˆ x+
?f
?y
ˆ y +
?f
?z
ˆ z.
– Cylindrical: ?f =
?f
??
ˆ ?+
1
?
?f
?? ˆ
? +
?f
?z
ˆ z.
– Spherical: ?f =
?f
?r
ˆ r +
1
r
?f
??
ˆ
? +
1
rsin?
?f
?? ˆ
? .
• Divergence (?·A):
– Cartesian: ?·A =
?Ax
?x
+
?Ay
?y
+
?Az
?z
.
– Cylindrical: ?·A =
1
?
?(?A?)
??
+
1
?
?A
? ?? +
?Az
?z
.
– Spherical: ?·A =
1
r
2
?(r
2
Ar)
?r
+
1
rsin?
?(sin?A
?
)
??
+
1
rsin?
?A
? ?? .
• Curl (?×A):
– Cartesian:
?×A =
 
?A
z
?y
-
?A
y
?z
!
ˆ x+
 
?A
x
?z
-
?A
z
?x
!
ˆ y +
 
?A
y
?x
-
?A
x
?y
!
ˆ z
– Cylindrical:
?×A =
 
1
?
?A
z
?? -
?A
? ?z
!
ˆ ?+
 
?A
?
?z
-
?A
z
??
!
ˆ
? +
1
?
 
?(?A
? )
??
-
?A
?
?? !
ˆ z
– Spherical:
?×A =
1
rsin?
 
?(sin?A
? )
??
-
?A
?
?? !
ˆ r+
1
r
 
1
sin?
?A
r
?? -
?(rA
? )
?r
!
ˆ
?+
1
r
 
?(rA
?
)
?r
-
?A
r
??
!
ˆ
? 2
5. Laplacian
• Scalar Laplacian (?
2
f):
– Cartesian: ?
2
f =
?
2
f
?x
2
+
?
2
f
?y
2
+
?
2
f
?z
2
.
– Cylindrical: ?
2
f =
1
?
?
??

?
?f
??

+
1
?
2
?
2
f
?? 2
+
?
2
f
?z
2
.
– Spherical: ?
2
f =
1
r
2
?
?r

r
2?f
?r

+
1
r
2
sin?
?
??

sin?
?f
??

+
1
r
2
sin
2
?
?
2
f
?? 2
.
6. Vector Identities
• ?·(?×A) = 0
• ?×(?f) = 0
• ?·(fA) = f(?·A)+A·(?f)
• ?×(fA) = f(?×A)+(?f)×A
7. Line, Surface, and Volume Integrals
• Line Integral:
R
C
A·dl.
• Surface Integral:
RR
S
A·dS.
• Volume Integral:
RRR
V
f dV.
8. Theorems
• Divergence Theorem:
ZZZ
V
?·AdV =
ZZ
S
A·dS
• Stokes Theorem: ZZ
S
(?×A)·dS =
I
C
A·dl
9. Coordinate Transformation
• Cartesian to Cylindrical Unit Vectors:
ˆ ? = cos? ˆ x+sin? ˆ y,
ˆ
? =-sin? ˆ x+cos? ˆ y, ˆ z = ˆ z
• Cartesian to Spherical Unit Vectors:
ˆ r = sin?cos? ˆ x+sin?sin? ˆ y +cos?ˆ z
ˆ
? = cos?cos? ˆ x+cos?sin? ˆ y-sin?ˆ z
ˆ
? =-sin? ˆ x+cos? ˆ y
3
10. Design Considerations
• Coordinate Choice: Use Cartesian for planar problems, cylindrical for radial
symmetry, spherical for point-source problems.
• Applications: Electric/magnetic?eldcalculations, chargedistributions, waveprop-
agation.
• Numerical Stability: Ensure proper scaling in cylindrical/spherical coordinates
to avoid singularities (? = 0, ? = 0,p).
4