Chapter One: Vector Analysis The use of vectors and vector analysis ...

Electromagnetic Theorem
(Dr. Omed Ghareb Abdullah) University of Sulaimani –College of Science – Physics Department
Then this vector is transformed to cylindrical coordinate as follows:
A ˆ ˆ ˆ ˆ ˆ ( ˆ ˆ
A a
A
x
( ax
a
) A
y
( a
y
a
) A
z
az
a
)
A aˆ
( ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ
A
A
x
ax
a
A
y
a
y
a
A
z
az
a
)
A A aˆ
A ( aˆ
aˆ
) A ( aˆ
aˆ
) A ( aˆ
aˆ
)
z
z
x
x
z
y
y
z
In matrix notation, we can write the transformation of vector (A) from A , A , A ) to
( A , A
, A
z
) as:
z
z
z
(
x y z
A
A
A
z
aˆ
a
ˆ
a
ˆ
x
x
x
aˆ
aˆ
aˆ
z
aˆ
aˆ
y
aˆ
y
y
aˆ
aˆ
aˆ
z
aˆ
z
aˆ
z
aˆ
z
aˆ
aˆ
aˆ
z
A
A
A
x
y
z
A
A
A
z
cos
sin
0
sin
cos
0
0
0
1
A
A
A
x
y
z
While when we have a vector ( A ) in cylindrical coordinate given by:
A( ,
,
z)
A aˆ
A aˆ
A ˆ , then this vector can be transformed to Cartesian
coordinate as:
A ˆ
x
A a
x
A
A ˆ
y
A a
y
A
A A aˆ
A
z
z
( aˆ
( aˆ
( aˆ
z
a z
aˆ
( ˆ
x
) A
a
aˆ
) ( ˆ
y
A
a
aˆ
) A ( aˆ
z
aˆ
x
) A
aˆ
y
) A
aˆ
) A
z
z
z
( aˆ
z
( aˆ
( aˆ
These equations in matrix notation can be written as:
z
z
z
aˆ
aˆ
aˆ
x
z
y
)
)
)
A
A
A
x
y
z
aˆ
a
ˆ
a
ˆ
aˆ
aˆ
aˆ
x
y
z
aˆ
aˆ
aˆ
aˆ
aˆ
aˆ
x
y
z
aˆ
aˆ
z
z
aˆ
z
aˆ
aˆ
aˆ
x
y
z
A
A
A
z
A
A
A
x
y
z
cos
sin
0
sin
cos
0
0
0
1
A
A
A
z
2. Cartesian to Spherical Transformation:
Point P in the figure has Cartesian coordinate ( x,
y,
z)
and spherical coordinate ( r , , )
.
The relation between the coordinates can be obtained as follows:
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