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
While when we have
a vector ( A ) in spherical coordinate given by:
A ( r , ,
) A ˆ
A
coordinate as:
A
x
A aˆ
x
A
A
y
A aˆ
y
A
A A aˆ
A
z
z
r
a r
r
r
r
( a
ˆ
(
a
ˆ
(
a
ˆ
r
r
r
â
A
These equations in matrix notation can be written as:
aˆ
x
) A
aˆ
y
) A
aˆ
) A
z
aâ , then this
vector can be transformed to Cartesian
( aˆ
ˆ
ax
) A ( ˆ ˆ
a
ax
)
( aˆ
aˆ
y
)
A ( ˆ ˆ
a
a
y
)
( aˆ
aˆ
) A ( aˆ
aˆ
)
z
z
A
A
A
aˆ
r
aˆ
a
ˆr
aˆ
a
ˆr
aˆ
x
y
z
x
y
z
aˆ
a
ˆ
aˆ
aˆ
aˆ
aˆ
x
y
z
aˆ
aˆ
x
aˆ
aˆ
y
aˆ
aˆ
z
A
A
A r
A
x
A
y
A
z
sin
cos
sin
sin
cos
cos
cos
cos
sin
sin
sin
cos
0
A
A
A r
3. Cylindrical to Spherical Transformation:
Point P in the figure has cylindrical coordinate ( ,
,
z)
and spherical coordinate ( r , ,
) .
The relation between
the coordinates can be
obtained as follows:
x
r sin
cos
y
r sin
sin
z
r cos
and
and
and
x cos
y
sin
z
z
r sin
z r cos
r
x
tan
tan
2
2
1
1
z
y
y
x
z
2
2