x = Ω cos ' Ω = p x + y y = Ω ' ' = arctan (y/x) z = z z = z
x = Ω cos ' Ω = p x + y y = Ω ' ' = arctan (y/x) z = z z = z
x = Ω cos ' Ω = p x + y y = Ω ' ' = arctan (y/x) z = z z = z
\
]
3!,",]
!
"
X!
X"
X
]
[
\
]
3
!
"
G!
!G"
G]
!G"
G"
G"
U
x = cos =
x
2
+ y
2
y = = arctan(y/x)
z = z z = z
u
= cos u
x
+ u
y
u
= u
x
+ cos u
y
u
z
= u
z
r = u
+ zu
z
dr = dl
+ dl
+ dl
z
u
z
= du
+ du
+ dzu
z
dS
= dl
dl
z
= ddz ; dS
= dl
dl
z
= ddz ; dS
z
= dl
dl
= dd
d = dl
dl
dl
z
= dddz
[
\
]
3
U sin !
"
X
U
X"
X
!
!
U cos !
[
\
]
3
"
!
U sin ! G"
U
G!
G"
U G!
U sin ! G"
U
G"
GU
x = r cos r =
x
2
+ y
2
+ z
2
y = r = arctan(
x
2
+ y
2
/z)
z = r cos = arctan(y/x)
u
r
= cos u
x
+ u
y
+ cos u
z
u
= cos u
x
+ u
y
u
z
u
= u
x
+ cos u
y
r = ru
r
dr = dl
r
u
r
+ dl
+ dl
= dru
r
+ rdu
+ r du
dS
r
= dl
= r
2
dd; dS
= dl
r
dl
= r drd; dS
= dl
r
dl
= rdrd
d = dl
r
dl
= r
2
drdd
f = f(x, y, z) A(x, y, z) = A
x
(x, y, z)u
x
+A
y
(x, y, z)u
y
+
A
z
(x, y, z)u
z
f =
f
x
u
x
+
f
y
u
y
+
f
z
u
z
A =
A
x
x
+
A
y
y
+
A
z
z
A =
A
z
y
A
y
z
!
u
x
+
A
x
z
A
z
x
!
u
y
+
A
y
x
A
x
y
!
u
z
(f)
2
f =
2
f
x
2
+
2
f
y
2
+
2
f
z
2
f = f(, , z) A(, , z) = A
(, , z)u
+A
(, , z)u
+
A
z
(, , z)u
z
f =
f
+
1
+
f
z
u
z
A =
1
(A
+
1
+
A
z
z
A =
A
z
z
!
u
z
A
z
!
u
+
1
(A
!
u
z
2
f =
1
!
+
1
2
f
2
+
2
f
z
2
f = f(r, , ) A(r, , ) = A
r
(r, , )u
r
+ A
(r, , )u
+
A
(r, , )u
f =
f
r
u
r
+
1
r
f
+
1
r
f
A =
1
r
2
(r
2
A
r
)
r
+
1
r
( A
+
1
r
A
A =
1
r
( A
!
u
r
+
1
r
A
r
(rA
)
r
!
u
+
1
r
(rA
)
r
A
r
!
u
2
f =
1
r
2
r
2
f
r
!
+
1
r
2
!
+
1
r
2 2
2
f
2
A B C
A (BC) = B (CA) = (AB) C
A(BC) = B(A C) C(A B)
f = f(r) g = g(r) A = A(r) B = B(r)
(f + g) = f +g
(fg) = f(g) + g(f)
(A+B) = A+ B
(fA) = f( A) +A (f)
(AB) = B (A) A(B)
(A+B) = A+B
(fA) = f(A) A(f)
(A) = 0
(f) = 0
(A) = ( A)
2
A