formula
DESCRIPTION
FormulaTRANSCRIPT
-
7
Formula
Gradient
R
zr
zyx
sin
11
1
f
R
f
RR
ff
z
ff
rr
ff
z
f
y
f
x
ff
Divergence
A
R
A
RR
AR
RA
z
AA
rr
rA
rA
z
A
y
A
x
AA
R
zr
zyx
sin
1sin
sin
11
11
2
2
Curl
R
zr
zyx
1
sin
11 sin
sin
1
1
1
RR
rzrz
xyzxyz
A
R
RA
RR
RAA
R
AA
RA
A
r
rA
rr
A
z
A
z
AA
rA
y
A
x
A
x
A
z
A
z
A
y
AA
Laplacian
2
2
222
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
sin
1sin
sin
11
11
f
R
f
RR
fR
RRf
z
ff
rr
fr
rrf
z
f
y
f
x
ff
-
8
Cartesian Cylindrical Spherical
Coordinate
parameters x, y, z r, , z R, ,
Vector A
zyx zyx AAA zr zr AAA R AAAR
Magnitude A
222
zyx AAA 222
zr AAA 222
AAAR
Position
vector, OP
zyx 111 zyx
for point P( 1x , 1y , 1z )
zr 11 zr
for point P( 1r , 1 , 1z )
R1R
for point P( 1R , 1 , 1 )
Unit vector
product
yxz
xzy
zyx
xzzyyx
zzyyxx
0
1
rz
rz
zr
rzzr
zzrr
0
1
R
R
R
RR
RR
0
1
Dot product
BA
zzyyxx BABABA zzrr BABABA BABABA RR
Cross product
BA
zyx
zyx
BBB
AAA
zyx
zr
zr
BBB
AAA
zr
BBB
AAA
R
R
R
Differential
length, d zyx dzdydx zr dzrddr R sin dRRddR
Differential
surface, ds z
y
x
dydxds
dzdxds
dzdyds
z
y
x
z
r
drdrds
dzdrds
dzrdds
z
r
R
sin
sin2
ddRRds
ddRRds
ddRds R
Differential
volume, dv dzdydx dzddrr dddRR sin2
-
9
Transformation Coordinate Variables Unit Vectors Vector Components
Cartesian to
Cylindrical
zz
xy
yxr
)/(tan 1
22
zz
yx
yxr
cossin
sincos
zz
yx
yxr
AA
AAA
AAA
cossin
sincos
Cylindrical to
Cartesian
zz
ry
rx
sin
cos
zz
ry
rx
cossin
sincos
zz
ry
rx
AA
AAA
AAA
cossin
sincos
Cartesian to
Spherical
)/(tan
)/(tan
1
221
222
xy
zyx
zyxR
cossin
sinsincos
coscos
cossinsin
cossin
yx
zy
x
zy
xR
cossin
sinsincos
coscos
cossinsin
cossin
yx
zy
x
zy
xR
AAA
AA
AA
AA
AA
Spherical to
Cartesian
cos
sinsin
cossin
Rz
Ry
Rx
sincos
cossincos
sinsin
sincoscos
cossin
Rz
Ry
Rx
sincos
cossincos
sinsin
sincoscos
cossin
AAA
AA
AA
AA
AA
Rz
Ry
Rx
Cylindrical to
Spherical
)/(tan 1
22
zr
zrR
zr
zrR
sincos
cossin
AA
AAA
AAA
zr
zrR
sincos
cossin
Spherical to
Cylindrical
cos
sin
Rz
Rr
sincos
cossin
Rz
Rr
sincos
cossin
AAA
AA
AAA
Rz
Rr
-
10
dSJI
SR
V
VE
E
dE
r
dV
r
QV
Q
WdEV
D
SdDQ
SdD
ED
aR
dvE
aR
dSE
aR
dE
aR
QE
,Q
FE
aR
QQF
dvQ
,dSQ
,dQ
B
A
AB
v
Senc
e
Rv
R
s
R
R
R
v
s
0
0
0
4
4
4
4
4
4
4
2
2
0
2
0
2
0
2
0
2
0
2112 12
112
112
4
0
0
4
2
2
0
2
0
3
j
t
DJ,Sd.JI
dBuV
Sdt
BV
tV
aISm
BmFrT
BIdFd
dt
udmBuEQF
JA
R
IdA
AB
HB
B
dA
SdB
SdB
JH
dSJIdH
dvJdSJId
R
RIdHd
ddd
emf
emf
emf
n
m
m
s
m
senc
s
22
2122
22
22
1
22
22
2122
21222322
212222322
17
0
112
0
4
12
1 2
2
21
21211
2
1
1
1
Hm 104
Fm 108548
1
2
1
4
21
cxcx
xdx
cxlncx
xdx
c
xtan
ccx
dx
cxxlncx
dx
cxcx
xdx
cxc
x
cx
dx
.
J
Jtan
tan
R
addIIF
ds
s
L L
R