Vector Notation:
Tr
q ""
scaler quantity (magnitude only)
vector quantity (magnitude and direction)
Cartesian Coordinates:
r
q "" = q"x" iˆ + q"y" ˆj + q"z" kˆ
Derivative (d):
dT =
"T
"T
"T
dx +
dy +
dz
"x
"y
"z
"T =
#T ˆ #T ˆ # T ˆ
i+
j+
k
#x
#y
#z
!
!
Gradient (∇):
!
" # ( k "T ) =
Divergence ( " # ):
!
2
Laplacian ( " = " # " ):
!
!
!
Cylindrical
Coordinates:
!
Derivative (d):
" 2T =
# 2T # 2T # 2T
+
+
#x 2 #y 2 #z 2
r
q "" = q"r" iˆ + q#"" ˆj + q"z" kˆ
dT =
"T
1 "T
"T
dr +
d# +
dz
"r
r "#
"z
"T =
#T ˆ 1 #T ˆ # T ˆ
i+
j+
k
#r
r #$
#z
!
Gradient (∇):
$ % $T ( $ % $T ( $ % $T (
'k * + 'k
* + 'k *
$x & $x ) $ y & $y ) $z & $z )
!
Divergence ( " # ):
" # ( k "T ) =
!
Laplacian ( " 2 = " # " ):
!
!
" 2T =
1 $ % $T ( 1 $ % $T ( $ % $T (
'k r * + 2 'k
* + 'k *
r $r &
$r ) r $+ & $+ ) $z & $z )
1 # $ #T ' 1 # 2T # 2T
+
&r ) +
r #r % #r ( r 2 #* 2 #z 2
!
Sperical
Coordinates:
!
r
q "" = q"r" iˆ + q#"" ˆj + q$"" kˆ
Derivative (d):
dT =
"T
1 "T
1 "T
dr +
d# +
d$
"r
r "#
r sin # "$
"T =
#T ˆ 1 #T ˆ
1 #T ˆ
i+
j+
k
#r
r #$
r sin $ #%
!
Gradient (∇):
!
" # ( k "T ) =
Divergence ( " # ):
!
Laplacian ( " 2 = " # " ):
!
!
!
!
1 $ % 2 $T (
1
$ %
$T (
1
$ % $T (
k
r
+
k
sin
+
+
'
*
'
*
'k
*
r 2 $r &
$r ) r 2 sin + $+ &
$+ ) r 2 sin 2 + $, & $, )
" 2T =
1 # $ 2 #T '
1
# $
#T '
1
# 2T
r
+
sin
*
+
&
)
&
)
r 2 #r % #r ( r 2 sin * #* %
#* ( r 2 sin 2 * #+ 2