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Useful Vector Equations and Identities
Dot (Scalar) Product: = cosBABA
Dist r ibut ive proper ty : CABA)CB(A +=+ Commutat ive property:
ABBA = Scal ing proper ty : )Bk(AB)Ak()BA(k ==
Cross (Vector) Product: nasinBABA =
Dist r ibut ive proper ty : CABA)CB(A +=+ Not Commutat ive: ABBA
= Scal ing proper ty : )Bk(AB)Ak()BA(k ==
Rectangular coordinate system:
Dot Product: zzyyxx BABABABA ++=
Cross Product:
zyx
zyx
zyx
BBBAAA
aaa
BA =
Magnitude of a vector : 2z2
y
2
x AAAAAA ++==
Differential Elements:
Length: zyx adzadyadxdl ++=
Volume: dzdydxdv =
Surfaces: xx adzdyds = yy adzdxds = zz adydxds =
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Cylindrical Coordinate System:
A point P(x,y,z) in the RCS system is given in the CCS system as
P(,,z )
where and z are the lengths and i s the angle as measured f rom
the x -axis.
= cosx , = siny , and zz =
22yx += and )x/y(tan 1=
Vector transformations one way:
=
zz
y
x
a
a
a
100
0cossin
0sincos
a
a
a
=
zZ
y
x
A
AA
100
0cossin
0sincos
A
AA
Vector transformations the other way:
=
z
y
x
z a
a
a
100
0cossin
0sincos
a
a
a
=
z
y
x
Z A
A
A
100
0cossin
0sincos
A
A
A
Differential elements:
Length: zadzadaddl ++=
Volume: dzdddv =
Surfaces: = adzdds = adzdds zz addds =
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Spherical Coordinate System:
A point P(x,y,z) in the RCS system is g iven in the SCS system
as P(r, ,)
where r i s the length, is measured from the posit ive z -ax is
and i s the angle
as measured from the x-axis.
Note that = cossinrx , = sinsinry , and = cosrz
222zyxr ++= , )x/y(tan 1= and )r/z(cos 1=
Vector transformations one way:
=
a
a
a
0sincos
cossincossinsin
sincoscoscossin
a
a
a r
z
y
x
=
A
A
A
0sincos
cossincossinsin
sincoscoscossin
A
A
A r
Z
y
x
Vector transformations the other way:
=
z
y
xr
a
a
a
0cossin
sinsincoscoscos
cossinsincossin
a
a
a
=
z
y
xr
A
A
A
0cossin
sinsincoscoscos
cossinsincossin
A
A
A
Differential elements:
Length:
++= adsinradradrdlr
Volume: = dddrsinrdv 2
Surfaces: r2
r addsinrds = = addrsinrds = addrrds
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Gradient of a scalar i s a vector:
RCS : zyx az
fa
y
fa
x
ff
+
+
=
CCS: zaz
fa
f1a
ff
+
+
=
SCS:
+
+
= a
f
sinr
1a
f
r
1a
r
ff r
The Divergence Theorem:
= SVdsFdvF
where S i s the sur face that complete ly encloses the volume V
.
Definition of divergence:
= s0v
dsFv
1limF
where s i s the sur face that complete ly encloses the d i f
ferent ia l vo lume v .
Divergence of a vector is a scalar quant i ty . I t is a measure
of net outward f low
of a f ie ld through a c losed sur face. I f the net outward f
low through a c losed
sur face is zero, the f ie ld is said to be continuous or
solenoidal. I t can be
computed at any point in space as fo l lows:
RCS :z
F
y
F
x
FF z
yx
+
+
=
CCS :z
FF1)F(1F z
+
+
=
SCS :( ) ( )
+
+
=
F
sinr
1Fsin
sinr
1
r
Fr
r
1F r
2
2
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Stokes' Theorem:
=CS
dFds)F(
where S i s the sur face that is complete ly enclosed by the
contour C .
Definition of Curl:
= c0s
dFs
1limF
where c i s the contour that complete ly encloses the d i f
ferent ia l sur face s .
Cur l is a measure of n et c i rculat ion of a vector f ie ld. I
f the net c i rculat ion around
a c losed path is zero, the f ie ld is said to be conservative
or irrotational .
Note that the cur l of a vector f ie ld is a lso a vector quant
i ty .
Cur l can be computed as fo l lows:
RCS :
zyx
zyx
FFF
zyx
aaa
F
=
CCS :
z
z
FFF
z
aaa
1F
=
SCS :
=
FsinrFrF
r
asinrara
sinr
1F
r
r
2
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The Laplacian of a scalar quantity is also a scalar
quantity.
I t is second order d i f ferent ia l operator .
Def in i t ion: ff2 =
Laplace's equation: 0f2 =
Poisson's equation: 0f2
I t can be computed as fo l lows:
RCS :2
2
2
2
2
22
z
f
y
f
x
ff
+
+
=
CCS :2
2
2
2
2
2
z
ff1f1f
+
+
=
SCS: 2
2
22
2
2
2 f
sinr
1fsinsinr
1
r
frrr
1f
+
+
=
Some Useful Vector Identities:
0f= (a)
0F = (b)
ff2 = (c)
FFF2
= (d)
( ) gfgf +=+ (e)
GF)GF( +=+ ( f)
GF)GF( +=+ (g)
gffg)gf( += (h)
fAAf)Af( += ( i )
AfAf)Af( += ( j )
BA)A(BBA = (k)
)BA(C)CA(B)CB(A = ( )
