10.1 Differentiation of vector
Chapter 10 Vector Calculus
kdu
udaj
du
udai
du
uda
u
uauua
du
uad
uuua
kuajuaiuaua
zyx
u
zyx
ˆ)(ˆ)(ˆ)(
)()(lim
)(
point someat continuous is )(
ˆ)(ˆ)(ˆ)()(
scoordinate Cartesian In
0
0
)(ua
)( uua
)()( uauuaa
able.differenti be may
vector basis the general in fixed, are ˆ ,ˆ ,ˆ s,coordinate Cartesian in :Note
ˆˆˆˆˆˆ)()(
ˆˆˆˆˆˆ)(
ˆ)(ˆ)(ˆ)()(
:physics In
2
2
2
2
2
2
kji
kdt
zdj
dt
ydi
dt
xdk
dt
dvj
dt
dvi
dt
dv
dt
tvdta
kvjvivkdt
dzj
dt
dyi
dt
dx
dt
rdtv
ktzjtyitxtr
zyx
zyx
For two-dimensional plane polar coordinates
. and as changes direction
their but magnitudesconstant have ˆ and ˆ (2)
direction and magnitude inconstant are ˆ and ˆ (1)
ee
ji
Chapter 10 Vector Calculus
x
y
i
j
ee
ejdt
di
dt
d
dt
edjie
ejdt
di
dt
d
dt
edjie
ˆˆsinˆcosˆˆcosˆsinˆ
ˆˆcosˆsinˆˆsinˆcosˆ
)2(ˆ)(ˆ
)ˆˆ()()(
ˆˆˆ
ˆ)()(
onaccelerati and velocity the find ,ˆ)()(vector position :Ex
..2
..
ee
eedt
dtvta
eedt
ede
dt
dtrtv
ettr
Differentiation of composite vector expressions
Chapter 10 Vector Calculus
TFrFrvmv
vmdt
drvm
dt
rdvmr
dt
d
dt
Ld
vmrLFrT
bdu
ad
du
bdaba
du
d
bdu
ad
du
bdaba
du
d
adu
d
du
ada
du
d
)()(
is momentumangular the , torque :Ex
)( (3)
)( (2)
)( (1)
du
ada
du
adaaa
du
daaa
du
ds
ds
ad
du
sadusssaa
02)(constant a is if *
)()( and )(vector a if *
2
Chapter 10 Vector Calculus
10.2 Integration of vector
)()()(
vectorconstant a is )()()(
)(
12
2
1
uAuAduua
bbuAduuadu
uAdua
u
u
motion. the ofconstant a is
that show , ˆ ngravitatio of law sNewton' :Ex 2
2
dt
rdrr
r
GMm
dt
rdm
2
||||
2
1 rate changing its 2/|| is area in change the
0 interval time malinfinitesifor
vectorconstant a is
0)(
0ˆ
2
2
22
2
c
dt
rdr
dt
dArdrdA
rdrrdt
cdt
rdr
dt
rd
dt
rd
dt
rdr
dt
rdr
dt
d
rrr
GM
dt
rdr
Chapter 10 Vector Calculus
10.3 Space curves
dudu
rd
du
rds
du
rd
du
rd
du
ds
du
rd
du
rd
du
ds
dzdydxrdrddskdzjdyidxurd
kuzjuyiuxur
u
u
2
1
is length arc The
)(
)()()()(ˆˆˆ)(
ˆ)(ˆ)(ˆ)()(
2
2222
Ex: A curve lying in the xy-plane is given by y=y(x), z=0. Evaluate the arc length along the curve between x=a and x=b.
xudxdx
dys
du
dy
du
rd
du
rd
jdu
dyi
du
rdjuyiuurxu
b
a
for )(1)(1
ˆˆˆ)(ˆ)(Set
22
Chapter 10 Vector Calculus
point. given aat tovector
tangentunit a is ˆ curve, the along (s) length arc the is If
. increasing of direction the inpoint that at
totangent vector a is ,)( by described is curve a If
C
tds
rdu
u
Cdu
rdurC
tds
rd ˆ
)(ur
system. cordinate retangular handed-right a form ˆ and ˆ ,ˆ
vector binormal ˆˆˆ
vector normal principal the is ˆ ˆˆ
ˆˆ , 0
ˆˆ1|ˆ|
/1 curvature of radius The
|||ˆ
| torespect with
ˆvector tangent unit the of rate changing the : Curvature (1)
2
2
bnt
ntb
nnds
td
ds
tdt
ds
tdtt
ds
rd
ds
tds
t
z
x y
C
)(ur
n
bt
Chapter 10 Vector Calculus
ˆˆˆˆˆˆˆˆˆ
ˆˆ
ˆˆˆ (3)
torsion the of radius the 1
curve a of torsion the ˆ
ˆˆˆ
ˆˆ
ˆ and ˆ tolar perpendicu ˆ
ˆˆˆ
ˆˆˆˆ
ˆˆˆˆˆ
)ˆˆ(00ˆˆfor
ˆˆ 0
ˆˆ1|ˆ| (2)
tbnbtnds
tdbt
ds
bd
ds
nd
tbn
ds
bdnn
ds
bd
nds
bdtb
ds
bd
ds
bdtt
ds
bdnbt
ds
bd
ds
tdbt
ds
bdtb
ds
dtb
ds
bdb
ds
bdbb
Frenet-Serret
formula:n
ds
bdtb
ds
ndn
ds
tdˆ
ˆ ˆˆˆ
ˆˆ
Chapter 10 Vector Calculus
curvature of radius the : normal principal the :ˆ
trajectory the totangent unit the :ˆ particle the of speed the :
ˆˆ)( by given is )(
trajectory a along travelling a of onaccelerati thethat Show :Ex2
n
tv
nv
tdt
dvtatr
tdt
dv ˆ
nv
ˆ2
onaccelerati lcentripeta :ˆ
onaccelerati tangential :ˆ
ˆˆ)(ˆˆˆˆ
ˆˆ)ˆ()(
ˆˆ)(
2
2
nv
tdt
dv
nv
tdt
dvtan
vnv
ds
td
dt
ds
dt
td
dt
tdvt
dt
dvtv
dt
d
dt
vdta
tvtdt
ds
dt
ds
ds
rd
dt
rdtv
2
22
2
2
2
2
2
2
)(
])([)]([)(
parameter some is ),( curve aFor
dt
ud
du
rd
dt
du
du
rd
dt
ud
du
rd
dt
du
dt
du
du
rd
du
d
dt
ud
du
rd
dt
du
du
rd
dt
d
dt
du
du
rd
dt
d
dt
vda
dt
du
du
rd
dt
rdvuur
Chapter 10 Vector Calculus
10.4 Vector functions of several arguments
j
n
j jn
n
n
jn
j j
jj
i
jn
j ji
n
niii
niin
duu
adu
u
adu
u
adu
u
aad
uuu
v
u
u
a
v
a
dv
ad
vuuva
v
u
u
a
v
u
u
a
v
u
u
a
v
u
u
a
v
a
vvvuuuuuaa
12
21
1
21
1
1
2
2
1
1
2121
.....
,..., variablesdependent vector a of aldifferenti The (3)
)( scalars and of functionexplicit an is If (2)
......
),...,( of function a also is ,),.....,( If (1)
2
1
21
curve ofvector tangent the is (b)
curve ofvector tangent the is a)(
curves. coordinate called are and
is )( totangent vector The
))(),(()()( and )(
by drepresente be can surface
the on )( curve any ,parameter aFor
ˆ),(ˆˆ),(),( (4)
0),,( (3)
),( (2)
ˆ),(ˆ),(ˆ),(),( (1)
:is equation surface the coordinate Cartesian In
cvu
r
cuv
r
cvcud
dv
v
r
d
du
u
r
d
rdr
vurrgvfu
r
kvufjviuvuryxfz
zyxg
yxfz
kvuzjvuyivuxvur
10.5 Surface
Chapter 10 Vector Calculus
Chapter 10 Vector Calculus
RRdudvndudv
v
r
u
rdSA
dudvndudvv
r
u
rdv
v
rdu
u
rdS
dvv
rdu
u
rrd
rdv
r
u
rn
v
r
u
r
||||
area Total
||||||
ramparallelog linfitesima an is Pat area ofelement The
is nt displacemevector malinfinitesi an P, of oodneighbourh the In
isvector normal a S, surface smooth the on Ppoint aFor
P.point theat T planetangent the define to
vectors two the use can we t,independen linearly are and If
Chapter 10 Vector Calculus
Ex: Find the element of area on the surface of a sphere of radius a,
and hence calculate the total surface area of the sphere.
2
0
2
0
222
2
1
4sinsinsin||
0cossinsinsin
sinsincoscoscos
ˆˆˆ
vector Normal
ˆcossinˆsinsin
is curve the totangent vector The (2)
ˆsinˆsincosˆcoscos
is curve the totangent vector The (1)
ˆcosˆsinsinˆcossin),(
addaAddadSan
aa
aaa
kjirr
n
jaiar
c
kajaiar
c
kajaiar
Chapter 10 Vector Calculus
10.7 Vector operator
vectortangent unit the is ˆ ˆ
curve the along length arc the , If
parameter a is ,)( If
)ˆˆˆ()ˆˆˆ(
to from in change The
ˆˆˆ grad),,( fieldscalar aFor
coordinate Cartesian in ˆˆˆ del
ttds
rd
ds
d
sudu
rd
du
duurr
ddzz
dyy
dxx
dzkdyjdxiz
ky
jx
ird
rdrr
zk
yj
xizyx
zk
yj
xi
Gradient of a scalar field
Chapter 10 Vector Calculus
||)()0(ˆ||
||||cos||ˆ
is vector particular a in distance
the torespect with of rate changing The
max
ds
da
ds
da
ds
d
as
Ex: For a function at a point (1,2-1), find its rate of change with distance in the direction . At the same point, what is the greatest possible rate of change with distance and in which direction does it occur?
yzyx 2kjia ˆ3ˆ2ˆ
20||)(ˆ2ˆ4 direction the In (2)
14
10)64(
14
1ˆ )ˆ3ˆ2ˆ(
14
1
||ˆ )1(
)1,2,1(point at ˆ2ˆ4ˆˆ)(ˆ2
max
2
ds
dki
ads
dkji
a
aa
kikyjzxixy
)( to from field electric in change the :Ex
ˆ
operator aldifferentiscalar the by found be could ˆ of direction
the in distance with field scalar)(or vector a of change of rate The
ErdEdrdrr
za
ya
xaa
a
zyx
Chapter 10 Vector Calculus
.derivative normal called is ,ˆ along rate changing || (3)
points. everyat ),,( surface the tovector normal a is ˆ (2)
point someat surface this tovector tangent a is ˆ (1)
ˆ0ˆ
)ˆˆˆ()ˆˆˆ(
and 0(constant) ),,( If
nn
czyxnn
t
ttds
rd
ds
dzk
ds
dyj
ds
dxi
zk
yj
xi
ds
dz
zds
dy
yds
dx
xds
dds
dczyx
yxjaxiay
kakazjyixkarr
azazakarr
akakzjyix
azyx
nrrn
r
nrr
n
kzjyixr
zyx
axis z the is 00ˆ2ˆ2
0ˆ2)ˆ)(ˆˆ(0ˆ2)(
a)(0,0,at sphere the normal line The )4(
0)(20ˆ2)( is planetangent The
)),0,0((at ˆ2ˆ2ˆ2ˆ2
surface theFor (3)
0)(
along P through passing linestraight the on is If (2)
0)( surfacetangent The
|vector Normal
ˆˆˆat PPoint (1)
0
0
2222
000
00
0,,
000
000
Chapter 10 Vector Calculus
Ex: Find the expression for the equations of the tangent plane and the line normal to the surface at the point P with the coordinates . Use the results to find the equations of the tangent plane and the line normal to the surface of the sphere at the point .
czyx ),,(000 , , zyx
2222 azyx ),0,0( a
z
yx
0n az
a
),0,0( a
Chapter 10 Vector Calculus
Divergence of a vector field
volumeunit per fluid of outflow of ratenet the is
fluid a in velocity local a is ),,( :Ex
Solenoidal 0 If
V
zyxV
a
z
a
y
a
x
aaadiv zyx
Scalar differential operator
zxyxz
zxy
zyx
232
32
2
2
2
2
2
2
22
62
fieldscalar the of Laplacian the Find :Ex
of Laplacian the
Curl of a vector
]ˆˆ)(ˆ[2
)20(ˆ)22(ˆ)20(ˆ
ˆˆˆ
ˆˆˆ :Ex
alIrrotation 0 If
ˆˆˆ
ˆ)(ˆ)(ˆ)(
222222
222222
2222222
2222222
kyzxjzyxxzizy-
zyxkyzxxzjzyi
zxzyzyxzyx
kji
a
kzxjzyizyxa
a
aaazyx
kji
ky
a
x
aj
x
a
z
ai
z
a
y
aaacurl
zyx
xyxxyz
Chapter 10 Vector Calculus
Chapter 10 Vector Calculus
0
0 wheel paddle small A
point.that of odneighborho the in fluid the of velocityangular
the to related is fluid, a in velocity local the is ),,( If
vrotatedoesnot
vrotate
vzyxv
2ˆ2
0
ˆˆˆ
ˆˆ)ˆˆˆ()ˆ(
. then ,ˆ andvector position the is If :Ex
k
xyzyx
kji
v
iyjxkzjyixkv
rvkr
Useful formulas:
Chapter 10 Vector Calculus
baababbaba
aaa
baabba
aaa
abbaabbaba
baba
baba
)()()()()( (9)
)( (8)
)()()( (7)
)( (6)
)()()()()( (5)
)( (4)
)( (3)
)( (2)
)( (1)
10.8 Vector operator formula
Chapter 10 Vector Calculus
kk
jji
iijkijkjji
i
ijjiji
ijkijkjji
ik
aa
aa
aaaa
aaa
)()(
)(
][)()]([
)(that Show :Ex
,,
,,
Useful special cases:
rdr
d
r
kzjyix
dr
d
kz
r
dr
dj
y
r
dr
di
x
r
dr
dk
zj
yi
xr
rdr
dr
zyxrrkzjyixr
ˆ)ˆˆˆ(
ˆˆˆˆˆˆ)(
ˆ)( 1)(
)(|| vector position a is ˆˆˆ 2/1222
Chapter 10 Vector Calculus
dr
d
r
zyx
dr
d
r
z
z
r
dr
d
r
y
y
r
dr
d
r
x
x
r
dr
d
r
z
dr
d
zr
y
dr
d
yr
x
dr
d
xr
dr
ddr
rd
dr
rd
rr
dr
dr
dr
dr
dr
drr
rrr
zyx
rr
r
x
rzyxx
x
zyx
z
zyx
y
yzyx
x
xr
dr
rdrrr
dr
dr
z
z
y
y
x
x
rrrr
2
2
2
222
2
2
2
2
2
2
2
2
2
22
3
222
3
22/1222
222222222
)(
)()()(
)()()(ˆ)( :term 2nd thefor
)()(2ˆ)()ˆ()ˆ())(()( (4)
213
r
3ˆ
similar are termsanother 1
])([ :term1st thefor
)()()(ˆ (3)
)()(3ˆ)(
])([ (2)
Chapter 10 Vector Calculus
rkzjyixrr
zk
r
yj
r
xi
z
rk
y
rj
x
rir
dr
d
r
yx
dr
d
r
xyk
dr
d
r
zx
dr
d
r
xzj
dr
d
r
zy
dr
d
r
yzir
dr
d
r
z
zdr
d
r
y
ydr
d
r
x
x
r
dr
d
x
yx
xyk
zx
xzj
zy
yzi
zyxzyx
kji
r
y
x
x
yk
z
x
x
zj
z
y
y
zi
zyxzyx
kji
r
rrrr
ˆ)ˆˆˆ(1ˆˆˆˆˆˆ (6)
0)(ˆ)(ˆ)(ˆ
for
)(ˆ)(ˆ)(ˆ
ˆˆˆ
(b)
0)(ˆ)(ˆ)(ˆ
ˆˆˆ
(a)
0])([ (5)
Chapter 10 Vector Calculus
],[ if 0
],[ if )()()(
for 0)(
function delta Dirac is )(
)(4)1
()1
()ˆ
( (9)
ˆ)
1( (8)
3 (7)
0
000
00
22
2
bax
baxxfdxxxxf
xxxx
r
rrrr
rr
r
r
r
b
a
Combinations of grad, div, and curl
Chapter 10 Vector Calculus
solenoidal is 0)( (4)
alirrotation is 0)( (3)
0///
///
)(
0)( (2)
0)(ˆ)(ˆ)(ˆ
///
///
ˆˆˆ
)(
0)( (1)
fieldvector : fieldscalar :
222222
babab
aaa
aaa
zyx
zyx
a
a
xyyxk
zxxzj
yzzyi
zyx
zyx
kji
a
zyx
�
Chapter 10 Vector Calculus
0)()()( (8)
them. onact can so ,
constantnot are vectorsunit the s,coordinate spherical and lcylindrica In
them. onact not does
operator the so constant, arevector unit the s,coordinate Cartesian In :
)ˆˆˆ)((
)()( (7)
)(ˆ)(ˆ)(ˆ
))(ˆˆˆ()( (6)
)( (5)
2
2
2
2
2
2
2
22
2
2
2222
2
2222
2
2
2
2
2
2
2
22
Note
kajaiazyx
a
aaa
z
a
yz
a
xz
ak
zy
a
y
a
xy
aj
zx
a
yx
a
x
ai
z
a
y
a
x
a
zk
yj
xia
zyx
zyx
zyxzyxzyx
zyx
10.9 Cylindrical and spherical polar coordinates
Chapter 10 Vector Calculus
A. Cylindrical polar coordinates
zz
zz
edzedededzeded
dzz
rd
rd
rrd
kekz
re
jiejir
e
jiejir
e
kzjir
z
zzyx
ˆˆˆ
ˆˆ ˆ
ˆcosˆsinˆ ˆcosˆsin
ˆsinˆcosˆ ˆsinˆcos
ˆˆsinˆcos:Ppoint for
20 0
sin cos
ly.respective ,ˆ and ,ˆ ,ˆ directions along
1 and , ,1 are factors Scale
is ˆ (3) is ˆ (2) is ˆ (1)
along distance the of change The
),,( to ),,( from Position
z
z
z
eee
hhh
dzedede
dzzddz
Chapter 10 Vector Calculus
z
aaaa
ez
ee
eaeaeaaz
dzddedzededdV
dddA
dzddrdrddsds
z
z
zz
z
1)(
1 (2)
ˆˆ1
ˆ )1(
ˆˆˆ fieldvector a and ),,,( fieldscalar aFor
|)ˆˆ(ˆ|
iselement volume The
is plane y-x in area The
])()()[()()( 2/122222/12
Chapter 10 Vector Calculus
.vector intoput are above The
sin cos
ˆˆ ˆcosˆsinˆ ˆsinˆcosˆ
ˆˆ ˆcosˆsinˆ ˆsinˆcosˆ
scoordinate lcylindrica in expressed ˆˆˆ :Ex
1)(
1 (4)
ˆˆˆ
1 (3)
2
2
2
2
2
22
a
zzyx
ek eejeei
kejiejie
kxzjyiyza
z
aaaz
eee
a
z
z
z
z
Chapter 10 Vector Calculus
B. Spherical polar coordinates
jieee
kjieee
kjiee
reree
jrirr
e
kjrirr
e
kjir
re
krjrirr
r
rzryrx
rr
r
r
ˆcosˆsin||/ˆ
ˆsinˆsincosˆcoscos||/ˆ
ˆcosˆsinsinˆcossinˆ
sin|| || 1||
ˆcossinˆsinsin
ˆsinˆsincosˆcoscos
ˆcosˆsinsinˆcossin
ˆcosˆsinsinˆcossin
20 0 0
cos sinsin cossin
ddrdredrerdedrdV
ddaeadedadA
drdrdr
rdrddsd
edrerdedr
edededr
dr
dr
drr
rrd
r
r
r
sin|)ˆsinˆ(ˆ|
:element volume The
sin|ˆˆsin|
a radius a with sphere a of surface the onelement area The
)(sin)()(
)(s
:ntdisplaceme of Magnitude
ˆsinˆˆ
:ntDisplaceme
2
2
222222
2
Chapter 10 Vector Calculus
Chapter 10 Vector Calculus
2
2
2222
22
2
22
sin
1)(sin
sin
1)(
1 (4)
sin
ˆˆˆ
sin
1 (3)
sin
1)(sin
sin
1)(
1 (2)
ˆsin
1ˆ
1ˆ (1)
ˆˆˆ fieldvector a ,),,( fieldscalar A
rrrr
rr
arraar
eee
ra
a
ra
rar
rra
er
er
er
eaeaeaar
r
r
r
r
rr
Chapter 10 Vector Calculus
10.10 General curvilinear coordinate
|| || || :factors Scale
1
ˆ 1
ˆ 1
ˆ :vectorsUnit
|| || ||
Pat curve- totangent vector a
Pat curve- totangent vector a
Pat curve- totangent vector a
Ppoint at vector position the is ),,(
),,( ),,( ),,(
),,( ),,( ),,(
33
22
11
333
222
111
332211
33
3
22
2
11
1
321
332211
321321321
u
rh
u
rh
u
rh
u
r
he
u
r
he
u
r
he
hehehe
uu
re
uu
re
uu
re
uuur
zyxuuzyxuuzyxuu
uuuzzuuuyyuuuxx
Chapter 10 Vector Calculus
333222111
332211
33
22
11
1
ˆˆˆ
sin 1 :coordinate Spherical
1 1 :coordinate lCylindrica
is changing ofelement distance The
eduheduheduh
edueduedu
duu
rdu
u
rdu
u
rrd
rhrhh
hhh
duhdu
r
z
ii
321321333222111
332211
23
23
22
22
21
21
2
|)ˆˆ(ˆ|
|)(| iselement volume The
3,2,1for ˆ)(
vector the by defined pedparallelpi malinfinitesi The
)()()()(
is length arc ofelement the lar,perpendicu
mutually are ˆ s,coordinater curvilinea orthogonalFor
dududuhhheduheduheduh
edueduedudV
ieduheduduu
r
duhduhduhrdrdds
e
iiiiiii
i
Chapter 10 Vector Calculus
3322113
32
21
1
332211332211
333
33
222
22
111
11
:vectors of sets Two
surface the to normal ||
ˆ
surface the to normal ||
ˆ
surface the to normal ||
ˆ
:vectorsunit three usefulAnother
uuuu
r
u
r
u
r
eeea
u
u
re
cuu
u
cuu
u
cuu
u
ii
ii
Chapter 10 Vector Calculus
vectors of system reciprocal are }{ and }{ iie
otherwise 0
if 1
)ˆˆˆ()ˆˆˆ(
jie
u
u
z
u
u
z
y
u
u
y
x
u
u
x
kz
uj
y
ui
x
uk
u
zj
u
yi
u
x
uu
re
ji
i
jj
i
j
i
j
i
jjj
iii
ji
ji
Chapter 10 Vector Calculus
Gradient
333
222
111
332211
33
22
11
ˆ1
ˆ1
ˆ1
)ˆˆˆ(
euh
euh
euh
kduhj duhiduhrd
duu
duu
duu
d
Divergence
)]()()([1
ˆˆˆ
3213
2132
1321321
332211
ahhu
ahhu
ahhuhhh
a
eaeaeaa
Proof:
)()()(
)()ˆ(
ˆˆˆ
3232132321
3232111
3322321
uuhhauuhha
uuhhaea
uhuheee
Chapter 10 Vector Calculus
)(1
)ˆ( and )(1
)ˆ(
)ˆ( and )ˆ( for same the
)(1
ˆ)()
ˆˆ()()ˆ(
0)(0)( 10.43 eq. From
3213321
332132321
22
3322
3211321
32
1321
3
3
2
232111
32
ahhuhhh
eaahhuhhh
ea
eaea
hhauhhh
hh
ehha
h
e
h
ehhaea
uu
Laplacian
)]()()([1
ˆ1
ˆ1
ˆ1
33
21
322
13
211
32
1321
2
333
222
111
uh
hh
uuh
hh
uuh
hh
uhhha
euh
euh
euh
a
Chapter 10 Vector Calculus
Curl
332211
321
332211
321332211
ˆˆˆ1
ˆˆˆ
ahahahuuu
eheheh
hhhaeaeaeaa
Proof:
)ˆ( and )ˆ(for same the
)(ˆ
)(ˆ
ˆ]ˆ)(
1ˆ)(
1ˆ)(
1[
ˆ)(
)()()ˆ(
3322
11221
311
313
2
1
1311
33211
22111
11
1
111
11111111111
eaea
hauhh
eha
uhh
e
h
eeha
uheha
uheha
uh
h
eha
uhauhauhaea