Derivative of any function f(x,y,z): Differential Calculus (revisited): dz z f dy y f dx x f df l d f dz k dy j dx i z f k y f j x f i df ˆ ˆ ˆ ˆ ˆ ˆ z f k y f j x f i f where ˆ ˆ ˆ Gradient of function f
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Derivative of any function f(x,y,z): Differential Calculus (revisited): Gradient of function f.
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Derivative of any function f(x,y,z):
Differential Calculus (revisited):
dzzf
dyyf
dxxf
df
ldf
dzkdyjdxizf
kyf
jxf
idf
ˆˆˆˆˆˆ
zf
kyf
jxf
ifwhere
ˆˆˆ
Gradient of function f
Gradient of a function
Change in a scalar function f corresponding to a change in position dr
rdfdf
f is a VECTOR
Geometrical interpretation of GradientZ
X
Y
P Qdr
f
Czyxf ),,(
change in f : rdfdf
=0=> f dr
Z
X
Y
P
Q
dr
1Cf
12 CCf
rdfCCdf
12
For a given |dr|, the change in scalar function f(x,y,z) is maximum when:
frd
||
=> f is a vector along the direction of maximum rate of change of the
function
Magnitude: slope along this maximal direction
If f = 0 at some point (x0,y0,z0)
(x0,y0,z0) is a stationary point of f(x,y,z)
=> df = 0 for small displacements about the point (x0,y0,z0)
The Operator
zk
yj
xi
ˆˆˆ
is NOT a vector,
but a VECTOR OPERATORVECTOR OPERATOR
Satisfies: •Vector rules
•Partial differentiation rules
On a scalar function f : f
can act:
GRADIENT
On a vector function F as: . F DIVERGENCE
On a vector function F as: × F CURL
Divergence of a vector
zF
y
F
xF
F zyx
zyx FkFjFiz
ky
jx
iF ˆˆˆˆˆˆ
.F is a measure of how much the vector F spreads out (diverges) from
the point in question.
Divergence of a vector is a scalar.
Physical interpretation of Divergence
Flow of a compressible fluid:
(x,y,z) -> density of the fluid at a point (x,y,z)
v(x,y,z) -> velocity of the fluid at (x,y,z)
Z
X
Y
dy
dxdz
A
DC
B
E F
HG
dydzv xx 0| dydzv dxxx |
dydzdxvx
vx
xx0
(rate of flow in)EFGH
(rate of flow out)ABCD
Net rate of flow out (along- x)
dxdydzvx x
Net rate of flow out through all pairs of surfaces (per unit time):
dxdydzvz
vy
vx zyx
dxdydzv
Net rate of flow of the fluid per unit volume per unit time:
v
DIVERGENCE
Curl
zyx FFF
zyx ///
kji
F
yf
x
fk
xf
zf
jz
f
yf
i xyzxyz ˆˆˆ
Curl of a vector is a vector
×F is a measure of how much the vector F “curls around” the point in question.
Physical significance of Curl
Circulation of a fluid around a loop:
X
Y
00 ,yx
dyyx 00 ,
00 ,ydxx
dyydxx 00 ,
1
4 2
3
yyxx
yyxx
dVdV
dVdV
43
21
Circulation (1234)
))(,()(),(
),(),(
0000
0000
dyyxvdxdyyV
yxv
dydxx
Vyxvdxyxv
yx
x
yyx
dxdyyV
x
V xy
Circulation per unit area = ( × V )|z
z-component of CURL
Curvilinear coordinates:
used to describe systems with symmetry.
Spherical coordinates (r, , Ø)
Cartesian coordinates in terms of spherical coordinates:
cossinrx
sinsinry
cosrz
Spherical coordinates in terms of Cartesian coordinates:
222 zyxr
zyx /tan 221
xy /tan 1
Unit vectors in spherical coordinates
cosˆsinsinˆcossinˆˆ kjir
sinˆsincosˆcoscosˆˆ kji
cosˆsinˆˆ ji r
Z
X
Y
r
Line element in spherical coordinates:
drrddrrld sinˆˆˆ
Volume element in spherical coordinates:
dddrrd sin2
Area element in spherical coordinates:
rddrad ˆsin21
ˆ2 ddrrad
on a surface of a sphere (r const.)
on a surface lying in xy-plane ( const.)
Gradient:
f
rf
rrf
rfsin1ˆ1ˆˆ
F
rF
rFr
rrF r sin
1sin
sin11 2
2
Divergence:
Curl:
r
r
FrFrr
rFr
Fr
FFr
rF
1ˆ
sin11ˆ
sinsin1ˆ
Fundamental theorem for gradient
We know df = (f ).dl
The total change in f in going from a(x1,y1,z1) to b(x2,y2,z2) along any path:
afbfldfb
a
Line integral of gradient of a function is given by the value of the
function at the boundaries of the line.
Corollary 1:
tindependenpathisldfb
a
Corollary 2: 0 ldf
Field from Potential
b
a
ldEaVbV
.
ldVaVbVb
a
From the definition of potential:
From the fundamental theorem of gradient:
E = - V
ldEldVb
a
b
a
Electric DipolePotential at a point due to dipole:
2
0
cos4
1,
r
prV
z
y
x
pr
Electric Dipole
f
rf
rrf
rfsin1ˆ1ˆˆ
ˆsin4
1ˆ13
0 r
pVr
E
rr
pr
rV
Er ˆcos24
1ˆ3
0
E = - V
Recall:
0sin1
V
rE
Electric Dipole
ˆsinˆcos24 3
0
rr
pE
ˆsinˆcos prpp
Using:
]ˆ3[1
41
30
prrpr
rEdip
Fundamental theorem for Divergence
adFdF
The integral of divergence of a vector over a volume is equal to the value of
the function over the closed surface that bounds the volume.
Gauss’ theorem, Green’s theorem
Fundamental theorem for Curl
Stokes’ theorem
ldFadF
Integral of a curl of a vector over a surface is equal to the value of the
function over the closed boundary that encloses the surface.
THE DIRAC DELTA FUNCTION
? F
F
rF
rFr
rrF r sin
1sin
sin11 2
2
Recall:
2
ˆ
r
rFLet
0
dF
0112
22
r
rrr
F
The volume integral of F:
Surface integral of F over a sphere of radius R:
S
AdF
rddRR
r ˆsinˆ 22
4From divergence theorem:
S
AdFdF
4
From calculation of Divergence:
0
dF
By using the Divergence theorem:
4 dF
Note: as r 0; F ∞
0,0
;,0
rat
buteverywhereF
4 dF
And integral of F over any volume containing the point r = 0
The Dirac Delta Function(in one dimension)
0
00
xif
xifx
1
dxxand
Can be pictured as an infinitely high, infinitesimally narrow “spike” with area 1