Generalized Coordinates
Cartesian coordinates ri , i = 1, 2, . . . D for Euclidean
space.
X
Distance by Pythagoras: (δs)2 =
(δri )2 .
iP
Unit vectors êi , displacement ∆~r = i ∆ri êi
Fields are functions of position, or of ~r or of {ri }.
~ (~r)
Scalar fields Φ(~r),
Vector fields V
X ∂Φ
~
êi ,
∇Φ
=
∂ri
i
X ∂Vi
~ ·V
~ =
∇
,
∂ri
i
X ∂2Φ
~ · ∇Φ
~
,
∇2 Φ = ∇
=
∂ri 2
i
X
∂Vk
~ ×V
~ =
∇
ijk j êi ,
∂r
ijk
3D only
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Other smooth coordinatization q i , i = 1, . . . D
q i (~r) and ~r({q i }) are well defined (in some domain)
mostly 1—1, so Jacobian det(∂q i /∂rj ) 6= 0.
Distance between P {q i } and P 0 {q i + δq i } is given
by

!
X
X X ∂rk
X ∂rk
(δs)2 =
(δrk )2 =
δq i 
δq j 
∂q i
∂q j
i
j
k
k
X
=
gij δq i δq j ,
ij
where
gij =
X ∂rk ∂rk
k
∂q i ∂q j
.
gij is a real symmetric matrix called the metric tensor.
In general a nontrivial function of the position, gij (q).
To repeat:
X
(δs)2 =
gij δq i δq j .
ij
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Functions (fields)
A scalar field f (P ) f (~r) can also be specified by a
function of the q’s, f˜(q) = f (~r(q)).
~ (~r) has a meaning
What about vector fields? V
independent of the coordinates used to describe it, but
components depend on the basis vectors. Should have
basis vectors ẽi aligned with the direction of q i . How to
define?
Consider
~r(q 1 + δq 1 , q 2 , q 3 ) − ~r(q 1 , q 2 , q 3 ) X ∂rk êk
=
ẽ1 = lim
√
δs
∂q 1 g11
δq 1 →0
k
and the similarly defined ẽ2 and ẽ3 . In general not good
orthonormal bases, because
ẽ1 · ẽ2 =
X ∂rk ∂rk √
√
/ g11 g22 = g12 / g11 g22 ,
1
2
∂q ∂q
k
which need not be zero.
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Another problem: ẽk is not normal to the surface q k =
constant.
What to do? Two approaches:
I
Shapiro
Give up on orthonormal basis vectors. Define
X ∂Φ
dxk . The
differential forms, such as dΦ =
∂xk
k
dxk
coefficients of
transform as covariant vectors.
Contravariant vectors may be considered coefficients
of directional derivatives ∂/∂q k . This is the favored
approach for working in curved spaces, differential
geometry and general relativity.
Restrict ourselves to orthogonal coordinate systems
— surface q i = constant intersects q j = constant at
~ i · ∇q
~ j = 0.
right angles. Then ∇q
X ∂q i ∂q j
~ i · ∇q
~ j=
.
In general define
g ij := ∇q
∂rk ∂rk
I
k
Note that g ij is not the same as gij .
Physics 504,
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Electricity
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Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
In fact
X X ∂q i ∂q ` X ∂rm ∂rm X ∂q i
X
∂rm
g i` g`j =
δ
=
km
∂q j
∂rk ∂rk m ∂q ` ∂q j
∂rk
`
= δij , so
`
g ..
k
km
Shapiro
is the inverse matrix to g.. .
~ i · ∇q
~ j = 0 for i 6= j, g ij = 0 for i 6= j, g .. is diagonal,
If ∇q
so g.. is also diagonal. And as (δs)2 > 0 for any non-zero
δ~r, g.. is positive definite, so for an orthogonal coordinate
system the diagonal elements are positive, gij = h2i δij ,
and g ij = h−2
i δij . Then the unit vectors are
ẽi = h−1
i
X
k
êk
X
∂rk
=:
Bki êk
∂q i
Physics 504,
Spring 2010
Electricity
and
Magnetism
(1)
k
with the inverse relation
X
X
X
X
∂q i
êk =
hi ẽi k =:
Aki ẽi =
Aki
B`i ê` .
(2)
∂r
i
i
i
`
P
As the êk are independent, this implies i Aki B`i = δk`
or AB T = 1I.
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
The set {êP
k } and the set {ẽi } are each orthonormal,
and êk = i Aki ẽi , so A is orthogonal, and ∴ B = A.
Can also check as
X
X
Aki Akj = hi hj
(∂q i /∂rk )(∂q j /∂rk ) = hi hj g ij = δij .
k
k
Thus A can be written two ways,
Aki = hi
k
∂q i
−1 ∂r
=
h
.
i
∂q i
∂rk
Note that Ajk is a function of position, not a constant. In
Euclidean space we say that êx is the same vector
1
regardless
X of which point ~r the vector is at . But then
ẽi =
Aji (P )êj is not the same vector at different
j
points P .
1
That is, Euclidean space comes with a prescribed parallel
transport, telling how to move a vector without changing it.
Physics 504,
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Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Vector Fields
So for a vector field
X
X
X
~ (P ) =
V
Ṽj (q)ẽj =
Vi (~r)êi =
Vi (~r)Aik ẽk ,
j
so Ṽk (q) =
i
Physics 504,
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Electricity
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Shapiro
ik
Generalized
Coordinates
X
Vi (~r(q))Aik (q).
iX
Also Vk (~r) =
Aki (~r)Ṽi (q(~r)).
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
i
Let’s summarize some of our previous relations:
X
X
êk =
Aki ẽi ,
ẽi =
Aji (P )êj
i
Aki = hi
j
∂rk
∂q i
= h−1
.
i
k
∂q i
∂r
gij = h2i δij ,
g ij = h−2
i δij
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Gradient of a scalar field
~
∇f
Physics 504,
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Electricity
and
Magnetism
!
X ∂ f˜ ∂q `
X ∂f
êk =
(Akm ẽm )
=
∂rk
∂q ` ∂rk
k`m
k
!
X ˜
X ∂ f˜ ∂q ` ∂q m
∂f
=
hm ẽm k =
hm ẽm g m`
`
k
∂q ∂r
∂r
∂q `
k`m
`m
`m
or
=
X
m
h−1
m
∂ f˜
ẽm .
∂q m
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
X
X ∂ f˜
∂ f˜
h−1
=
hm ẽm h−2
ẽm ,
m
m δm` =
`
∂q m
∂q
m
~
∇f
Shapiro
(3)
Both f (~r) and f˜(q) represent the same function f (P ) on
space, so we often carelessly leave out the twiddle.
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Velocity of a particle
~v =
X drk
k
=
X X ∂rk dq i
êk =
dt
∂q i dt
ij
i
k
X dq i
dt
hj δij ẽj =
X
j
hj
dq j
dt

!
X ∂q j

hj k ẽj 
∂r
j
ẽj .
Note it is hj dq j which has the right dimensions for an
infinitesimal length, while dq j by itself might not.
Example, spherical coordinates.

spherical shells
 r
θ
cone, vertex at 0
Surfaces of constant

φ
plane containing z
with the shells centered at the origin. These intersect at
right angles, so they are orthogonal coords.
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Physics 504,
Spring 2010
Electricity
and
Magnetism
By looking at distances from varying one coordinate,
comparing to (ds)2 = h2r (dr)2 + h2θ (dθ)2 + h2φ (dφ)2 ,
we see that
hr = 1,
hθ = r,
hφ = r sin θ.
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Thus
~v = ṙẽr + rθ̇ẽθ + r sin θφ̇ẽφ
and
v 2 = ṙ2 + r2 θ̇2 + r2 sin2 θφ̇2 .
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Derivatives of Vectors
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
∂êi
Euclidean parallel transport:
=0
∂rj
h−1
j
X ∂rk ∂
∂
−1
ẽ
=
h
i
j
∂q j
∂q j ∂rk
k`
= h−1
j
X
k`
`
−1 ∂r
hi ê` i
∂q
Akj
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
X ∂rk ∂A`i
k`
=
Generalized
Coordinates
∂q j ∂rk
∂A`i
ê` .
∂rk
ê`
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Differential Forms
A general 1-form over a space coordinatized by q i is
X
ω=
Ai (P )dq i ,
and is associated with a vector. But if we are using
orthogonal curvilinear coordinates, it is more natural to
express the coefficients as multiplying
the “normalized”
P
i
i
1-forms ωi := hi dq , with ω = i Ai dq = Ṽi ωi .
~ = P Ṽi ẽi .
Then ω is associated with the vector V
i
Note that if
∂f i X −1 ∂f
dq =
hi
ωi ,
∂q i
∂q i
i
~ =
the associated vector is V
Shapiro
Generalized
Coordinates
i
ω = df = σi
Physics 504,
Spring 2010
Electricity
and
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X
i
h−1
i
∂f
~ .
ẽi = ∇f
∂q i
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
2-forms
General 2-form: ω (2)
1X
Bij ωi ∧ ωj , with Bij = −Bji .
=
2
Physics 504,
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ij
In three dimensions, this can be associated with a vector
X
X
1X
~ =
B
B̃i ẽi with B̃i =
ijk Bjk , Bjk =
ijk B̃i .
2
i
~ ω (1) and ω (2) =
If V
ω (2) =
jk
dω (1) ,
i
then
X
1X
Bij ωi ∧ ωj = d
Ṽi hi dq i
2
ij
=
X ∂(Ṽi hi )
ij
=
i
X
∂q j
−1
h−1
i hj
ij
1
1
Thus Bij = h−1
h−1
2
2 i j
dq j ∧ dq i
∂(Ṽi hi )
ωj ∧ ωi ,
∂q j
!
∂(Ṽj hj ) ∂(Ṽi hi )
−
.
∂q i
∂q j
!
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Physics 504,
Spring 2010
Electricity
and
Magnetism
The associated vector has coefficients
1
1X
∂
∂
B̃k =
ijk
Ṽj hj − j Ṽi hi
2
hi hj ∂q i
∂q
ij
X
1 ∂
=
ijk
Ṽj hj .
hi hj ∂q i
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
ij
For cartesian coordinates hi = 1, and we recognize this as
~ , so dω (1) ∇
~ ×V
~ , which is a
the curl of V
coordinate-independent statement. Thus we have for any
orthogonal curvilinear coordinates
~ ×
∇
X
i
Ṽi ẽi =
X
ijk
ijk
1 ∂ h
Ṽ
ẽk .
j
j
hi hj ∂q i
(4)
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
~ ·B
~ and 3-forms
∇
Finally, let’s consider a vector
associated 2-form
1X
ijk B̃i ωj ∧ ωk =
ω (2) =
2
i
~ = P B̃i ẽi and its
B
i
Physics 504,
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Shapiro
1X
ijk B̃i hj hk dq j ∧ dq k .
2
i
The exterior derivative is a three-form
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
dω (2) =
=
=
1X
2
ijk`
ijk
∂ B̃i hj hk `
dq ∧ dq j ∧ dq k
∂q`
∂ B̃i hj hk 1
dq ∧ dq 2 ∧ dq 3
2
∂q`
ijk`
X ∂ h1 h2 h3 1
B̃i ω1 ∧ ω2 ∧ ω3 .
h1 h2 h3
∂qi
hi
1X
ijk` `jk
i
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
A 3-form in three dimensions is associated with a scalar
function f (P ) by
1 X
ω (3) = f dr1 ∧ dr2 ∧ dr3 = f
abc dra ∧ drb ∧ drc
6
=
=
abc
∂ra ∂rb ∂rc
1 X
f
abc i j k dq i ∧ dq j ∧ dq k
6
∂q ∂q ∂q
abcijk
b
c
1 X
∂ra
−1 ∂r
−1 ∂r
f
abc h−1
h
h
i
j
k ∂q k
6
∂q i
∂q j
abcijk
=
ωi ∧ ωj ∧ ωk
X
1
ijk ωi ∧ ωj ∧ ωk
f det A
6
ijk
= f det A ω1 ∧ ω2 ∧ ω3
= f ω1 ∧ ω2 ∧ ω3 ,
where det A = 1 because A is orthogonal, but also we
assume the ẽi form a right handed coordiate system.
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
~ dω (2) f , with
So we see that if ω (2) B,
X ∂ h1 h2 h3 1
f=
B̃i .
h1 h2 h3
∂qi
hi
i
Shapiro
~ · B,
~ but
In cartesian coordinates this just reduces to ∇
this association is coordinate-independent, so we see that
in a general curvilinear coordinate system,
X ∂ h1 h2 h3 1
~
~
∇·B =
B̃i .
h1 h2 h3
∂q i
hi
i
Finally, let’s examine the Laplacian on a scalar:
X ∂ h1 h2 h3
1
−1 ∂Φ
2
~
~
f = ∇ Φ = ∇ · ∇Φ =
hi
h1 h2 h3
∂q i
hi
∂q i
i
X ∂ h1 h2 h3 ∂Φ 1
=
h1 h2 h3
∂q i
h2i ∂q i
i
Physics 504,
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Electricity
and
Magnetism
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Summary
~v =
X
hj
j
dq j
ẽj ,
dt
!
~ ×
∇
X
Ṽi ẽi
=
i
~ ·B
~ =
∇
∇2 Φ =
~ =
∇f
X
h−1
m
m
∂ f˜
ẽm ,
∂q m
1 ∂ h
Ṽ
ẽk ,
ijk
j
j
hi hj ∂q i
ijk
X ∂ h1 h2 h3 1
B̃i ,
h1 h2 h3
∂q i
hi
i
X ∂ h1 h2 h3 ∂Φ 1
X
h1 h2 h3
i
∂q i
h2i
∂q i
For the record, even for generalized coordinates that are
not orthogonal, we can write
1 X ∂ ij √ ∂
∇2 = √
g
g j,
g
∂q i
∂q
ij
where g := det g.. .
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Cylindrical Polar Coordinates
Although we have developed this to deal with esoteric
orthogonal coordinate systems, such as those for your
homework, let us here work out the familiar cylindrical
polar and spherical coordinate systems.
Cylindrical Polar: r, φ, z,
(δs)2 = (δr)2 + r2 (δφ)2 + (δz)2 ,
so hr = hz = 1, hφ = r. Then
~
∇f
=
∂f
1 ∂f
∂f
ẽr +
ẽφ +
ẽz ,
∂r
r ∂φ
∂z
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Polar, continued
!
∂rV
∂V
∂V
∂V
φ
z
r
z
~ ×
−
ẽr +
−
ẽφ
∇
Ṽi ẽi
∂φ
∂z
∂z
∂r
i
1 ∂rVφ ∂Vz
+
−
ẽz
r
∂r
∂φ
∂Vφ
1 ∂Vz
∂Vr
∂Vz
=
−
ẽr +
−
ẽφ
r ∂φ
∂z
∂z
∂r
∂Vφ 1 ∂Vz
1
+
−
+ Vφ ẽz
∂r
r ∂φ
r
!
X
∂V
∂(rV
)
1
∂(rV
)
φ
r
z
~ ·
∇
Ṽi ẽi
=
+
+
r
∂r
∂φ
∂z
X
1
=
r
i
=
∂Vr
1 ∂Vφ ∂Vz
1
+
+
+ Vr
∂r
r ∂φ
∂z
r
Physics 504,
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Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Polar, continued further
Physics 504,
Spring 2010
Electricity
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Shapiro
Generalized
Coordinates
2
∇ Φ =
=
∂Φ
∂ 1 ∂Φ
r
+
∂r
∂φ r ∂φ
∂
∂Φ i
+
r
∂z
∂z
1 ∂
∂Φ
1 d2 Φ d2 Φ
r
+ 2 2 + 2
r ∂r
∂r
r dφ
dz
1h ∂
r ∂r
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Spherical Coordinates
Spherical Coordinates:
r radius, θ polar angle, φ
azimuth
z
θ
(δs)2 = (δr)2 +r2 (δθ)2 +r2 sin2 θ(δφ)2 ,
so hr = 1, hθ = r, hφ = r sin θ.
~
∇f
=
~ ×V
~
∇
=
Physics 504,
Spring 2010
Electricity
and
Magnetism
ϕ
Shapiro
L
y
x
∂f
1 ∂f
1 ∂f
ẽr +
ẽθ +
ẽφ
∂r
r ∂θ
r sin θ ∂φ
∂
∂
1
r sin θVφ −
rVθ ẽr
r2 sin θ ∂θ
∂φ
1
∂
∂
+
Vr −
r sin θVφ ẽθ
r sin θ ∂φ
∂r
1 ∂
∂
+
rVθ −
Vr ẽφ
r ∂r
∂θ
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Spherical Coordinates, continued
~ ×V
~
∇
=
~ ·B
~ =
∇
=
∂
∂
1
(sin θVφ ) −
Vθ ẽr
r sin θ ∂θ
∂φ
1
∂
1 ∂
+
Vr −
(rVφ ) ẽθ
r sin θ ∂φ
r ∂r
1 ∂
∂
+
(rVθ ) −
Vr ẽφ ,
r ∂r
∂θ
1 h∂ 2
∂ r
sin
θ
B̃
+
r
sin
θ
B̃
r
θ
r2 sin θ ∂r
∂θ
i
∂
+
rB̃φ
∂φ
1 ∂ 2 1 ∂ r
B̃
+
sin
θ
B̃
r
θ
r2 ∂r
r sin θ ∂θ
1
∂
+
B̃φ
r sin θ ∂φ
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Spherical, continued further
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
∇2 Φ =
=
∂Φ
∂
∂Φ
1 ∂ 2
r sin θ
+
sin θ
2
r sin θ ∂r
∂r
∂θ
∂θ
∂ 1 ∂Φ +
∂φ sin θ ∂φ
∂
1 ∂
1
∂Φ
2 ∂Φ
r
+ 2
sin θ
r2 ∂r
∂r
r sin θ ∂θ
∂θ
2
1
∂ Φ
+ 2 2
.
r sin θ ∂φ2
Generalized
Coordinates
Cartesian
coords for E D
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical