Generalized Coordinates
Physics 504,
Spring 2010
Electricity
and
Magnetism
ri , i
Cartesian coordinates
= 1, 2, . . . D for Euclidean
space.
X
(δri )2 .
Distance by Pythagoras: (δs)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
~
∇Φ
=
êi ,
∂ri
i
X ∂Vi
~ ·V
~ =
∇
,
∂ri
Shapiro
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
i
X ∂2Φ
,
∂ri 2
i
X
∂Vk
=
ijk j êi ,
∂r
2
~ · ∇Φ
~
∇ Φ=∇
=
~ ×V
~
∇
3D only
ijk
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
2
k 2
i 
j
(δs) =
(δr ) =
δq
δq
∂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
gij δq i δq j .
(δs)2 =
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
ij
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 =
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
X ∂rk ∂rk √
√
/ g11 g22 = g12 / g11 g22 ,
∂q 1 ∂q 2
k
km
= δij , so g .. is the inverse matrix to g.. .
X
k
êk
X
∂rk
=:
Bki êk
i
∂q
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
~ 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
k
I
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
Note that g ij is not the same as gij .
In fact
X
X X ∂q i ∂q ` X ∂rm ∂rm X ∂q i
∂rm
g i` g`j =
=
δkm j
∂q
∂rk ∂rk m ∂q ` ∂q j
∂rk
(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`
T
or AB = 1I.
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
k
Thus A can be written two ways,
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Aki = hi
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
k
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
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Give up on orthonormal basis vectors. Define
X ∂Φ
differential forms, such as dΦ =
dxk . The
∂xk
coefficients of dxk 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.
Derivatives
which need not be zero.
`
What to do? Two approaches:
I
k
`
Another problem: ẽk is not normal to the surface q k =
constant.
∂q i
∂rk
= h−1
.
i
k
∂q i
∂r
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
Aji (P )êj is not the same vector at different
ẽi =
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,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for
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
i
Generalized
Coordinates
Vi (~r(q))Aik (q).
iX
Also Vk (~r) =
Aki (~r)Ṽi (q(~r)).
j
X drk
=
X X ∂rk dq i
êk =
dt
∂q i dt
X dq i
ij
i
k
dt
hj δij ẽj =
X
j

!
X ∂q j

hj k ẽj 
∂r
j
dq j
hj
ẽj .
dt
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.
Derivatives of Vectors
Shapiro
Generalized
Coordinates
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
Shapiro
h−1
j
∂
ẽi
∂q j
∂êi
=0
∂rj
X ∂rk ∂ ∂r`
h−1
= h−1
j
i ê`
j
i
k
∂q ∂r
∂q
k`
= h−1
j
=
X
k`
X ∂rk ∂A`i
k`
Akj
∂q j
∂rk
∂A`i
ê` .
∂rk
or
~
∇f
=
X
h−1
m
∂ f˜
ẽm .
∂q m
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
`m
(3)
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,
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
hθ = r,
hφ = r sin θ.
~v = ṙẽr + rθ̇ẽθ + r sin θφ̇ẽφ
and
Shapiro
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
Thus
v 2 = ṙ2 + r2 θ̇2 + r2 sin2 θφ̇2 .
Differential Forms
A general 1-form over a space coordinatized by q i is
X
ω=
Ai (P )dq i ,
i
Generalized
Coordinates
Derivatives
ê`
`m
Physics 504,
Spring 2010
Electricity
and
Magnetism
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Note it is hj dq j which has the right dimensions for an
infinitesimal length, while dq j by itself might not.
Euclidean parallel transport:
Shapiro
Generalized
Coordinates
Both f (~r) and f˜(q) represent the same function f (P ) on
space, so we often carelessly leave out the twiddle.
g ij = h−2
i δij
Velocity of a particle
k
k`m
m
∂q i
∂rk
= hi k = h−1
.
i
∂q i
∂r
gij = h2i δij ,
~v =
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
!
X ∂f
X ∂ f˜ ∂q `
(Akm ẽm )
ê
=
k
∂rk
∂q ` ∂rk
k
k`m
!
X ˜
X ∂ f˜ ∂q `
∂q m
∂f
=
hm ẽm k =
hm ẽm g m`
`
k
∂q ∂r
∂r
∂q `
Physics 504,
Spring 2010
Electricity
and
Magnetism
=
X ∂ f˜
X
∂ f˜
=
hm ẽm h−2
h−1
ẽ ,
m δm` =
m
m m
∂q
∂q `
m
Derivatives
Let’s summarize some of our previous relations:
X
X
êk =
Aki ẽi ,
ẽi =
Aji (P )êj
Aki
~
∇f
Cartesian
coords for
Generalized
Coordinates
Vector Fields
i
i
Gradient of a scalar field
Shapiro
ik
X
so Ṽk (q) =
Physics 504,
Spring 2010
Electricity
and
Magnetism
∂f i X −1 ∂f
dq =
hi
ωi ,
∂q i
∂q i
i
~ =
the associated vector is V
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
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
1-forms ωi := hi dq i , with ω = i Ai dq i = Ṽi ωi .
~ = P Ṽi ẽi .
Then ω is associated with the vector V
i
Note that if
ω = df = σi
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
X
i
h−1
i
∂f
~ .
ẽi = ∇f
∂q i
Cartesian
coords for
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
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
i
jk
~ ω (1) and ω (2) = dω (1) , then
If V
ω (2) =
X
1X
Ṽi hi dq i
Bij ωi ∧ ωj = d
2
ij
=
=
i
X ∂(Ṽi hi )
ij
X
ij
!
∂q j
−1
h−1
i hj
dq j ∧ dq i
∂(Ṽi hi )
ωj ∧ ωi ,
∂q j
~ ·B
~ and 3-forms
∇
~ = P B̃i ẽi and its
Finally, let’s consider a vector B
i
associated 2-form
1X
1X
ω (2) =
ijk B̃i ωj ∧ ωk =
ijk B̃i hj hk dq j ∧ dq k .
2
2
i
i
The exterior derivative is a three-form
∂ B̃i hj hk `
1X
ijk
dq ∧ dq j ∧ dq k
2
∂q`
ijk`
=
=
Shapiro
Generalized
Coordinates
Cartesian
coords for
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 =
Ṽj hj − j Ṽi hi
ijk
i
2
hi hj ∂q
∂q
ij
X
1 ∂ =
Ṽj hj .
ijk
hi hj ∂q i
Shapiro
Generalized
Coordinates
Cartesian
coords for
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
X
Ṽi ẽi =
i
ijk
ijk
1 ∂ hj Ṽj ẽk .
hi hj ∂q i
(4)
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
!
∂(Ṽj hj ) ∂(Ṽi hi )
.
−
∂q i
∂q j
1
1
Thus Bij = h−1
h−1
2
2 i j
dω (2) =
Physics 504,
Spring 2010
Electricity
and
Magnetism
∂ B̃i hj hk 1
1X
ijk` `jk
dq ∧ dq 2 ∧ dq 3
2
∂q`
ijk`
X ∂ h 1 h2 h3 1
B̃i ω1 ∧ ω2 ∧ ω3 .
h 1 h2 h 3
∂qi
hi
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
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
abc i j k dq i ∧ dq j ∧ dq k
f
6
∂q ∂q ∂q
abcijk
∂ra
∂rb
∂rc
1 X
h−1
h−1
abc h−1
f
i
j
k
i
j
k
6
∂q
∂q
∂q
abcijk
=
ωi ∧ ωj ∧ ωk
X
1
ijk ωi ∧ ωj ∧ ωk
f det A
6
ijk
= f det A ω1 ∧ ω2 ∧ ω3
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
= f ω1 ∧ ω 2 ∧ ω 3 ,
i
where det A = 1 because A is orthogonal, but also we
assume the ẽi form a right handed coordiate system.
~ dω (2) f , with
So we see that if ω (2) B,
X ∂ h1 h 2 h 3 1
f=
B̃i .
h1 h2 h 3
∂qi
hi
i
Physics 504,
Spring 2010
Electricity
and
Magnetism
Summary
~v =
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 ∂ h 1 h2 h3 1
~ ·B
~ =
∇
B̃i .
h1 h 2 h 3
∂q i
hi
i
Finally, let’s examine the Laplacian on a scalar:
X ∂ h1 h 2 h 3
∂Φ
1
~ · ∇Φ
~ =
f = ∇2 Φ = ∇
h−1
i
i
i
h 1 h2 h 3
∂q
hi
∂q
i
X ∂ h1 h2 h3 ∂Φ 1
=
h1 h 2 h 3
∂q i
h2i ∂q i
i
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
X
hj
j
~ ×
∇
X
dq j
ẽj ,
dt
!
Ṽi ẽi
=
i
~ ·B
~ =
∇
∇2 Φ =
~ =
∇f
X
m
h−1
m
∂ f˜
ẽm ,
∂q m
1 ∂ hj Ṽj ẽk ,
hi hj ∂q i
ijk
X ∂ h1 h2 h3 1
B̃i ,
h 1 h2 h 3
∂q i
hi
i
X ∂
h1 h2 h3 ∂Φ
1
X
ijk
h 1 h2 h 3
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
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
Physics 504,
Spring 2010
Electricity
and
Magnetism
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,
2
2
2
2
so hr = hz = 1, hφ = r. Then
=
Shapiro
Generalized
Coordinates
~ ×
∇
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
2
(δs) = (δr) + r (δφ) + (δz) ,
~
∇f
Polar, continued
∂f
1 ∂f
∂f
ẽr +
ẽφ +
ẽz ,
∂r
r ∂φ
∂z
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
~ ·
∇
!
∂rVφ
∂Vz
∂Vr
∂Vz
ẽr +
ẽφ
−
−
∂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
1 ∂(rVr ) ∂Vφ ∂(rVz )
=
Ṽi ẽi
+
+
r
∂r
∂φ
∂z
X
=
Ṽi ẽi
Physics 504,
Spring 2010
Electricity
and
Magnetism
Shapiro
∇2 Φ =
=
∂Φ
∂ 1 ∂Φ
1h ∂
r
+
r ∂r
∂r
∂φ r ∂φ
∂
∂Φ i
+
r
∂z
∂z
1 ∂
∂Φ
1 d2 Φ d 2 Φ
r
+ 2 2 + 2
r ∂r
∂r
r dφ
dz
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 θB̃r +
r sin θB̃θ
r2 sin θ ∂r
∂θ
i
∂ +
rB̃φ
∂φ
1 ∂ 2 1 ∂ r B̃r +
sin θB̃θ
2
r ∂r
r sin θ ∂θ
1
∂
+
B̃φ
r sin θ ∂φ
1
r
Shapiro
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
i
=
Polar, continued further
Physics 504,
Spring 2010
Electricity
and
Magnetism
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Spherical Coordinates
Spherical Coordinates:
r radius, θ polar angle, φ
azimuth
so hr = 1, hθ = r, hφ = r sin θ.
Physics 504,
Spring 2010
Electricity
and
Magnetism
~
∇f
=
~ ×V
~
∇
=
Physics 504,
Spring 2010
Electricity
and
Magnetism
z
θ
(δs)2 = (δr)2 +r2 (δθ)2 +r2 sin2 θ(δφ)2 ,
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
∂Vr
1 ∂Vφ ∂Vz
1
+
+
+ Vr
∂r
r ∂φ
∂z
r
ϕ
Shapiro
L
y
x
∂f
1 ∂f
1 ∂f
ẽr +
ẽθ +
ẽφ
∂r
r ∂θ
r sin θ ∂φ
∂
∂
1
ẽr
−
r
sin
θV
rV
φ
θ
r2 sin θ ∂θ
∂φ
∂
1
∂
+
Vr −
r sin θVφ ẽθ
r sin θ ∂φ
∂r
1 ∂
∂
+
rVθ −
Vr ẽφ
r ∂r
∂θ
Spherical, continued further
Shapiro
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Cartesian
coords for
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
Shapiro
∇2 Φ =
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical
Generalized
Coordinates
=
1 ∂ 2
∂Φ
∂
∂Φ
r sin θ
+
sin θ
r2 sin θ ∂r
∂r
∂θ
∂θ
∂ 1 ∂Φ +
∂φ sin θ ∂φ
1 ∂
1
∂Φ
∂
2 ∂Φ
r
+
sin
θ
r2 ∂r
∂r
r2 sin θ ∂θ
∂θ
1
∂2Φ
+ 2 2
.
r sin θ ∂φ2
Generalized
Coordinates
Cartesian
coords for
Generalized
Coordinates
Vector Fields
Derivatives
Gradient
Velocity of a
particle
Derivatives of
Vectors
Differential
Forms
2-forms
3-forms
Cylindrical
Polar and
Spherical