mapping of probabilities - upmc · these pages are from a book in preparation: mapping of...
TRANSCRIPT
These pages are from a book in preparation:
Mapping of Probabilities
Theory for the Interpretation ofUncertain Physical Measurements
by Albert Tarantola
(to be submitted to Cambridge University Press)
The aim of the book is to develop the mathematical bases necessaryfor a proper treatment of measurement uncertainties
(and this includes the formulation of Inverse Problems).I don’t think that the right setting for this kind of problems
is the usual one (based on conditional probabilities). This is why I develop here some new notions: image of a probability,
reciprocal image of a probability, intersection of probabilities [these are generalizations of the usual operations on sets]).
This text is still confidential:you can read it and perhaps, learn some new things, but you are not allowed
to publish results based on the notions presented in this text,unless you ask me ([email protected]) for a permission.
Please send me any comment you may have.
Chapter 2
Manifolds
Old text begins.Probability densities play an important role in physics. To handle them properly,
we must have a clear notion of what ‘integrating a scalar function over a manifold’means.
While mathematicians may assume that a manifold has a notion of ‘volume’ de-fined, physicists must check if this is true in every application, and the answer is notalways positive. We must understand how far can we go without having a notion ofvolume, and we must understand which is the supplementary theory that appearswhen we do have such a notion.
It is my feeling that every book of probability theory should start with a chapterexplaining all the notions of tensor calculus that are necessary to develop an intrinsictheory of probability. This is the role of this chapter. In it, in addition to ‘ordinary’tensors, we shall find the tensor capacities and tensor densities that were common inthe books of a certain epoch, but that are not in fashion today (wrongly, I believe).
Old text ends.
24 Manifolds
2.1 Manifolds and Coordinates
In this first chapter, the basic notions of tensor calculus and of integration theory areintroduced. I do not try to be complete. Rather, I try to develop the minimum theorythat is necessary in order to develop probability theory in subsequent chapters.
The reader is assumed to have a good knowledge of tensor calculus, the goalof the chapter being more to fix terminology and notations than to advance in thetheory.
Many books on tensor calculus exist. Among the many books on tensor calculus,the best are (of course) in French, and Brillouin (1960) is the best among them. Manyother books contain introductory discussions on tensor calculus. Weinberg (1972) isparticularly lucid.
Perhaps original in this text is a notation proposed to distinguish between den-sities and capacities. While the trick of using indices in upper or lower positionto distinguish between vectors or forms (or, in metric spaces, to distinguish between‘contravariant’ or ‘covariant’ components) makes formulas intuitive, I propose to usea bar (in upper or lower position) to distinguish between densities (like a probabilitydensity) or capacities (like the capacity element of integration theory), this also lead-ing to intuitive results. In particular the bijection existing between these objects inmetric spaces becomes as ‘natural’ as the one just mentioned between contravariantand covariant components.
All through this book the implicit sum convention over repeated indices is used: anexpression like ti j n j means ∑ j ti j n j .
2.1 Manifolds and Coordinates 25
2.1.1 Linear Spaces
Consider a finite-dimensional linear space L , with vectors denoted u , v . . . If {ei}is a basis of the linear space, any vector v can be (uniquely) decomposed as
v = vi ei , (2.1)
this defining the components {vi} of the vector v in the basis {ei} .A linear form over L is a linear application from L into the set of real numbers, i.e.,
a linear application that to every vector v ∈ L associates a real number. Denoting byf a linear form, the number λ associated by f to an arbitrary vector v is denoted
λ = 〈 f , v 〉 . (2.2)
For any given linear form, say f , there is a unique set of quantities { fi} such thatfor any vector v ,
〈 f , v 〉 = fi vi . (2.3)
It is easy to see that the set of linear forms over a linear space L is itself a linearspace, that is denoted L∗ . The quantities { fi} can then be seen as being the compo-nents of the form f on a basis of forms {ei} , that is called the dual of the vector basis{ei} , and that may be defined by the condition
〈 ei , e j 〉 = δij (2.4)
(where δij is the ‘symbol’ that takes the value ‘one’ when i = j and ‘zero’ when
i 6= j ).The two linear space L and L∗ are the ‘building blocks’ of an infinite series of
more complex linear spaces. For instance, a set of coefficients tijk can be used to
define the linear application
{vi} , { fi} , {gi} 7→ λ = tijk vi f j gk . (2.5)
As it is easy to define the sum of two such linear applications, and the multiplicationof such a linear application by a real number, we can say that the coefficients {ti
jk}define an element of a linear space, denoted L∗ ⊗ L⊗ L . The coefficients {ti
jk} canthen be seen as the components of an element t of the linear space L∗ ⊗ L⊗ L on abasis that is denoted {ei ⊗ e j ⊗ ek} , and one writes
t = tijk ei ⊗ e j ⊗ ek . (2.6)
26 Manifolds
2.1.2 Manifolds
Grossly speaking, a manifold is a ‘space of points’. The physical 3D space is anexample of a three-dimensional manifold, and the surface of a sphere is an exampleof a two-dimensional manifold. In our theory, we shall consider manifolds with anarbitrary —but finite— number of dimensions. Those manifolds may be flat or not(although the ‘curvature’ of a manifold will appear only in one of the appendixes[note: what about the curvature of the sphere?]).
We shall examine ‘smooth manifolds’ only. For instance, the surface of a sphereis a smooth manifold. The surface of a cone is smooth everywhere, excepted at thetip of the cone.
The points inside well chosen portions of a manifold can be designated by theircoordinates: a coordinate system with n coordinates defines a one-to-one applicationbetween a portion of a manifold and a portion of <n . We then say that the manifoldhas n dimensions. The term ‘portion’ is used here to stress that many manifoldscan not be completely covered by a single coordinate system: any single coordinatesystem on the surface of the sphere will be pathological at least at one point (thespherical coordinates are pathological at two points, the two poles).
In what follows, smooth manifolds shall be denoted by symbols like M and N ,and the points of a manifold by symbols like P and Q . A coordinate system isdenoted, for instance, by {xi} .
At each point P of an n-dimensional manifold M one can introduce the lineartangent space, and all the vectors and tensors that can exist1 at that point. When asystem of coordinates {xi} is defined over the manifold M , at each point P of themanifold there is the natural basis (of the tangent linear space at P ). Actual tensorscan be defined at any point independently of any coordinate system (and of any localbasis), but their components are, of course, only defined when a basis is chosen. Usu-ally, this basis is the natural basis associated to a coordinate system. When changingcoordinates, the natural basis changes, so the components of the tensors change too.The formulas describing the change of components of a tensor under a change ofcoordinates are recalled below.
While tensors are intrinsic objects, it is sometimes useful to introduce ‘tensordensities’ and ‘tensor capacities’, that depend on the coordinates being used in anessential way. These densities and capacities are useful, in particular, to develop thenotion of volume (or of ‘measure’) on a manifold, and, therefore, to introduce thebasic concept of integral. It is for this reason that, in addition to tensors, densitiesand capacities are also considered below.
1The vectors belong to the tangent linear space, and the tensors belong to the different linear spacesthat can be built at point P using the different tensor products of the tangent linear space and its dual.
2.1 Manifolds and Coordinates 27
2.1.3 Changing Coordinates
Consider, over a finite-dimensional (smooth) manifold M , a first system of coordi-nates {xi} ; (i = 1, . . . , n) and a second system of coordinates {xi′} ; (i′ = 1, . . . , n)(putting the ‘primes’ in the indices rather than in the x’s greatly simplifies manytensor equations).
One may write the coordinate transformation using any of the two equivalentfunctions
xi′ = xi′(x1, . . . , xn) ; (i′ = 1, . . . , n)
xi = xi(x1′ , . . . , xn′) ; (i = 1, . . . , n) .(2.7)
We shall need the two sets of partial derivatives2
Xi′i =
∂xi′
∂xi ; Xii′ =
∂xi
∂xi′ . (2.8)
One hasXi′
k Xkj′ = δi′
j′ ; Xik′ Xk′
j = δij . (2.9)
To simplify language and notations, it is useful to introduce two matrices of partialderivatives, ranging the elements Xi
i′ and Xi′i as follows,
X =
X11′ X1
2′ X13′ · · ·
X21′ X2
2′ X23′ · · ·
......
... . . .
; X′ =
X1′1 X1′
2 X1′3 · · ·
X2′1 X2′
2 X2′3 · · ·
......
... . . .
.
(2.10)Then, equations 2.9 just tell that the matrices X and X′ are mutually inverses:
X′ X = X X′ = I . (2.11)
The two matrices X and X′ are called Jacobian matrices. As the matrix X′ is obtainedby taking derivatives of the variables xi′ with respect to the variables xi , one obtainsthe matrix {Xi′
i} as a function of the variables {xi} , so we can write X′(x) ratherthan just writting X′ . The reciprocal argument tels that we can write X(x′) ratherthan just X . We shall later use this to make some notations more explicit.
Finally, the Jacobian determinants of the transformation are the determinants of thetwo Jacobian matrices:
X′ = det X′ ; X = det X . (2.12)
Of course, X X′ = 1 .2Again, the same letter X is used here, the ‘primes’ in the indices distinguishing the different
quantities.
28 Manifolds
2.1.4 Tensors, Capacities, and Densities
Consider a finite-dimensional manifold M with some coordinates {xi} . Let P be apoint of the manifold, and {ei} a basis of the linear space tangent to M at P , thisbasis being the natural basis associated to the coordinates {xi} at point P .
Let T = Ti j...k`... ei ⊗ e j · · · ek ⊗ e` · · · be a tensor at point P . The Ti j...
k`... are,therefore, the components of T on the basis ei ⊗ e j · · · ek ⊗ e` · · · .
On a change of coordinates from {xi} into {xi′} , the natural basis will change,and, therefore, the components of the tensor will also change, becoming Ti′ j′ ...
k′`′ ... .It is well known that the new and the old components are related through
Ti′ j′ ...k′`′ ... =
∂xi′
∂xi∂x j′
∂x j · · ·∂xk
∂xk′∂x`
∂x`′· · · Ti j...
k`... , (2.13)
or, using the notations introduced above,
Ti′ j′ ...k′`′ ... = Xi′
i X j′j · · ·Xk
k′ X``′ · · · Ti j...
k`... . (2.14)
In particular, for totally contravariant and totally covariant tensors,
Ti′ j′ ... = Xi′i X j′
j · · · Ti j··· ; Ti′ j′ ... = Xii′ X j
j′ · · · Ti j... . (2.15)
In addition to actual tensors, we shall encounter other objects, that ‘have indices’also, and that transform in a slightly different way: densities and capacities (see forinstance Weinberg [1972] and Winogradzki [1979]). Rather than a general expositionof the properties of densities and capacities, let us anticipate that we shall only findtotally contravariant densities and totally covariant capacities (like the Levi-Civitacapacity, to be introduced below). From now on, in all this text,
• a density is denoted with an overline, like in a ;
• a capacity is denoted with an underline, like in b .
Let me now give what we can take as defining properties: Under the consideredchange of coordinates, a totally contravariant density a = a i j... ei ⊗ e j . . . changescomponents following the law
a i′ j′ ... =1
X′ Xi′i X j′
j · · · a i j... , (2.16)
or, equivalently, a i′ j′ ... = X Xi′i X j′
j · · · a i j... . Here X = det X and X′ = det X′
are the Jacobian determinants introduced in equation 2.12. This rule for the changeof components for a totally contravariant density is the same as that for a totallycontravariant tensor (equation at left in 2.15), excepted that there is an extra factor,the Jacobian determinant X = 1/X′ .
2.1 Manifolds and Coordinates 29
Similarly, a totally covariant capacity b = b i j... ei ⊗ e j . . . changes componentsfollowing the law
b i′ j′ ... =1X
Xii′ X j
j′ · · · b i j... , (2.17)
or, equivalently, b i′ j′ ... = X′ Xii′ X j
j′ · · · b i j... . Again, this rule for the change ofcomponents for a totally covariant capacity is the same as that for a totally covarianttensor (equation at right in 2.15), excepted that there is an extra factor, the Jacobiandeterminant Y = 1/X .
The most notable examples of tensor densities and capacities are the Levi-Civitadensity and Levi-Civita capacity (examined in section 2.1.8 below).
The number of terms in equations 2.16 and 2.17 depends on the ‘variance’ ofthe objects considered (i.e., in the number of indices they have). We shall find, inparticular, scalar densities and scalar capacities, that do not have any index. Thenatural extension of equations 2.16 and 2.17 is (a scalar can be considered to be atotally antisymmetric tensor)
a′ =1
X′ a = X a (2.18)
for a scalar density, and
b′ =1X
b = X′ b (2.19)
for a scalar capacity.The most notable example of a scalar capacity is the capacity element (as ex-
plained in section 2.1.11, this is the equivalent of the ‘volume’ element that can bedefined in metric manifolds). Scalar densities abound; for example, a probabilitydensity.
Let us write the two equations 2.18–2.19 more explicitly. Using x′ as variable,
a′(x′) = X(x′) a(x(x′)) ; b′(x′) =1
X(x′)b(x(x′)) , (2.20)
or, equivalently, using x as variable,
a′(x′(x)) =1
X′(x)a(x) ; b′(x′(x)) = X′(x) b(x) . (2.21)
For completeness, let me mention here that densities and capacities of higherdegree are also usually introduced (they appear briefly below). For instance, undera change of variables, a second degree (totally contravariant) tensor density wouldnot satisfy equation 2.16, but, rather,
a i′ j′ ... =1
(X′)2 Xi′i X j′
j · · · a′ i j... , (2.22)
30 Manifolds
where the reader should note the double bar used to indicate that a i′ j′ ... is a seconddegree tensor density. Similarly, under a change of variables, a second degree (totallycovariant) tensor capacity would not satisfy equation 2.17, but, rather,
b i′ j′ ... =1
X2 Xii′ X j
j′ · · · b i j... . (2.23)
The multiplication of tensors is one possibility for defining new tensors, like inti j
k = f j sik . Using the rules of change of components given above it is easy to
demonstrate the following properties:
• the product of a density by a tensor gives a density (like in pi = ρ vi );
• the product of a capacity by a tensor gives a capacity (like in si j = ti u j );
• the product of a capacity by a density gives a tensor (like in dσ = g dτ ).
Therefore, in a tensor equality, the total number of bars in each side of the equality mustbe balanced (counting upper and lower bars with opposite sign).
2.1 Manifolds and Coordinates 31
2.1.5 Kronecker Tensors (I)
There are two Kronecker’s ‘symbols’, δij and δi
j . They are defined similarly:
δij =
{1 if i and j are the same index0 if i and j are different indices , (2.24)
and
δij =
{1 if i and j are the same index0 if i and j are different indices . (2.25)
It is easy to verify that these are more than simple ‘symbols’: they are tensors. Forunder a change of variables we should have, using equation 2.14, δi′
j′ = Xi′i X j
j′ δij ,
i.e., δi′j′ = Xi′
i Xij′ , which is indeed true (see equation 2.9). Therefore, we shall say
that δij and δi
j are the Kronecker tensors.Warning: a common error in beginners is to give the value 1 to the symbol δi
i .In fact, the right value is n , the dimension of the space, as there is an implicit sumassumed: δi
i = δ11 + δ2
2 + · · ·+ δnn = 1 + 1 + · · ·+ 1 = n .
32 Manifolds
2.1.6 Orientation of a Coordinate System
The Jacobian determinants associated to a change of variables x y have beendefined in section 2.1.2. As their product must equal +1, they must be both posi-tive or both negative. Two different coordinate systems x = {x1, x2, . . . , xn} andy = {y1, y2, . . . , yn} are said to have the ‘same orientation’ (at a given point) if theJacobian determinants of the transformation, are positive. If they are negative, it issaid that the two coordinate systems have ’opposite orientation’.
Note: what follows is a useless complication!Note: what about
ñ det g ?
Nota: hablar con Tolo (tiene las ideas muy claras sobre el asunto. . .As changing the orientation of a coordinate system simply amounts to change
the order of two of its coordinates, in what follows we shall assume that in all ourchanges of coordinates, the new coordinates are always ordered in a way that theorientation is preserved. The special one-dimensional case (where there is only onecoordinate) is treated in an ad-hoc way.
Example 2.1 In the Euclidean 3D space, a positive orientation is assigned to a Cartesiancoordinate system {x, y, z} when the positive sense of the z is obtained from the positivesenses of the x axis and the y axis following the screwdriver rule. Another Cartesian coor-dinate system {u, v, w} defined as u = y , v = x , w = z , then would have a negativeorientation. A system of theee spherical coordinates, if taken in their usual order {r,θ,ϕ} ,then also has a positive orientation, but when changing the order of two coordinates, like in{r,ϕ,θ} , the orientation of the coordinate system is negative. For a system of geographicalcoordinates3, the reverse is true, while {r,ϕ, λ} is a positively oriented system, {r, λ,ϕ} isnegatively oriented.
3The geographical coordinate λ (latitude) is related to the spherical coordinate θ as λ+θ = π/2 .Therefore, cosϑ = sinθ .
2.1 Manifolds and Coordinates 33
2.1.7 Totally Antisymmetric Tensors
A tensor is completely antisymmetric if any even permutation of indices does notchange the value of the components, and if any odd permutation of indices changesthe sign of the value of the components:
tpqr... =
{+ti jk... if i jk . . . is an even permutation of pqr . . .−ti jk... if i jk . . . is an odd permutation of pqr . . .
(2.26)
For instance, a fourth rank tensor ti jkl is totally antisymmetric if
ti jkl = tikl j = til jk = t jilk = t jkil = t jlki
= tki jl = tk jli = tkli j = tlik j = tl jik = tlki j
= −ti jlk = −tik jl = −tilk j = −t jikl = −t jkli = −t jlik
= −tkil j = −tk jil = −tkl ji = −tli jk = −tl jki = −tlki j
(2.27)
a third rank tensor ti jk is totally antisymmetric if
ti jk = t jki = tki j = −tik j = −t jik = −tk ji , (2.28)
a second rank tensor ti j is totally antisymmetric if
ti j = −t ji . (2.29)
By convention, a first rank tensor ti and a scalar t are considered to be totally anti-symmetric (they satisfy the properties typical of other antisymmetric tensors).
34 Manifolds
2.1.8 Levi-Civita Capacity and Density
When working in a manifold of dimension n , one introduces two Levi-Civita ‘sym-bols’, εi1i2 ...in and εi1i2 ...in (having n indices each). They are defined similarly:
εi jk... =
+1 if i jk . . . is an even permutation of 12 . . . n
0 if some indices are identical−1 if i jk . . . is an odd permutation of 12 . . . n ,
(2.30)
and
εi jk... =
+1 if i jk . . . is an even permutation of 12 . . . n
0 if some indices are identical−1 if i jk . . . is an odd permutation of 12 . . . n .
(2.31)
In fact, these are more than ‘symbols’: they are respectively a capacity and a den-sity. Let us check this, for instance, for εi jk... . In order for εi jk... to be a capacity, oneshould verify that, under a change of variables over the manifold, expression 2.17holds, so one should have ε i′ j′ ... = 1
X Xii′ X j
j′ · · · ε i j... . That this is true, followsfrom the property Xi
i′ X jj′ · · · ε i j... = Xεi′ j′ ... that can be demonstrated using the
definition of a determinant (see equation 2.33). It is not obvious a priori that a prop-erty as strong as that expressed by the two equations 2.30–2.31 is conserved throughan arbitrary change of variables. We see that this is due to the fact that the verydefinition of determinant (equation 2.33) contains the Levi-Civita symbols.
Therefore, εi jk... is to be called the Levi-Civita capacity, and εi jk... is to be calledthe Levi-Civita density. By definition, these are totally antisymmetric.
In a space of dimension n the following properties hold
εs1 ...sn εs1 ...sn = n!
εi1s2 ...sn εj1s2 ...sn = (n− 1)! δ j1
i1
εi1i2s3 ...sn εj1 j2s3 ...sn = (n− 2)! ( δ j1
i1δ
j2i2− δ j2
i1δ
j1i2
)
· · · = · · · ,
(2.32)
the successive equations involving the ‘Kronecker determinants’, whose theory isnot developed here.
2.1 Manifolds and Coordinates 35
2.1.9 Determinants
The Levi-Civita’s densities and capacities can be used to define determinants. Forinstance, in a space of dimension n , the determinants of the tensors Qi j, Ri
j, Sij ,
and Ti j are defined by
Q =1n!εi1i2 ...in ε j1 j2 ... jn Qi1 j1 Qi2 j2 . . . Qin jn
R =1n!εi1i2 ...in ε j1 j2 ... jn Ri1
j1 Ri2j2 . . . Rin
jn
S =1n!εi1i2 ...in ε
j1 j2 ... jn Si1 j1 Si2 j2 . . . Sinjn
T =1n!εi1i2 ...in ε j1 j2 ... jn Ti1 j1 Ti2 j2 . . . Tin jn .
(2.33)
In particular, it is the first of equations 2.33 that is used below (equation 2.73) todefine the metric determinant.
36 Manifolds
2.1.10 Dual Tensors and Exterior Product of Vectors
In a space of dimension n , to any totally antisymmetric tensor Ti1 ...in of rank n oneassociates the scalar capacity
t =1n!εi1 ...in Ti1 ...in , (2.34)
while to any scalar capacity t we can associate the totally antisymmetric tensor ofrank n
Ti1 ...in = εi1 ...in t . (2.35)
These two equations are consistent when taken together (introducing one into theother gives an identity). We say that the capacity t is the dual of the tensor T , andthat the tensor T is the dual of the capacity t .
In a space of dimension n , given n vectors v1 , v2 . . . vn , one defines the scalarcapacity w = εi1 ...in (v1)i1 (v2)i2 . . . (vn)in , or, using simpler notations,
w = εi1 ...in vi11 vi2
2 . . . vinn , (2.36)
that is called the exterior product of the n vectors. This exterior product is usuallydenoted
w = v1 ∧ v2 ∧ · · · ∧ vn , (2.37)
and we shall sometimes use this notation, although it is not manifestly covariant (thenumber of ‘bars’ at the left and the right is not balanced).
The exterior product changes sign if the order of two vectors is changed, and iszero if the vectors are not linearly independent.
2.1 Manifolds and Coordinates 37
2.1.11 Capacity Element
Consider, at a point P of an n-dimensional manifold M , n vectors {dr1, dr2, . . . , drn}of the tangent linear space (the notation dr is used to suggest that, later on, a limitwill be taken, where all these vectors will tend to the zero vector). Their exteriorproduct is
dv = εi1 ...in dri11 dri2
2 . . . drinn , (2.38)
or, equivalently,dv = dr1 ∧ dr2 ∧ · · · ∧ drn . (2.39)
Let us see why this has to be interpreted as the capacity element associated to then vectors {dr1, dr2, . . . , drn} .
Assume that some coordinates {xi} have been defined over the manifold, andthat we choose the n vectors at point P each tangent to one of the coordinate linesat this point:
dr1 =
dx1
0...0
; dr2 =
0
dx2
...0
; · · · ; drn =
00...
dxn
. (2.40)
The n vectors, then, can be interpreted as the ‘perturbations’ of the n coordinates.The definition in equation 2.38 then gives
dv = ε12...n dx1 dx2 . . . dxn . (2.41)
Using a notational abuse, this capacity element is usually written, in mathematicaltexts, as
dv(x) = dx1 ∧ dx2 ∧ · · · ∧ dxn , (2.42)
while in physical texts, using more elementary notations, one simply writes
dv = dx1 dx2 . . . dxn . (2.43)
This is the usual capacity element that appears in elementary calculus to developthe notion of integral. I say ‘capacity element’ and not ‘volume element’ becausethe ‘volume’ spanned by the vectors {dr1, dr2, . . . , drn} shall only be defined whenthe manifold M shall be a ‘metric manifold’, i.e., when the ‘distance’ between twopoints of the manifold is defined.
The capacity element dv can be interpreted as the ‘small hyperparallepiped” de-fined by the ‘small vectors’ {dr1, dr2, . . . , drn} , as suggested in figure 2.1 for a three-dimensional space.
Under a change of coordinates (see an explicit demonstration in appendix 5.2.1)one has
dx1′ ∧ · · · ∧ dxn′ = det X′ dx1 ∧ · · · ∧ dxn . (2.44)
This, of course, is just a special case of equation 2.19 (that defines a scalar capacity).
38 Manifolds
Figure 2.1: From three ‘small vectors’ in a three-dimensional space one defines the three-dimensionalcapacity element dv = εi jk dri
1dr j2drk
3 , that can be in-terpreted as representing the ‘small parallelepiped’defined by the three vectors. To this parallelepipedthere is no true notion of ‘volume’ associated, unlessthe three-dimensional space is metric.
dr1
dr2
dr3
2.1 Manifolds and Coordinates 39
2.1.12 Integral
Consider an n-dimensional manifold M , with some coordinates {xi} , and assumethat a scalar density f (x1, x2, . . . ) has been defined at each point of the manifold(this function being a density, its value at each point depends on the coordinatesbeing used; an example of practical definition of such a scalar density is given insection 2.2.10).
Dividing each coordinate line in ‘small increments’ ∆xi divides the manifold M
(or some domain D of it) in ‘small hyperparallelepipeds’ that are characterized, aswe have seen, by the capacity element (equations 2.41–2.43)
∆v = ε12...n ∆x1 ∆x2 . . . ∆xn = ∆x1 ∆x2 . . . ∆xn . (2.45)
At every point, we can introduce the scalar ∆v f (x1, x2, . . . ) and, therefore, for anydomain D ⊂ M , the discrete sum ∑ ∆v f (x1, x2, . . . ) can be considered, where onlythe ‘hyperparallelepipeds’ that are inside the domain D (or at the border of thedomain) are taken into account (as suggested by figure 2.2).
Figure 2.2: The volume of an arbitrarily shaped, smooth,domain D of a manifold M , can be defined as the limit ofa sum, using elementary regions adapted to the coordinates(regions whose elementary capacity is well defined).
The integral of the scalar density f over the domain D is defined as the limit(when it exists)
I =∫D
dv f (x1, x2, . . . ) ≡ lim ∑ ∆v f (x1, x2, . . . ) , (2.46)
where the limit corresponds, taking smaller and smaller ‘cells’, to consider an infinitenumber of them.
This defines an invariant quantity: while the capacity values ∆v and the densityvalues f (x1, x2, . . . ) essentially depend on the coordinates being used, the integraldoes not (the product of a capacity times a density is a tensor).
This invariance is trivially checked when taking seriously the notation∫D dv f .
In a change of variables x x′ , the two capacity elements dv(x) and dv′(x′) arerelated via (equation 2.19)
dv′(x′) =1
X(x′)dv( x(x′) ) (2.47)
40 Manifolds
(where X(x′) is the Jacobian determinant det{∂xi/∂xi′} ), as they are tensorial ca-pacities, in the sense of section 2.1.4. Also, for a density we have
f ′(x′) = X(x′) f ( x(x′) ) . (2.48)
In the coordinates x we have
I(D) =∫
x∈Ddv(x) f (x) , (2.49)
and in the coordinates x′ ,
I(D)′ =∫
x′∈Ddv′(x′) f ′(x′) . (2.50)
using the two equations 2.47–2.48, we imediately obtain I(D) = I(D)′ this showingthat the integral of a density (integrated using the capacity element) is an invariant.
2.1 Manifolds and Coordinates 41
2.1.13 Capacity Element and Change of Coordinates
Note: real text yet to be written, this is a first attempt to the demonstration. Thedemonstration os possibly wrong, as I have not cared to well define the new capacityelement.
At a given point of an n-dimensional manifold we can consider the n vectors{dr1, . . . , drn} , associated to some coordinate system {x1, . . . , xn} , and we have thecapacity element
dv = εi1 ...in (dr1)i1 . . . (drn)in . (2.51)
In a change of coordinates {xi} 7→ {xi′} , each of the n vectors shall have his com-ponents changed according to
(dr)i′ =∂xi′
∂xi (dr)i = Xi′i (dr)i . (2.52)
Reciprocally,
(dr)i =∂xi
∂xi′ (dr)i′ = Xi′i (dr)i . (2.53)
The capacity element introduced above can now be expressed as
dv = εi1 ...in Xi1i′1
. . . Xini′n (dr1)i′1 . . . (drn)i′n . (2.54)
I guess that I can insert here the factor 1n! ε
i′1 ...i′n ε j′1 ... j′n , to obtain
dv = εi1 ...in Xi1i′1
. . . Xini′n ( 1
n! εi′1 ...i′n ε j′1 ... j′n ) (dr1) j′1 . . . (drn) j′n . (2.55)
If yes, then I would have
dv = ( 1n! εi1 ...in ε
i′1 ...i′n Xi1i′1
. . . Xini′n )ε j′1 ... j′n (dr1) j′1 . . . (drn) j′n , (2.56)
i.e., using the definition of determinant (third of equations 2.33),
dv = det X ε j′1 ... j′n (dr1) j′1 . . . (drn) j′n . (2.57)
We recognize here the capacity element dv′ = ε j′1 ... j′n (dr1) j′1 . . . (drn) j′n associated tothe new coordinates. Therefore, we have obtained
dv = det X dv′ . (2.58)
This, of course, is consistent with the definition of a scalar capacity (equation 2.19).
42 Manifolds
2.2 Volume
2.2.1 Metric
OLD TEXT BEGINS.In some parameter spaces, there is an obvious definition of distance between
points, and therefore of volume. For instance, in the 3D Euclidean space the distancebetween two points is just the Euclidean distance (which is invariant under transla-tions and rotations). Should we choose to parameterize the position of a point by itsCartesian coordinates {x, y, z} , then,
Note: I have to talk about the conmensurability of distances,
ds2 = ds2r + ds2
s , (2.59)
every time I have to define the Cartesian product of two spaces each with its ownmetric.
OLD TEXT ENDS.A manifold is called a metric manifold if there is a definition of distance between
points, such that the distance ds between the point of coordinates x = {xi} and thepoint of coordinates x + dx = {xi + dxi} can be expressed as4
ds2 = gi j(x) dxi dx j , (2.60)
i.e., if the notion of distance is ‘of the L2 type’5. At every point of a metric manifold,therefore, there is a symmetric tensor gi j defined, the metric tensor.
The inverse metric tensor, denoted gi j , is defined by the condition
gi j g jk = δik . (2.61)
It can be demonstrated that under a change of variables, its components change likethe components of a contravariant tensor, from where the notation gi j . Therefore,the equations defining the change of components of the metric and of the inversemetric are (see equations 2.15)
gi′ j′ = Xii′ X j
j′ gi j and gi′ j′ = X′i′i X′ j′
j gi j . (2.62)
In section 2.1.2, we introduced the matrices of partial derivatives. It is useful toalso introduce two metric matrices, with respectively the covariant and contravariantcomponents of the metric:
g =
g11 g12 g13 · · ·g21 g22 g23 · · ·
......
... . . .
; g-1 =
g11 g12 g13 · · ·g21 g22 g23 · · ·
......
... . . .
, (2.63)
4This is a property that is valid for any coordinate system that can be chosen over the space.5As a counterexample, in a two-dimensional manifold, the distance defined as ds = |dx1|+ |dx2|
is not of the L2 type (it is L1 ).
2.2 Volume 43
the notation g-1 for the second matrix being justified by the definition 2.61, that nowreads
g-1 g = I . (2.64)
In matrix notation, the change of the metric matrix under a change of variables,as given by the two equations 2.62, is written
g′ = Xt g X ; g′-1 = X′ g-1 X′t . (2.65)
If an every point P of a manifold M there is a metric gi j defined, then, themetric can be used to define a scalar product over the linear space tangent to M atP : given two vectors v and w , their scalar product is
v ·w ≡ gi j vi w j . (2.66)
One can also define the scalar product of two forms f and h at P (forms that belongto the dual of the linear space tangent to M at P ):
f · h ≡ gi j fi h j . (2.67)
The norm of a vector v and the norm of a form f are respectively defined as ‖ v ‖ =√
v · v =√
gi j vi v j and ‖ f ‖ =√
f · f =√
gi j vi v j .
44 Manifolds
2.2.2 Bijection Between Forms and Vectors
Let {ei} be the basis of a linear space, and {ei} the dual basis (that, as we haveseen, is a basis of the dual space).
In the absence of a metric, there is no natural association between vectors andforms. When there is a metric, to a vector vi ei we can associate a form whose com-ponents on the dual basis {ei} are
vi ≡ gi j v j . (2.68)
Similarly, to a form f = fi ei , one can associate the vector whose components on thevector basis ei are
f i ≡ gi j f j . (2.69)
[Note: Give here some of the properties of this association (that the scalar productis preserved, etc.).]
2.2 Volume 45
2.2.3 Kronecker Tensor (II)
The Kronecker’s tensors δij and δi
j are defined that the space has a metric or not.When one has a metric, one can raise and lower indices. Let us, for instance, lowerthe first index of δi
j :δi j ≡ gik δ
kj = gi j . (2.70)
Equivalently, let us raise one index of gi j :
gij ≡ gik gk j = δi
j . (2.71)
These equations demonstrate that when there is a metric, the Kronecker tensor and themetric tensor are identical. Therefore, when there is a metric, we can drop the symbolsδi
j and δij , and use the symbols gi
j and gij instead.
46 Manifolds
2.2.4 Fundamental Density
Let g the metric tensor of the manifold. For any (positively oriented) system ofcoordinates, we define the quantity g , that we call the metric density (in the givencoordinates) as
g =√
det g . (2.72)
More explicitly, using the definition of determinant in the first of equations 2.33,
g =√
1n! ε
i1i2 ...in ε j1 j2 ... jn gi1 j1 gi2 j2 . . . gin jn . (2.73)
This equation immediately suggests what it is possible to prove: the quantity g sodefined is a scalar density (at the right, we have two upper bars under a square root).
The quantityg = 1/g (2.74)
is obviously a capacity, that we call the metric capacity. It could also have been definedas
g =√
det g-1 =√
1n! εi1i2 ...in ε j1 j2 ... jn gi1 j1 gi2 j2 . . . gin jn . (2.75)
2.2 Volume 47
2.2.5 Bijection Between Capacities, Tensors, and Densities
As mentioned in section 2.1.4, (i) the product of a capacity by a density is a tensor,(ii) the product of a tensor by a density is a density, and (iii) the product of a tensorby a capacity is a capacity. So, when there is a metric, we have a natural bijectionbetween capacities and tensors, and between tensors and densities.
For instance, to a tensor capacity ti j...k`... we can associate the tensor
ti j...k`... ≡ g ti j...
k`... (2.76)
to a tensor si j...k`... we can associate the tensor density
si j...k`... ≡ g si j...
k`... (2.77)
and the tensor capacitysi j...
k`... ≡ g si j...k`... , (2.78)
and to a tensor density ri j...k`... we can associate the tensor
ri j...k`... ≡ g ri j...
k`... . (2.79)
Equations 2.76–2.79 introduce an important notation (that seems to be novel): in thebijections defined by the metric density and the metric capacity, we keep the sameletter for the tensors, and we just put bars or take out bars, much like in the bijectionbetween vectors and forms defined by the metric, where we keep the same letter,and we raise or lower indices.
48 Manifolds
2.2.6 Levi-Civita Tensor
From the Levi-civita capacity εi j...k we can define the Levi-Civita tensor εi j...k as
εi j...k = g εi j...k . (2.80)
Explicitly, this gives
εi jk... =
+√
det g if i jk . . . is an even permutation of 12 . . . n0 if some indices are identical
−√
det g if i jk . . . is an odd permutation of 12 . . . n .(2.81)
Alternatively, from the Levi-civita density εi j...k we could have defined the con-travariant tensor εi j...k as
εi j...k = g εi j...k . (2.82)
It can be shown that εi j...k can be obtained from εi j...k using the metric to raise theindices, so εi j...k and εi j...k are the same tensor (from where the notation).
2.2 Volume 49
2.2.7 Volume Element
We may here start by remembering equation 2.38,
dv = εi1 ...in dri11 dri2
2 . . . drinn , (2.83)
that expresses the capacity element defined by n vectors dr1 , dr2 . . . drn .In the special situation where n vectors are taken successively along each of the
n coordinate lines, this gives (equation 2.43) dv = dx1 dx2 . . . dxn . The dxi in thisexpression are mere coordinate increments, that bear no relation to a length. Aswe are now working under the hypothesis that we have a metric, we know thatthe length associated to the coordinate increment, say, dx1 , is6 ds = √
g11 dx1 . If thecoordinate lines where orthogonal at the considered point, then, the volume element,say dv , associated to the capacity element dv = dx1 dx2 . . . dxn would be dv =√
g11 dx1 √g22 dx2 . . .√
gnn dxn . If the coordinates are not necessarily orthogonal,this expression needs, of course, to be generalized.
One of the major theorems of integration theory is that the actual volume as-sociated to the hyperparallelepiped characterized by the capacity element dv , asexpressed by equation 2.83, is
dv = g dv , (2.84)
where g is the metric density introduced above. [Note: Should I give a demon-stration of this property here?] We know that dv is a capacity, and g a density.Therefore, the volume element dv , being the product of a density by a capacity isa true scalar. While dv has been called a ‘capacity element’, dv is called a volumeelement.
The overbar in g is to remember that the determinant of the metric tensor is adensity, in the tensorial sense of section 2.1.4, while the underbar in dv is to remem-ber that the ‘capacity element’ is a capacity in the tensorial sense of the term. Inequation 2.84, the product of a density times a capacity gives the volume elementdv , that is a true scalar (i.e., a scalar whose value is independent of the coordinatesbeing used). In view of equation 2.84, we can call g(x) the ‘density of volume’ in thecoordinates x = {x1, . . . , xn} . For short, we shall call g(x) the volume density7. It isimportant to realize that the values g(x) do not represent any intrinsic property ofthe space, but, rather, a property of the coordinates being used.
Example 2.2 In the Euclidean 3D space, using spherical coordinates x = {r,θ,ϕ} , as thelength element is ds2 = dr2 + r2 sin2θ dϕ2 + r2 dθ2, the metric matrix is grr grθ grϕ
gθr gθθ gθϕgϕr gϕθ gϕϕ
=
1 0 00 r2 00 0 r2 sin2θ
. (2.85)
6Because the length of a general vector with components dxi is ds2 = gi j dxi dx j .7So we now have two names for g , the ‘metric density’ and the ‘volume density’.
50 Manifolds
and the metric determinant is
g =√
det g = r2 sinθ . (2.86)
As the capacity element of the space can be expressed (using notations that are not manifestlycovariant)
dv = dr dθ dϕ , (2.87)
the expression dv = g dv gives the volume element
dv = r2 sinθ dr dθ dϕ . (2.88)
2.2 Volume 51
2.2.8 Volume Element and Change of Variables
Assume that one has an n-dimensional manifold M and two coordinate systems,say {x1, . . . , xn} and {x1′ , . . . , xn′} . If the manifold is metric, the components of themetric tensor can be expressed both in the coordinates x and in the coordinates x′ .The (unique) volume element, say dv , accepts the two different expressions
dv =√
det gx dx1 ∧ · · · ∧ dxn =√
det gx′ dx1′ ∧ · · · ∧ dxn′ . (2.89)
The Jacobian matrices of the transformation (matrices with the partial derivatives),X and X′ , have been introduced in section 2.1.3. The components of the metric arerelated through
gi′ j′ = Xii′ X j
j′ gi j , (2.90)
or, using matrix notations, gx′ = Xt gx X . Using the identities det gx′ = det(Xt gx X) =(det X)2 det gx , one arrives at√
det gx′ = det X√
det gx . (2.91)
This is he relation between the two fundamental densities associated to each of thetwo coordinate systems. Of course, this corresponds to equation 2.18 (page 29), usedto define scalar densities.
52 Manifolds
2.2.9 Volume of a Domain
With the volume element available, we can now define the volume of a domain D ofthe manifold M , that we shall denote as
V(D) =∫D
dv , (2.92)
by the expression ∫D
dv ≡∫D
dv g . (2.93)
This definition makes sense because we have already defined the integral of a den-sity in equation 2.46. Note that the (finite) capacity of a finite domain D cannot bedefined, as an expression like
∫D dv would make any (invariant) sense.
We have here defined equation 2.92 in terms of equation 2.93, but it may wellhappen that, in numerical evaluations of an integral, the division of the space intosmall ‘hyperparallelepideds’ that is implied by the use of the capacity element is notthe best choice. Figure 2.3 suggests a division of the space into ‘cells’ having grosslysimilar volumes (to compared with figure 2.4). If the volume ∆vp of each cell isknown, the volume of a domain D can obviously be defined as a limit
V(D) = lim∆vp→0
∑p
∆vp . (2.94)
We will discuss this point further in later chapters.
Figure 2.3: The volume of an arbitrarily shaped, smooth,domain D of a manifold M , can be defined as the limit of asum, using elementary regions whose individual volume isknown (for instance, triangles in this 2D illustration). Thisway of defining the volume of a region does not require thedefinition of a coordinate system over the space.
Figure 2.4: For the same shape of figure 2.3, the volume can beevaluated using, for instance, a polar coordinate system. In a nu-merical integration, regions near the origin may be oversampled,while regions far from the orign may be undersampled. In somesituation, this problem may become crucial, so this sort of ‘coor-dinate integration’ is to be reserved to analytical developmentsonly.
The finite volume obviously satisfies the following two properties:
2.2 Volume 53
• for any domain D of the manifold, V(D) ≥ 0 ;
• if D1 and D2 are two disjoint domains of the manifold, then V(D1 ∪ D2) =V(D1) + V(D2) .
54 Manifolds
2.2.10 Example: Mass Density and Volumetric Mass
Imagine that a large number of particles of equal mass are distributed in the phys-ical space (assimilated to an Euclidean 3D space) and that, for some reason, wechose to work with cylindrical coordinates {r,ϕ, z} . Choosing small increments{∆r, ∆ϕ, ∆z} of the coordinates, we divide the space into cells of equal capacity, that(using notations that are not manifestly covariant) is given by
∆v = ∆r ∆ϕ∆z . (2.95)
We can count how many particles are inside each cell (see figure 2.5), and, thereforewhich is the mass ∆m inside each cell. The quantity δm/∆v , being the ratio of ascalar by a capacity is a density. In the limit of an infinite number of particles, wecan take the limit where ∆r , ∆ϕ , and ∆z all tend to zero and the limit
ρ(r,ϕ, z) = lim∆r→0 , ∆ϕ→0 , ∆z→0
∆m∆v
(2.96)
is the mass density at point {r,ϕ, z} .
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Figure 2.5: We consider, in an Euclidean 3D space, a cylinder with a circular basis ofradius 1, and cylindrical coordinates (r,ϕ, z) . Only a section of the cylinder is rep-resented in the figure, with all its thickness, ∆z , projected on the drawing plane. Atthe left, we have represented a ‘map’ of the corresponding circle, and, at the middle,the coordinate lines on a ‘metric representation’ of the space. By construction, all the‘cells’ in the middle have the same capacity ∆v = ∆r ∆ϕ∆z . The points representparticles with given masses. As explained in the text, counting how many particlesare inside each cell directly gives an estimation of the ‘mass density’ ρ(r,ϕ, z) . Tohave, instead, a direct estimation of the ‘volumetric mass’ ρ(r,ϕ, z) , a division ofthe space into cells of equal volume (not equal capacity) should have been done, assuggested at the right.
Given the mass density ρ(r,ϕ, z) , the total mass inside a domain D of the spaceis to be obtained as
M(D) =∫D
dv ρ , (2.97)
2.2 Volume 55
where the capacity element dv appears, not the volume element dv .If instead of dividing the space into cells of equal capacity ∆v , we divide it into
cells of equal volume, ∆v (as suggested at the right of the figure 2.5), then the limit
ρ(r,ϕ, z) = lim∆v→0
∆m∆v
(2.98)
gives the volumetric mass ρ(r,ϕ, z) , different from the mass density ρ(r,ϕ, z) . Giventhe volumetric mass density ρ(r,ϕ, z) , the total mass inside a domain D of the spaceis to be obtained as
M(D) =∫D
dv ρ , (2.99)
where the volume element dv appears, not the capacity element dv . The relationbetween the mass density ρ and the volumetric mass ρ is the universal relationbetween any scalar density and a scalar,
ρ = gρ , (2.100)
where g is the metric density. As in cylindrical coordinates, g = r , the relationbetween mass density and volumetric mass is
ρ(r,ϕ, z) = rρ(r,ϕ, z) . (2.101)
It is unfortunate that in common physical terminology the terms ‘mass density’and ‘volumetric mass’ are used as synonymous. While for common applications,this does not pose any problem, there is a sometimes a serious misunderstanding inprobability theory about the meaning of a ‘probability density’ and of a ‘volumetricprobability’.
56 Manifolds
2.3 Mappings
Note: explain here that we consider a mapping from a p-dimensional manifold M
into a q-dimensional manifold N .
2.3.1 Image of the Volume Element
This section is very provisional. I do not know yet how much of it I will need.We have p-dimensional manifold, with coordinates {xa} = {x1, . . . , xp} . At a
given point, we have p vectors {dx1, . . . , dxp} . The associated capacity element is
dv = εa1 ...ap (dx1)a1 . . . (dxp)ap . (2.102)
We also have a second, q-dimensional manifold, with coordinates {ψα} = {ψ1, . . . ,ψq} .At a given point, we have q vectors {dψ1, . . . , dψq} . The associated capacity ele-ment is
dω = εα1 ...αq (dψ1)α1 . . . (dψq)
αq . (2.103)
Consider now an application
x 7→ ψ = ψ(x) (2.104)
from the first into the second manifold. I examine here the case
p ≤ q , (2.105)
i.e., the case where the dimension of the first manifold is smaller or equal than thatof the second manifold. When transporting the p vectors {dx1, . . . , dxp} from thefirst into the second manifold (via the application ψ(x) ), this will define on the q-dimensional manifold a p-dimensional capacity element, dωp . We wish to relatedωp and dv .
It is shown in the appendix (check!) that one has
dωp =√
det(Ψt Ψ) dv . (2.106)
Let us be interested in the image of the volume element. We denote by g the met-ric tensor of the first manifold, and by γ the metric tensor of the second manifold.
Bla, bla, bla, and it follows from this that when letting dωp be the (p-dimensional)volume element obtained in the (q-dimensional) second manifold by transport of thevolume element dv of the first manifold, one has
dωp
dv=
√det(Ψt γΨ)√
det g. (2.107)
2.3.2 Reciprocal Image of the Volume Element
I do not know yet if I will need this section.