Mapping of Probabilities

These pages are from a book in preparation: Mapping of Probabilities

Theory for the Interpretation of Uncertain Physical Measurements

by Albert Tarantola

(to be submitted to Cambridge University Press)

The aim of the book is to develop the mathematical bases necessary for 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 not always positive. We must understand how far can we go without having a notion of volume, and we must understand which is the supplementary theory that appears when we do have such a notion. It is my feeling that every book of probability theory should start with a chapter explaining all the notions of tensor calculus that are necessary to develop an intrinsic theory 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 in the 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 are introduced. I do not try to be complete. Rather, I try to develop the minimum theory that is necessary in order to develop probability theory in subsequent chapters. The reader is assumed to have a good knowledge of tensor calculus, the goal of the chapter being more to ﬁx terminology and notations than to advance in the theory. 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. Many other books contain introductory discussions on tensor calculus. Weinberg (1972) is particularly lucid. Perhaps original in this text is a notation proposed to distinguish between densities and capacities. While the trick of using indices in upper or lower position to distinguish between vectors or forms (or, in metric spaces, to distinguish between ‘contravariant’ or ‘covariant’ components) makes formulas intuitive, I propose to use a bar (in upper or lower position) to distinguish between densities (like a probability density) or capacities (like the capacity element of integration theory), this also lead- ing to intuitive results. In particular the bijection existing between these objects in metric spaces becomes as ‘natural’ as the one just mentioned between contravariant and covariant components. All through this book the implicit sum convention over repeated indices is used: an i j i j expression like t n j means ∑ j t 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

i v = v ei , (2.1)

i this deﬁning the components {v } 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 by f a linear form, the number λ associated by f to an arbitrary vector v is denoted

λ = h f , v i . (2.2)

For any given linear form, say f , there is a unique set of quantities { fi} such that for any vector v , i h f , v i = fi v . (2.3) It is easy to see that the set of linear forms over a linear space L is itself a linear ∗ space, that is denoted L . The quantities { fi} can then be seen as being the components 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

i i h e , e j i = δ j (2.4)

i (where δ j 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 jk more complex linear spaces. For instance, a set of coefficients ti can be used to define the linear application

i jk i {v } , { fi} , {gi} 7→ λ = ti v f j gk . (2.5)

As it is easy to define the sum of two such linear applications, and the multiplication jk of 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 } can then be seen as the components of an element t of the linear space L∗ ⊗ L ⊗ L on a i basis that is denoted {e ⊗ e j ⊗ ek} , and one writes

jk i t = ti e ⊗ 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 an example of a three-dimensional manifold, and the surface of a sphere is an example of a two-dimensional manifold. In our theory, we shall consider manifolds with an arbitrary —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 sphere is a smooth manifold. The surface of a cone is smooth everywhere, excepted at the tip of the cone. The points inside well chosen portions of a manifold can be designated by their coordinates: a coordinate system with n coordinates defines a one-to-one application between a portion of a manifold and a portion of measure’) on a manifold, and, therefore, to introduce the basic concept of integral. It is for this reason that, in addition to tensors, densities and capacities are also considered below.

1The vectors belong to the tangent linear space, and the tensors belong to the different linear spaces that 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- 0 nates {xi} ; (i = 1, . . . , n) and a second system of coordinates {xi } ; (i0 = 1, . . . , n) (putting the ‘primes’ in the indices rather than in the x’s greatly simplifies many tensor equations). One may write the coordinate transformation using any of the two equivalent functions

0 0 xi = xi (x1,..., xn) ; (i0 = 1, . . . , n) 0 0 (2.7) xi = xi(x1 ,..., xn ) ; (i = 1, . . . , n) .

We shall need the two sets of partial derivatives2

i0 i i0 ∂x i ∂x X = ; X 0 = . (2.8) i ∂xi i ∂xi0 One has i0 k i0 i k0 i X k X j0 = δ j0 ; X k0 X j = δ j . (2.9) To simplify language and notations, it is useful to introduce two matrices of partial i i0 derivatives, ranging the elements X i0 and X i as follows,

 1 1 1   10 10 10  X 10 X 20 X 30 ··· X 1 X 2 X 3 ··· 2 2 2 0 0 0 X 0 X 0 X 0 ···  0 X2 X2 X2 ···  X =  1 2 3  ; X =  1 2 3  ...... (2.10) Then, equations 2.9 just tell that the matrices X and X0 are mutually inverses:

X0 X = XX0 = I . (2.11)

The two matrices X and X0 are called Jacobian matrices. As the matrix X0 is obtained 0 by taking derivatives of the variables xi with respect to the variables xi , one obtains i0 i 0 the matrix {X i} as a function of the variables {x } , so we can write X (x) rather than just writting X0 . The reciprocal argument tels that we can write X(x0) rather than just X . We shall later use this to make some notations more explicit. Finally, the Jacobian determinants of the transformation are the determinants of the two Jacobian matrices:

X0 = det X0 ; X = det X . (2.12)

Of course, XX0 = 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 a point of the manifold, and {ei} a basis of the linear space tangent to M at P , this basis being the natural basis associated to the coordinates {xi} at point P . i j... k ` i j... Let T = T k`... ei ⊗ e j ··· e ⊗ e ··· be a tensor at point P . The T k`... are, k ` therefore, the components of T on the basis ei ⊗ e j ··· e ⊗ e ··· . 0 On a change of coordinates from {xi} into {xi } , the natural basis will change, i0 j0... and, therefore, the components of the tensor will also change, becoming T k0`0... . It is well known that the new and the old components are related through

i0 j0 k ` i0 j0... ∂x ∂x ∂x ∂x i j... T 0`0 = ··· 0 0 ··· T ` , (2.13) k ... ∂xi ∂x j ∂xk ∂x` k ... or, using the notations introduced above,

i0 j0... i0 j0 k ` i j... T k0`0... = X i X j ··· X k0 X `0 ··· T k`... . (2.14)

In particular, for totally contravariant and totally covariant tensors,

i0 j0... i0 j0 i j··· i j T = X i X j ··· T ; Ti0 j0... = X i0 X j0 ··· 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 for instance Weinberg [1972] and Winogradzki [1979]). Rather than a general exposition of the properties of densities and capacities, let us anticipate that we shall only ﬁnd totally contravariant densities and totally covariant capacities (like the Levi-Civita capacity, 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 considered i j... change of coordinates, a totally contravariant density a = a ei ⊗ e j . . . changes components following the law

0 0 1 0 0 a i j ... = Xi X j ··· a i j... , (2.16) X0 i j

i0 j0... i0 j0 i j... 0 0 or, equivalently, a = XX i X j ··· a . Here X = det X and X = det X are the Jacobian determinants introduced in equation 2.12. This rule for the change of components for a totally contravariant density is the same as that for a totally contravariant tensor (equation at left in 2.15), excepted that there is an extra factor, the Jacobian determinant X = 1/X0 . 2.1 Manifolds and Coordinates 29

i j Similarly, a totally covariant capacity b = b i j... e ⊗ e . . . changes components following the law

1 i j b 0 0 = X 0 X 0 ··· b , (2.17) i j ... X i j i j... 0 i j or, equivalently, b i0 j0... = X X i0 X j0 ··· b i j... . Again, this rule for the change of components for a totally covariant capacity is the same as that for a totally covariant tensor (equation at right in 2.15), excepted that there is an extra factor, the Jacobian determinant Y = 1/X . The most notable examples of tensor densities and capacities are the Levi-Civita density 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’ of the objects considered (i.e., in the number of indices they have). We shall ﬁnd, in particular, scalar densities and scalar capacities, that do not have any index. The natural extension of equations 2.16 and 2.17 is (a scalar can be considered to be a totally antisymmetric tensor)

1 a0 = a = X a (2.18) X0 for a scalar density, and

1 b0 = b = X0 b (2.19) X for a scalar capacity. The most notable example of a scalar capacity is the capacity element (as explained in section 2.1.11, this is the equivalent of the ‘volume’ element that can be deﬁned in metric manifolds). Scalar densities abound; for example, a probability density. Let us write the two equations 2.18–2.19 more explicitly. Using x0 as variable, 1 a0(x0) = X(x0) a(x(x0)) ; b0(x0) = b(x(x0)) , (2.20) X(x0) or, equivalently, using x as variable, 1 a0(x0(x)) = a(x) ; b0(x0(x)) = X0(x) b(x) . (2.21) X0(x) For completeness, let me mention here that densities and capacities of higher degree are also usually introduced (they appear brieﬂy below). For instance, under a change of variables, a second degree (totally contravariant) tensor density would not satisfy equation 2.16, but, rather,

0 0 1 0 j0 a i j ... = Xi X ··· a0 i j... , (2.22) (X0)2 i j 30 Manifolds

0 0 where the reader should note the double bar used to indicate that a i j ... is a second degree tensor density. Similarly, under a change of variables, a second degree (totally covariant) tensor capacity would not satisfy equation 2.17, but, rather,

1 i j b 0 0 = X 0 X 0 ··· b . (2.23) i j ... X2 i j i j... The multiplication of tensors is one possibility for deﬁning new tensors, like in k k ti j = f j si . 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 must be balanced (counting upper and lower bars with opposite sign). 2.1 Manifolds and Coordinates 31

2.1.5 Kronecker Tensors (I) j i There are two Kronecker’s ‘symbols’, δi and δ j . They are deﬁned similarly:

1 if i and j are the same index δ j = (2.24) i 0 if i and j are different indices , and 1 if i and j are the same index δi = (2.25) j 0 if i and j are different indices . It is easy to verify that these are more than simple ‘symbols’: they are tensors. For i0 i0 j i under a change of variables we should have, using equation 2.14, δ j0 = X i X j0 δ j , i0 i0 i i.e., δ j0 = X i X j0 , which is indeed true (see equation 2.9). Therefore, we shall say j i that δi and δ j are the Kronecker tensors. i Warning: a common error in beginners is to give the value 1 to the symbol δi . In fact, the right value is n , the dimension of the space, as there is an implicit sum i 1 2 n assumed: δi = δ1 + δ2 + ··· + δn = 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 been deﬁned in section 2.1.2. As their product must equal +1, they must be both positive or both negative. Two different coordinate systems x = {x1, x2,..., xn} and y = {y1, y2,..., yn} are said to have the ‘same orientation’ (at a given point) if the Jacobian determinants of the transformation, are positive. If they are negative, it is said that the two coordinate systems have ’opposite orientation’. Note: what follows is a useless complication! p Note: what about ± 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 our changes of coordinates, the new coordinates are always ordered in a way that the orientation is preserved. The special one-dimensional case (where there is only one coordinate) is treated in an ad-hoc way.

Example 2.1 In the Euclidean 3D space, a positive orientation is assigned to a Cartesian coordinate system {x, y, z} when the positive sense of the z is obtained from the positive senses of the x axis and the y axis following the screwdriver rule. Another Cartesian coordinate system {u, v, w} deﬁned as u = y , v = x , w = z , then would have a negative orientation. 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 geographical coordinates3, the reverse is true, while {r,ϕ, λ} is a positively oriented system, {r, λ,ϕ} is negatively oriented.

3The geographical coordinate λ (latitude) is related to the spherical coordinate θ as λ +θ = π/2 . Therefore, cos ϑ = 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 not change the value of the components, and if any odd permutation of indices changes the sign of the value of the components: ( +ti jk... if i jk . . . is an even permutation of pqr ... tpqr... = (2.26) −ti jk... if i jk . . . is an odd permutation of pqr ...

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 (2.27) = −ti jlk = −tik jl = −tilk j = −t jikl = −t jkli = −t jlik

= −tkil j = −tk jil = −tkl ji = −tli jk = −tl jki = −tlki j 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 antisymmetric (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- i1i2...in bols’, i1i2...in and (having n indices each). They are defined similarly:  +1 if i jk . . . is an even permutation of 12 . . . n  i jk... = 0 if some indices are identical (2.30)  −1 if i jk . . . is an odd permutation of 12 . . . n , and  +1 if i jk . . . is an even permutation of 12 . . . n  i jk... = 0 if some indices are identical (2.31)  −1 if i jk . . . is an odd permutation of 12 . . . n . In fact, these are more than ‘symbols’: they are respectively a capacity and a density. Let us check this, for instance, for i jk... . In order for i jk... to be a capacity, one should verify that, under a change of variables over the manifold, expression 2.17 1 i j holds, so one should have i0 j0... = X X i0 X j0 ··· i j... . That this is true, follows i j from the property X i0 X j0 ··· i j... = X i0 j0... that can be demonstrated using the definition of a determinant (see equation 2.33). It is not obvious a priori that a property as strong as that expressed by the two equations 2.30–2.31 is conserved through an arbitrary change of variables. We see that this is due to the fact that the very definition of determinant (equation 2.33) contains the Levi-Civita symbols. i jk... Therefore, i jk... is to be called the Levi-Civita capacity, and is to be called the Levi-Civita density. By definition, these are totally antisymmetric. In a space of dimension n the following properties hold

s1...sn s1...sn = n! j j1s2...sn = (n − 1)! δ 1 i1s2...sn i1 j j j j (2.32) j1 j2s3...sn = (n − 2)! ( δ 1 δ 2 − δ 2 δ 1 ) i1i2s3...sn i1 i2 i1 i2 ··· = ··· , the successive equations involving the ‘Kronecker determinants’, whose theory is not 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. For j i instance, in a space of dimension n , the determinants of the tensors Qi j, Ri , S j , and Ti j are deﬁned by

1 Q = i1i2...in j1 j2...jn Q Q ... Q n! i1 j1 i2 j2 in jn 1 = i1i2...in j1 j2 jn R j1 j2...jn Ri1 Ri2 ... Rin n! (2.33) 1 S = j1 j2...jn Si1 Si2 ... Sin n! i1i2...in j1 j2 jn 1 T = Ti1 j1 Ti2 j2 ... Tin jn . n! i1i2...in j1 j2...jn In particular, it is the ﬁrst of equations 2.33 that is used below (equation 2.73) to deﬁne 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 one associates the scalar capacity

1 t = Ti1...in , (2.34) n! i1...in while to any scalar capacity t we can associate the totally antisymmetric tensor of rank n Ti1...in = i1...in t . (2.35) These two equations are consistent when taken together (introducing one into the other gives an identity). We say that the capacity t is the dual of the tensor T , and that 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 scalar = ( )i1 ( )i2 ( )in capacity w i1...in v1 v2 ... vn , or, using simpler notations,

= i1 i2 in w i1...in v1 v2 ... vn , (2.36) that is called the exterior product of the n vectors. This exterior product is usually denoted w = v1 ∧ v2 ∧ · · · ∧ vn , (2.37) and we shall sometimes use this notation, although it is not manifestly covariant (the number 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 is zero 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 limit will be taken, where all these vectors will tend to the zero vector). Their exterior product is = i1 i2 in dv i1...in dr1 dr2 ... drn , (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 the n vectors {dr1, dr2,..., drn} . Assume that some coordinates {xi} have been defined over the manifold, and that we choose the n vectors at point P each tangent to one of the coordinate lines at this point:       dx1 0 0  0  dx2  0  dr =   dr =   ··· dr =   1  .  ; 2  .  ; ; n  .  . (2.40)  .   .   .  0 0 dxn The n vectors, then, can be interpreted as the ‘perturbations’ of the n coordinates. The definition in equation 2.38 then gives 1 2 n dv = 12...n dx dx ... dx . (2.41) Using a notational abuse, this capacity element is usually written, in mathematical texts, 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 develop the notion of integral. I say ‘capacity element’ and not ‘volume element’ because the ‘volume’ spanned by the vectors {dr1, dr2,..., drn} shall only be defined when the manifold M shall be a ‘metric manifold’, i.e., when the ‘distance’ between two points of the manifold is defined. The capacity element dv can be interpreted as the ‘small hyperparallepiped” defined 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 0 0 dx1 ∧ · · · ∧ dxn = det X0 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- dr3 dimensional space one deﬁnes the three-dimensional i j k capacity element dv = i jk dr1dr2dr3 , that can be interpreted as representing the ‘small parallelepiped’ dr2 deﬁned by the three vectors. To this parallelepiped there is no true notion of ‘volume’ associated, unless dr1 the three-dimensional space is metric. 2.1 Manifolds and Coordinates 39

2.1.12 Integral

Consider an n-dimensional manifold M , with some coordinates {xi} , and assume that 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 coordinates being used; an example of practical deﬁnition of such a scalar density is given in section 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, as we have seen, by the capacity element (equations 2.41–2.43)

1 2 n 1 2 n ∆v = 12...n ∆x ∆x ... ∆x = ∆x ∆x ... ∆x . (2.45)

At every point, we can introduce the scalar ∆v f (x1, x2,... ) and, therefore, for any domain D ⊂ M , the discrete sum ∑ ∆v f (x1, x2,... ) can be considered, where only the ‘hyperparallelepipeds’ that are inside the domain D (or at the border of the domain) 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 of a 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)

Z I = dv f (x1, x2,... ) ≡ lim ∑ ∆v f (x1, x2,... ) , (2.46) D where the limit corresponds, taking smaller and smaller ‘cells’, to consider an inﬁnite number of them. This deﬁnes an invariant quantity: while the capacity values ∆v and the density values f (x1, x2,... ) essentially depend on the coordinates being used, the integral does not (the product of a capacity times a density is a tensor). R This invariance is trivially checked when taking seriously the notation D dv f . 0 0 0 In a change of variables x x , the two capacity elements dv(x) and dv (x ) are related via (equation 2.19)

1 dv0(x0) = dv( x(x0)) (2.47) X(x0) 40 Manifolds

0 (where X(x0) is the Jacobian determinant det{∂xi/∂xi } ), as they are tensorial capacities, in the sense of section 2.1.4. Also, for a density we have

f 0(x0) = X(x0) f ( x(x0)) . (2.48)

In the coordinates x we have Z I(D) = dv(x) f (x) , (2.49) x∈D and in the coordinates x0 , Z I(D)0 = dv0(x0) f 0(x0) . (2.50) x0∈D using the two equations 2.47–2.48, we imediately obtain I(D) = I(D)0 this showing that 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. The demonstration os possibly wrong, as I have not cared to well deﬁne the new capacity element. At a given point of an n-dimensional manifold we can consider the n vectors 1 n {dr1,..., drn} , associated to some coordinate system {x ,..., x } , and we have the capacity element = ( )i1 ( )in dv i1...in dr1 ... drn . (2.51) 0 In a change of coordinates {xi} 7→ {xi } , each of the n vectors shall have his components changed according to

i0 0 ∂x 0 (dr)i = (dr)i = Xi (dr)i . (2.52) ∂xi i Reciprocally, i ∂x 0 0 (dr)i = (dr)i = Xi (dr)i . (2.53) ∂xi0 i The capacity element introduced above can now be expressed as

0 0 i1 in i in dv = X 0 ... X 0 (dr1) 1 ... (drn) . (2.54) i1...in i1 in

1 i0 ...i0 I guess that I can insert here the factor 1 n 0 0 , to obtain n! j1...jn

0 0 0 0 i1 in 1 i ...in j jn dv = X 0 ... X 0 ( 1 0 0 )(dr1) 1 ... (drn) . (2.55) i1...in i1 in n! j1...jn If yes, then I would have

0 0 0 0 1 i ...in i1 in j jn dv = ( 1 X 0 ... X 0 ) 0 0 (dr1) 1 ... (drn) , (2.56) n! i1...in i1 in j1...jn i.e., using the deﬁnition of determinant (third of equations 2.33),

0 0 j jn dv = det X 0 0 (dr1) 1 ... (drn) . (2.57) j1...jn

0 j0 j0 We recognize here the capacity element dv = 0 0 (dr1) 1 ... (drn) n associated to j1...jn the new coordinates. Therefore, we have obtained

dv = det X dv0 . (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 distance between 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 its Cartesian coordinates {x, y, z} , then, Note: I have to talk about the conmensurability of distances, 2 2 2 ds = dsr + dss , (2.59) every time I have to define the Cartesian product of two spaces each with its own metric. 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 the point of coordinates x + dx = {xi + dxi} can be expressed as4 2 i j ds = gi j(x) dx dx , (2.60) 5 i.e., if the notion of distance is ‘of the L2 type’ . 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 i j i g g jk = δ k . (2.61) It can be demonstrated that under a change of variables, its components change like the 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 inverse metric are (see equations 2.15)

i j i0 j0 0i0 0j0 i j gi0 j0 = X i0 X j0 gi j and g = X i X j g . (2.62) In section 2.1.2, we introduced the matrices of partial derivatives. It is useful to also introduce two metric matrices, with respectively the covariant and contravariant components of the metric:    11 12 13  g11 g12 g13 ··· g g g ··· 21 22 23 g = g21 g22 g23 ···  ; g-1 = g g g ···  , (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 now reads 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

g0 = Xt g X ; g0-1 = X0 g-1 X0t . (2.65)

If an every point P of a manifold M there is a metric gi j deﬁned, then, the metric can be used to deﬁne a scalar product over the linear space tangent to M at P : given two vectors v and w , their scalar product is

i j v · w ≡ gi j v w . (2.66)

One can also deﬁne the scalar product of two forms f and h at P (forms that belong to the dual of the linear space tangent to M at P ):

i j f · h ≡ g fi h j . (2.67)

The norm of a vector v and the norm of a form f are respectively deﬁned as k v k = √ q √ q i j i j v · v = gi j v v and k f k = f · f = g vi v j . 44 Manifolds

2.2.2 Bijection Between Forms and Vectors

i Let {ei} be the basis of a linear space, and {e } the dual basis (that, as we have seen, is a basis of the dual space). In the absence of a metric, there is no natural association between vectors and i forms. When there is a metric, to a vector v ei we can associate a form whose components on the dual basis {ei} are

j vi ≡ gi j v . (2.68)

i Similarly, to a form f = fi e , one can associate the vector whose components on the vector basis ei are i i j f ≡ g f j . (2.69) [Note: Give here some of the properties of this association (that the scalar product is preserved, etc.).] 2.2 Volume 45

2.2.3 Kronecker Tensor (II) i j The Kronecker’s tensors δ j and δi 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, lower i the ﬁrst index of δ j : k δi j ≡ gik δ j = gi j . (2.70)

Equivalently, let us raise one index of gi j :

i ik i g j ≡ g gk j = δ j . (2.71)

These equations demonstrate that when there is a metric, the Kronecker tensor and the metric tensor are identical. Therefore, when there is a metric, we can drop the symbols i j i j δ j and δi , and use the symbols g j and gi instead. 46 Manifolds

2.2.4 Fundamental Density Let g the metric tensor of the manifold. For any (positively oriented) system of coordinates, we define the quantity g , that we call the metric density (in the given coordinates) as p g = det g . (2.72) More explicitly, using the definition of determinant in the first of equations 2.33, q = 1 i1i2...in j1 j2...jn g n! gi1 j1 gi2 j2 ... gin jn . (2.73)

This equation immediately suggests what it is possible to prove: the quantity g so deﬁned is a scalar density (at the right, we have two upper bars under a square root). The quantity g = 1/g (2.74) is obviously a capacity, that we call the metric capacity. It could also have been deﬁned as q q = -1 = 1 i1 j1 i2 j2 in jn g det g n! i1i2...in j1 j2...jn g g ... g . (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 tensor by a capacity is a capacity. So, when there is a metric, we have a natural bijection between capacities and tensors, and between tensors and densities. i j... For instance, to a tensor capacity t k`... we can associate the tensor

i j... i j... t k`... ≡ g t k`... (2.76)

i j... to a tensor s k`... we can associate the tensor density

i j... i j... s k`... ≡ g s k`... (2.77) and the tensor capacity i j... i j... s k`... ≡ g s k`... , (2.78) i j... and to a tensor density r k`... we can associate the tensor

i j... i j... r k`... ≡ g r k`... . (2.79)

Equations 2.76–2.79 introduce an important notation (that seems to be novel): in the bijections deﬁned by the metric density and the metric capacity, we keep the same letter for the tensors, and we just put bars or take out bars, much like in the bijection between 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 i j...k we can deﬁne the Levi-Civita tensor i j...k as

i j...k = g i j...k . (2.80)

Explicitly, this gives  p + det g if i jk . . . is an even permutation of 12 . . . n  i jk... = 0 if some indices are identical (2.81)  p − det g if i jk . . . is an odd permutation of 12 . . . n .

Alternatively, from the Levi-civita density i j...k we could have deﬁned the contravariant tensor i j...k as i j...k = g i j...k . (2.82)

i j...k It can be shown that can be obtained from i j...k using the metric to raise the i j...k indices, so and i 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,

= i1 i2 in dv i1...in dr1 dr2 ... drn , (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 this expression are mere coordinate increments, that bear no relation to a length. As we are now working under the hypothesis that we have a metric, we know that 1 6 √ 1 the length associated to the coordinate increment, say, dx , is ds = g11 dx . If the coordinate 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 = √ 1 √ 2 √ n g11 dx g22 dx ... gnn dx . 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 associated to the hyperparallelepiped characterized by the capacity element dv , as expressed by equation 2.83, is

dv = g dv , (2.84) where g is the metric density introduced above. [Note: Should I give a demonstration 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 is a true scalar. While dv has been called a ‘capacity element’, dv is called a volume element. The overbar in g is to remember that the determinant of the metric tensor is a density, in the tensorial sense of section 2.1.4, while the underbar in dv is to remember that the ‘capacity element’ is a capacity in the tensorial sense of the term. In equation 2.84, the product of a density times a capacity gives the volume element dv , that is a true scalar (i.e., a scalar whose value is independent of the coordinates being used). In view of equation 2.84, we can call g(x) the ‘density of volume’ in the coordinates x = {x1,..., xn} . For short, we shall call g(x) the volume density7. It is important to realize that the values g(x) do not represent any intrinsic property of the 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 the length element is ds2 = dr2 + r2 sin2 θ dϕ2 + r2 dθ2, the metric matrix is     grr grθ grϕ 1 0 0 2 gθr gθθ gθϕ  = 0 r 0  . (2.85) 2 2 gϕr gϕθ gϕϕ 0 0 r sin θ

6 i 2 i j Because the length of a general vector with components dx is ds = gi j dx dx . 7So we now have two names for g , the ‘metric density’ and the ‘volume density’. 50 Manifolds and the metric determinant is p g = det g = r2 sinθ . (2.86)

As the capacity element of the space can be expressed (using notations that are not manifestly covariant) 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, 0 0 say {x1,..., xn} and {x1 ,..., xn } . If the manifold is metric, the components of the metric tensor can be expressed both in the coordinates x and in the coordinates x0 . The (unique) volume element, say dv , accepts the two different expressions

q 1 n p 10 n0 dv = det gx dx ∧ · · · ∧ dx = det gx0 dx ∧ · · · ∧ dx . (2.89)

The Jacobian matrices of the transformation (matrices with the partial derivatives), X and X0 , have been introduced in section 2.1.3. The components of the metric are related through i j gi0 j0 = X i0 X j0 gi j , (2.90) t t or, using matrix notations, gx0 = X gx X . Using the identities det gx0 = det(X gx X) = 2 (det X) det gx , one arrives at p p det gx0 = det X det gx . (2.91)

This is he relation between the two fundamental densities associated to each of the two coordinate systems. Of course, this corresponds to equation 2.18 (page 29), used to 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 of the manifold M , that we shall denote as Z V(D) = dv , (2.92) D by the expression Z Z dv ≡ dv g . (2.93) D D This definition makes sense because we have already defined the integral of a density in equation 2.46. Note that the (finite) capacity of a finite domain D cannot be R defined, 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 well happen that, in numerical evaluations of an integral, the division of the space into small ‘hyperparallelepideds’ that is implied by the use of the capacity element is not the best choice. Figure 2.3 suggests a division of the space into ‘cells’ having grossly similar volumes (to compared with figure 2.4). If the volume ∆vp of each cell is known, the volume of a domain D can obviously be defined as a limit

V(D) = lim ∑ ∆vp . (2.94) ∆vp→0 p

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 a sum, using elementary regions whose individual volume is known (for instance, triangles in this 2D illustration). This way of defining the volume of a region does not require the definition of a coordinate system over the space.

Figure 2.4: For the same shape of ﬁgure 2.3, the volume can be evaluated using, for instance, a polar coordinate system. In a numerical integration, regions near the origin may be oversampled, while regions far from the orign may be undersampled. In some situation, this problem may become crucial, so this sort of ‘coordinate integration’ is to be reserved to analytical developments only.

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 physical space (assimilated to an Euclidean 3D space) and that, for some reason, we chose 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, therefore which is the mass ∆m inside each cell. The quantity δm/∆v , being the ratio of a scalar by a capacity is a density. In the limit of an inﬁnite number of particles, we can take the limit where ∆r , ∆ϕ , and ∆z all tend to zero and the limit ∆m ρ(r,ϕ, z) = lim (2.96) ∆r→0 , ∆ϕ→0 , ∆z→0 ∆v 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 of radius 1, and cylindrical coordinates (r,ϕ, z) . Only a section of the cylinder is represented in the ﬁgure, with all its thickness, ∆z , projected on the drawing plane. At the 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 represent particles with given masses. As explained in the text, counting how many particles are inside each cell directly gives an estimation of the ‘mass density’ ρ(r,ϕ, z) . To have, instead, a direct estimation of the ‘volumetric mass’ ρ(r,ϕ, z) , a division of the space into cells of equal volume (not equal capacity) should have been done, as suggested at the right.

Given the mass density ρ(r,ϕ, z) , the total mass inside a domain D of the space is to be obtained as Z M(D) = dv ρ , (2.97) D 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

∆m ρ(r,ϕ, z) = lim (2.98) ∆v→0 ∆v gives the volumetric mass ρ(r,ϕ, z) , different from the mass density ρ(r,ϕ, z) . Given the volumetric mass density ρ(r,ϕ, z) , the total mass inside a domain D of the space is to be obtained as Z M(D) = dv ρ , (2.99) D where the volume element dv appears, not the capacity element dv . The relation between the mass density ρ and the volumetric mass ρ is the universal relation between any scalar density and a scalar,

ρ = g ρ , (2.100) where g is the metric density. As in cylindrical coordinates, g = r , the relation between mass density and volumetric mass is

ρ(r,ϕ, z) = r ρ(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 in probability theory about the meaning of a ‘probability density’ and of a ‘volumetric probability’. 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 = ( )a1 ( )ap dv a1...ap dx1 ... dxp . (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 element is ω = ( ψ )α1 ( ψ )αq d α1...αq d 1 ... d q . (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 that of the second manifold. When transporting the p vectors {dx1,..., dxp} from the first 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 relate dωp and dv . It is shown in the appendix (check!) that one has q t dωp = det(Ψ Ψ) dv . (2.106) Let us be interested in the image of the volume element. We denote by g the metric 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 the volume element dv of the first manifold, one has q t dωp det(Ψ γ Ψ) = p . (2.107) dv det g

2.3.2 Reciprocal Image of the Volume Element I do not know yet if I will need this section.