Displacement Calculus ∗

In this work we establish a theory of Calculus based on the new concept of displacement . We develop all the concepts and results necessary to go from the deﬁnition to differential equations, starting with topology and measure and moving on to differentiation and integration. We ﬁnd interesting notions on the way, such as the integral with respect to a path of measures or the displacement derivative. We relate both of these two concepts by a Fundamental Theorem of Calculus. Finally, we develop the necessary framework in order to study displacement equations by relating them to Stieltjes differential equations.


Introduction
Derivatives are, in the classical sense of Newton [23], infinitesimal rates of change of one (dependent) variable with respect to another (independent) variable.Formally, the derivative of f with respect to x is The symbol ∆ represents what we call the variation, that is, the change of magnitude underwent by a given variable 1 .This variation is, in the classical setting, defined in the most simple possible way as ∆x = x − x, where x is the point at which we want to compute the derivative (the point of departure) and x another point which we assume close enough to x.From this, it follows naturally that the variation of the dependent variable has to be expressed as ∆ f = f ( x) − f (x).This way, when x tends to x, that is, when ∆x tends to zero, we have Of course, this naïve way of defining the variation is by no means the unique way of giving meaning to such expression.The intuitive idea of variation is naturally linked to the mathematical concept of distance.After all, in order to measure how much a quantity has varied it is enough to see how far apart the new point x is from the first x that is, we have to measure, in some sense, the distance between them.This manner of extending the notion of variation -and thus of derivative-has been accomplished in different ways.The most crude of these if what is called the absolute derivative.

Definition 1.1 ([7, expression (1)]
).Let (X , d X ) and (Y, d Y ) be two metric spaces and consider f : X → Y and x ∈ X .We say f is absolutely differentiable at x if and only if the following limit -called absolute derivative of f at xexists: , x) .
In the case of differentiable functions f : → we have that, as expected, Hence, this result conveys the true meaning of the absolute derivative -it is the absolute value of the derivative-and it extends the notion of derivative to the broader setting of metric spaces.Even so, this definition may seem somewhat unfulfilling as a generalization.For instance, in the case of the real line, it does not preserve the spirit of the intuitive notion of 'infinitesimal rates of change': changes of rate have, of necessity, to be allowed to be negative.
A more subtle extension of differentiability to the realm of metric spaces can be achieved through mutational analysis where the affine structure of differentials is changed by a family of functions, called mutations, that mimic the properties and behavior of derivatives.We refer the reader to [22] for more information on the subject.
The considerations above bring us to another possible extension of the notion of derivative: that of the Stieltjes derivative, also known as g-derivative.Here we present the definition used in [21].However, a lot of previous work exists on the topic of differentiation with respect to a function, such as the work Averna and Preiss, [2], Daniell [8,9] or even more classical references like [16].

Definition 1.2 ([21]
).Let g : → be a monotone nondecreasing function which is continuous from the left.The Stieltjes derivative with respect to g -or g-derivativeof a function f : → at a point x ∈ is defined as follows, provided that the corresponding limits exist: if g is continuous at x, or if g is discontinuous at x.
Clearly, we have defined ∆x through a rescaling of the abscissae axis by g.Observe that, although d(x, x) = |g( x) − g(x)| is a pseudometric [10], ∆x = g( x) − g(x) is allowed to change sign.
The aim of this paper is to take this generalization one step further in the following sense.The definition of ∆x does not have to depend on a rescaling, but its absolute value definitely has to Section 3 deals with the construction of a measure associated to displacement spaces.We restrict ourselves to the real line, where we first construct its associated measure as a Lebesgue-Stieltjes measure.Then, we construct a theory of integration for displacement spaces.Here we define the concept of integral with respect to a path of measures which will be the key to defining an integral associated to a displacement.Section 4 is devoted to the definition and properties of a displacement derivative which will be later be proven to be compatible with the displacement measure in that we can provide a Fundamental Theorem of Calculus relating both of them (Theorems 4. 6 and 4.14).Later, in Section 5, we study the connection existing between this type of derivatives and Stieltjes derivatives and, in Section 6, we propose a diffusion model on smart surfaces based on displacements.
The last section is devoted to the conclusions of this work and the open problems lying ahead.

Displacement spaces
In this section we focus on the definition of displacement spaces.This new framework is then illustrated with some examples which show, for example, that every set equipped with a metric map is a displacement space.We also study a topological structure that displacement spaces can be endowed with.

Definitions and properties
Let us make explicit the basic definition of this paper.Definition 2.1.Let X = be a set.A displacement is a function ∆ : X 2 → such that the following properties hold: (H1) ∆(x, x) = 0, x ∈ X .
(H2) For all x, y ∈ X , lim All limits occurring in this work will be considered with the usual topology of .A pair (X , ∆) is called a displacement space.
Remark 2.2.Why (H1) and (H2)?These two hypotheses are of prominent topological flavor.(H1) will guarantee that open balls are nonempty in the to-be-defined non-necessarily-metric topology related to ∆.On the other hand, (H2) will be sufficient (and indeed necessary) to show that open balls are, indeed, open (Lemma 2.13) and that the ∆-topology is second countable (Lemma 2.17).We will later discuss (Remark 2.18) whether or not we can forestall (H2) when we restrict ourselves to displacement calculus.
Remark 2.3.Note that, for (H2) to be satisfied, it is enough to show that lim for all x, y ∈ X , as the reverse inequality always holds.
The following lemma gives a useful sufficient condition for (H2) to be satisfied.
Since ϕ(0) = 0, ϕ is continuous at 0 and lim n→∞ |∆( y, z n )| = 0, lim sup n→∞ ψ( y, z n ) = 0, so Indeed, by definition of lim inf, we have that for any ∈ + there exists Since ϕ is strictly increasing, for each ∈ + there exists n 0 ∈ such that for n ≥ n 0 we have Thus, for any > 0 we have that which, using the left-continuity of ϕ, leads to (2.3).Hence, it follows from (2.2) and (2.3) that which, together with the fact that ϕ is strictly increasing, yields that Since this holds for any (z n ) n∈ in X such that lim n→∞ |∆( y, z n )| = 0, we have the desired result.
Lemma 2.4 illustrates that condition (H2) is a way of avoiding the triangle inequality -or more general versions of it-which is common to metrics and analogous objects.We can find similar conditions in the literature.For instance, in [14, Definition 3.1], they use, while defining an RS-generalized metric space (X , ∆), the condition (D 3 ) There exists C > 0 such that if x, y ∈ X and More complicated conditions can be found in [22, (H3) Section 3.1, (H3') Section 4.1].
Last, we remark that the same statement as (H2'), but dropping the left-continuity, is actually sufficient to prove the results in this work.
In the next examples we use the sufficient condition provided by Lemma 2.4.
Example 2.5.Consider the sphere 1 and define the following map: ) is a displacement that measures the minimum counter-clockwise angle necessary to move from x to y.From a real life point of view, this map describes the way cars move in a roundabout.Suppose that a car enters the roundabout at a point x and wants to exit at a point y.In that case, circulation rules force the car to move in a given direction, which happens to be counter-clockwise in most of the countries around the world.In this case, drivers are assumed to take the exit y as soon as they reach it.
Example 2.6.Let (X , E) be a complete weighted directed graph, that is, X = {x 1 , . . ., x n } is a finite set of n ∈ vertices and E ∈ n ( ) is a matrix with zeros in the diagonal and positive numbers everywhere else.The element e j,k of the matrix E denotes the weight of the directed edge from vertex x j to vertex x k .This kind of graph can represent, for instance, the time it takes to get from one point in a city to another by car, as Figure 2 , and the map ∆(x j , x k ) := e j,k .It can be checked that ∆ is subadditive -which is to be expected since, if we could get faster from a point to another through a third one Google Maps would have chosen that option.Hence, (H2') holds for ϕ(r) = r, and so ∆ is a displacement.Example 2.7 (Zermelo's navigation problem).In 1931, Zermelo solved the following navigation problem [29].Let F = (u, v) ∈ ( 2 , 2 ) be a vector field, for instance, the velocity field of the wind on top of a body of water, or the velocity field of the water itself.Assume an object that moves with constant celerity V on that body of water wants to go from a point A (which we can assume at the origin) to a point B. Which is the least time consuming path to take?
We are going to assume that 2 , that is, the object can navigate against wind.Zermelo proved, using variational methods, that the solution of the problem satisfies the following system of partial differential equations: being the last equation known as Zermelo's equation.Observe that, if u, v ∈ 2 ( 2 ), there exists a unique solution of the system.Through the change of variables ( x, y) = B − (x, y), instead of going from the origin to the point B we go from B to the origin, and the equations will provide a different time.This illustrates the fact that, when measuring how far apart something is in terms of time, symmetry is not generally satisfied.For instance, if we measure the distance between two points of a river by the time it takes to get from one point to another it is not the same to go upstream than downstream.
Let A, B ∈ 2 .If ∆(A, B) is the smallest time necessary to arrive from A to B in Zermelo's navigation problem, ∆ ≥ 0 is a displacement on 2 , for ∆ is subadditive and (H1)-(H2') are clearly satisfied.
In the symmetric setting -that is, ∆(x, y) = ∆( y, x)this problem is a paradigmatic example of Finslerian length space.The theory regarding these spaces has been thoroughly developed but, as stated in [5], although "one could modify the definitions to allow non symmetric length structures and metrics", this case has not been studied yet.What we present in this paper might be an starting point for a theory of non symmetric length spaces.
Example 2.8.The map ∆ : × → defined as ∆(x, y) = g( y) − g(x) for a nondecreasing and left-continuous function g (cf.[21]) is a displacement as it satisfies (H2') for ϕ = Id.In what follows, we will refer to these displacements as Stieltjes displacements.Furthermore, the following lemma shows a way to ensure that a displacement is a Stieltjes displacement.Lemma 2.9.Let (X , ∆) be a displacement space.Then there exists g : X → such that ∆(x, y) = g( y) − g(x) for every x, y ∈ X if and only if, for every x, y, z ∈ X , Proof.Necessity is straightforward.In order to prove sufficiency, take x 0 ∈ X and define g(x) = ∆(x 0 , x) for x ∈ X .Then, For the interest of the work ahead, we include the following example of a non-Stieltjes displacement.

Displacement topologies
It is a well-known result that a set equipped with a metric map generates a topology through the definition of open balls.The same thing happens with displacement spaces.However, fewer nice properties can be obtained from just the definition.Definition 2.11.Given a displacement space (X , ∆), x ∈ X and r ∈ + , we define the ∆-ball or simply ball) of center x and radius r as Also, we define the ∆-topology in the following way: Clearly, τ ∆ is a topology.We denote by ∆ the set of ∆-balls in X and by τ u the usual euclidean topology of n for any n ∈ .

2⇔3. Just observe that |∆
Remark 2.14.Note that hypothesis (H1) is not necessary for the previous result or the definition of the topology itself.In fact, it has only been used so far to show that open balls are nonempty.
Allowing the open balls to be the empty set changes nothing as it always belongs to the topology, making the result true in any case.However, hypothesis (H1) will be key in the definition of the displacement derivative in Section 4.

Every element of τ ∆ is union of elements in
By definition of τ ∆ , we know that for every x ∈ U there exists r x ∈ + such that B(x, r x ) ⊂ U. Since x ∈ B(x, r x ), we have that X = x∈X B(x, r x ).

By (H2) we have that
Example 2.16.The conditions obtained in Lemma 2.15 do not suffice to obtain both (H1) and (H2).Consider the space X = {0, 1} together with de function ∆ given by ∆(0, In this case ∆ = τ ∆ and the topology coincides with that of the Sierpiński space.Observe that ∆ is not a displacement, although it satisfies the theses 1-3 of Lemma 2.15.
It is also worth to observe that the map ∆(0, This means that, if we want to find sufficient conditions in order for a ∆ to be a displacement, those conditions cannot be purely topological.Furthermore, since the Sierpiński space is not regular we deduce that it is not uniformizable, and thus we conclude that not every displacement space is uniformizable. In the particular context of the real line, some further results can be obtained.In order to archieve them we ask for the following hypothesis for ∆ : X 2 ⊂ 2 → .(H3) ∆(x, y) ≤ ∆(x, z) for every x, y, z ∈ X such that y ≤ z.Lemma 2.17.Let ∆ : 2 → satisfy (H1)-(H3).Then ( , τ ∆ ) is a second-countable topological space.
Proof.First of all, given x ∈ and r ∈ + , we can express B ∆ (x, r) as follows: Moreover, since ∆ x is non-decreasing, due to the bounded completeness of ( , ≤), by definition of open set, and so, since each B ∆ (x, r x ) is a interval, we can write for some sets of indices , , , , where each of those intervals is an open ball of τ ∆ .
The set A = i∈ (a i , b i ) is an open set in ( , τ u ) and therefore second countable, which implies that A is Lindelöf [13, p. 182] and, hence, there exists a countable subcover of A, i.e., A = n∈ (a i n , b i n ) for some set of indices {i n } n∈ .Similarly, the set B = j∈ [a j , b j ) is an open set in the Sorgenfrey line, which is hereditarily Lindelöf [13, p. 79], and so and once again, arguing as for the sets B and C, we obtain that for some sets of indices {l n } n∈ , {l n } n∈ .However, by the definition of D we have that a l , b l ∈ D for all l ∈ , so which is clearly countable.Therefore, U is the countable union of open balls, i.e., τ ∆ is a secondcountable topology.
Remark 2.18.This last proof relies heavily on the fact that the real number system, with its usual order, is bounded complete, that is, that every bounded (in the order sense) set has an infimum and a supremum.Observe also that the interaction between the topologies τ u and τ ∆ plays a mayor role in the proof.Finally, hypothesis (H2) is necessary in this result through Lemma 2.13, which implies that open ∆-balls are, indeed, open.
Related to this last point, the authors would like to comment on the fact that hypothesis (H2) will not be necessary in the particular setting of the displacement calculus.However, it provides -as illustrated before with Lemmas 2.13 and 2.17-some information about the relation between τ ∆ and the displacement calculus we are yet to develop.In particular, Lemma 2.17 shows that, for the real line, every τ u -Borel σ-algebra is, in particular, a τ ∆ -Borel σ-algebra so the integration theory that will follow, when considering (H2), will be valid for the open sets of τ ∆ .Nevertheless, while studying specific problems -like differential equations-we will deal, in general, with intervals or other elements of the τ u -Borel σ-algebra without worrying about the specifics of the τ ∆ topology, which, as said before, makes (H2) unneeded.Definition 2.19.Given displacement spaces (X , ∆ 1 ) and (Y, ∆ 2 ), a function f : We say that a map f : As usual, continuity can be characterized using open balls, as it is shown in the following result. (2.6) 3 Displacement measure theory on the real line In this section we aim to define a measure over a non-degenerate interval [a, b] ⊂ .To do so, we will use "local" measures µ z , for z ∈ [a, b], to construct a measure µ which does not depend on a specific point z.In order to achieve that, we will consider ([a, b], ≤, ∆) satisfying hypotheses (H1)-(H3), and two extra conditions: (H4) There exists γ : is left-continuous (with the usual topology of ) at x. Remark 3.1.Note that under hypothesis (H3), it is enough to check that there exists γ : to confirm that (H4) holds.
First of all, note that the set of maps ∆ : [a, b] 2 → that satisfy hypotheses (H1)-(H5) is not empty, as any Stieltjes displacement satisfies all of them.Moreover, there exist non-Stieltjes displacements that also satisfy all of the hypotheses.To show that this is the case, we will need the following result.
The function γ is well-defined and bounded because the three variable mapping is continuous on a compact domain.In particular, (H4, iii) holds.
Finally, (H4, ii) is a consequence of the fact that D 2 ∆ is continuous on [a, b] 2 , and, therefore, uniformly continuous on [a, b] 2 .Indeed, let z ∈ [a, b] be fixed; for > 0 we can find δ > 0 such that for each z ∈ [a, b], |z − z| < δ, we have that Therefore, if |z − z| < δ, we have that We have just proven that It clearly has continuous partial derivatives, and Hence, ∆ satisfies (H1)-(H5) for γ defined as in (3.1).
Although hypothesis (H5) might seem harmless, when combined with (H4), we obtain leftcontinuity everywhere.Proof.Let > 0, x, y ∈ [a, b] and γ be the map on (H4).Let us show that ∆ x is left-continuous at y. Since, by (H5), ∆ y is left-continuous at y, there exists δ > 0 such that, for 0 With the previous result in mind, we can define the "local" measures µ z as the Lebesgue-Stieltjes measure associated with the non-decreasing left-continuous map ∆ z .We shall denote by z the σ-algebra over which µ z is defined.

Let us denote by
the Borel σ-algebra (for τ u ) and by := z∈[a,b] z .Note that ⊂ as ⊂ z for all z ∈ [a, b].Moreover, is a σ-algebra as it is an arbitrary intersection of σ-algebras.Hence, we can consider the restriction of µ z , z ∈ [a, b], to .We will still denote it by µ z .A set A ∈ is said to be ∆-measurable.
Recall that a function f : for all U ∈ .We will say in that case that f is ∆-measurable.This notation will be consistent with the ∆-measure that we will introduce later.Observe that f is ∆-measurable if and only f : Thus, we have that µ z µ z µ z for all z, z ∈ X .Hence, if a property holds µ z -everywhere, it holds µ x -everywhere for all x ∈ [a, b].Again, in order to simplify the notation, we will say that such property holds ∆-everywhere.Analogously, this expression will be consistent with the ∆-measure presented later in this paper.
Then, given z, z ∈ [a, b] we can apply the Radon-Nikodým Theorem [3] to these measures, so there exist two ∆-measurable From these expressions it is clear that h z,z = 1 and Further properties are shown in the next results.Hence, which is a contradiction.Therefore, For the other inequality, take h z,z as in (3.3).Using (3.5), we have that from which the result follows.
Note that this result yields that, for z, z ∈ [a, b] fixed, we have that Then, it follows from (3.4) that Thus, we can assume without loss of generality that the functions in (3.3) satisfy (3.6).Given this consideration, we can obtain the following result.
Proposition 3.6.For all t ∈ [a, b], we have that Hence it is enough to consider the limit when z → z in the previous inequalities, together with hypothesis (H3, ii), to obtain the result.
Remark 3.7.Note that, given z ∈ [a, b], we also have that there exist m z , M z > 0 such that Now the result follows from (H3, iii).
We will now focus on the definition of the ∆-measure which is based on the integrals defined by the measures µ z , z ∈ [a, b].We first will show that a bigger family of maps is well-defined.
Proof.In order to show that h(•, α(•)) is µ z -measurable, let us define the map h z : [a, b] 2 → [0, +∞) given by h z (t, x) = h x,z (t).We will first show that h z is a ∆ z -Carathéodory in the sense of [10, Definition 7.1] adapted to our notation, that is: (iii) for every r > 0, there exists f r ∈ The map µ α is a measure, [25, Theorem 1.29], and it will received the name of ∆ α -measure.In particular, when α is the identity map, it will be called the ∆-measure, and it will be denoted by µ ≡ µ Id .
The following result shows that µ α does not depend on a specific point of Observe that the notation we have used so far is consistent with the definition of the ∆measure µ.Indeed, for example, f is ∆-measurable if and only if it is µ-measurable; as µ and µ z , z ∈ [a, b] are both defined, after due restriction of µ z to , over the same σ-algebra.Also, by definition, we have that µ µ z .The converse is also true thanks to (H4, iii).Indeed, for z ∈ [a, b], there exists Thus, if µ(A) = 0 then µ z (A) = 0, i.e., µ z µ for all z ∈ [a, b].Therefore, a property holds ∆-everywhere if and only if it holds µ-everywhere.
As a final comment, note that µ : → [0, +∞] is a Borel measure that assigns finite measure to bounded sets.As it can be seen in [1, Chapter 1, Section 3, Subsection 2], this means that it can be thought of as a Lebesgue-Stieltjes measure, µ g , given by .We define the integral of a µ α -measurable function f over X with respect to the path of ∆-measures α as provided the integral exists.This definition does not depend on the z chosen.As usual, we define the set of µ α -integrable functions on X as , we can define the set of ∆integrable functions over X ∈ as We now study the relationship between 1 ∆ (X ) and 1 µ (X ).First of all, recall that, in this framework, µ and µ z are defined over the same σ-algebra , so the concepts of µ-measurable and ∆-measurable are equivalent.Let f ∈ 1 ∆ (X ) and z ∈ [a, b].Hypothesis (H4, iii) implies that there exist M > 0 such that |γ(x, z)| < M for all x ∈ [a, b].Thus, using (3.6) we have Finally, we study the behavior of µ over some interesting sets related to the map ∆.These sets will be fundamental in the definition of the ∆-derivative.Let us define the sets C ∆ and D ∆ as Proposition 3.12.Let g be as in (3.10) and let C g and D g be as in [21], that is, Then C ∆ = C g and D ∆ = D g for the Stieltjes displacement given by g.
Proof.For the equality D ∆ = D g , it is enough to note that for any t ∈ [a, b] we have that Now, in order to see that C ∆ = C g , let t ∈ C ∆ .Then ∆ t (•) = 0 on (t − , t + ) for some ∈ + .Let r, s ∈ (t − , t + ), r < s.Then, by Remark 3.7, we have that That is, ∆ t is constant on (t − , t + ), and since ∆ t (t) = 0, it follows that t ∈ D ∆ .
The first consequence of Proposition 3.12 is that D ∆ is at most countable since it is the set of discontinuities of a monotone function.Further properties can be obtain from Propositions 2.5 and 2.6 in [21].

Displacement derivatives
We now introduce the concept of displacement derivative of a function defined over a compact interval endowed with a displacement structure in the real line equipped with the usual topology.
We chose this setting because some nice properties, such as the linearity of the derivative, are quite helpful in order to study the relationship between the displacement derivative and its integral.
Observe that this definition does not require ∆ to be symmetric.Furthermore, this definition is a more general setting than g-derivatives (and therefore time-scales, as pointed out in [21]).
Finally, one might think that the natural choice for the definition of the derivative would be by taking the limit in the τ ∆ topology.However, if x ∈ D ∆ , x is a continuity point of ∆ x , and it is easy to see that such limit can be translated into a limit in the usual topology, which is far more convenient for the theory that follows.It is at this point that the importance of (H1) arises as commented in Remark 2.14.Without this hypothesis we would not be able to assure that the balls of center x and any radii are nonempty, so considering the τ ∆ limit might not be well-defined.

Fundamental Theorem of Calculus
In this section we will make explicit the relationship between the ∆-derivative of f and its integral with respect to the ∆-measure.In particular, our first goal now is to show that, for f In the particular setting of Stieltjes derivatives, this result has been proven in different ways and can be found in [8] and, more extensively, in [11].
In order to do so, we will follow an approach similar to that of [21], starting by guaranteeing the differentiability of monotone functions.For that matter, we will use the following two results that are direct consequences of Lemmas 4.2 and 4.3 in [21] adapted to our framework.We can now prove the ∆-differentiability of monotone functions.To do so, we will follow the ideas of [4].Proof.First of all, note that, since f is nondecreasing, f is regulated so for any y = x, we can define the Dini upper and lower ∆-derivatives as Furthermore, since f is monotone, it has a countable number of discontinuity points, so Thus, it is enough to show that the sets both have ∆-measure zero (and therefore µ-measure zero).
We first show that F is a null ∆-measure set.Fix z ∈ (a, b) \ (D ∆ ∪ O ∆ ) and define the Dini upper and lower ∆ z -derivatives as Note that (H4) implies that and, analogously, Hence, F is a subset of so, it suffices to show that µ z (F n ) = 0 for all n ∈ .By contradiction, assume that there exists n 0 ∈ such that µ z (F n 0 ) > 0. In that case, we rewrite F n 0 as the countable union of sets F n 0 ,r,s with r, s ∈ , r > s > 0, and Thus, there exist r 0 , s 0 ∈ , r 0 > s 0 > 0, and ∈ + such that µ z (F n 0 ,r 0 ,s 0 ) = .Now let α = r 0 −s 0 2n 0 , β = r 0 +s 0 2n 0 and h(x) = f (x) − β∆ z (x).Then F n 0 ,r 0 ,s 0 = H with Note that h is of bounded variation as it is the difference of two nondecreasing functions.Therefore, the set . By doing this, we obtain a collection of open subintervals of (a, b), Now let Q = { y 0 , y 1 , . . ., y q } be the partition of [a, b] determined by the points of P and the endpoints of the intervals I 1 , I 2 , . . ., I N .For each [x k−1 , x k ] containing at least one of the intervals in {I 1 , I 2 , . . ., I N }, Proposition 4.2 yields that where the summation is taken over the closed intervals determined by and L k is the sum of the ∆ z -measures of those intervals I 1 , I 2 , . . ., By taking the previous inequality and summing over k, we obtain which contradicts the definition of T .
Hence, all that is left to do is to show that the set Let which is a contradiction.
Finally, a key result for the proof of the Fundamental Theorem of Calculus is Fubini's Theorem on almost everywhere differentiation of series for ∆-derivatives.We now state such result but we omit its proof as it is essentially the one provided in [26] We now have all the necessary tools to state and prove the first part of the Fundamental Theorem of Calculus for ∆-derivatives.

Theorem 4.6 (Fundamental Theorem of Calculus
Proof.Without loss of generality we can assume that f ≥ 0, as the general case can be reduced to the difference of two such functions.Since f ≥ 0, the function F is nondecreasing and therefore ∆-differentiable.We consider several cases separately: where the integral is to be understood as a Lebesgue-Stieltjes integral.Note that H is welldefined.Then were the equality H ∆ x (x) = h x,x (x) follows from the Fundamental Theorem of Calculus for the Stieltjes derivative (see [21,Theorem 2.4]).
Case 2: Let M 0 (∆) be the set of all step functions whose discontinuities are not in Case 3: There exists a nondecreasing sequence and then it follows from the Lebesgue's Monotone Convergence Theorem for measures that for all x ∈ [a, b].Since each summand is a nondecreasing step function of x, we can apply Case 2 and Proposition 4.5 to deduce that for µ-a.a.x ∈ [a, b) we have , where each of the f i 's is the limit of a nondecreasing sequence of step functions in the conditions of Case 3. Definition 4.7.Let x ∈ and F : [a, b] → .We shall say that F is ∆ x -absolutely continuous if for every > 0, there exists δ > 0 such that for every open pairwise disjoint family of subintervals A map F : [a, b] → n is ∆ x -absolutely continuous if each of its components is a ∆ x -absolutely continuous function.
Remark 4.8.Note that as a consequence of (H4), if F is ∆ x -absolutely continuous, it is ∆ yabsolutely continuous for all y ∈ .Hence, we will just say that F is ∆-absolutely continuous.
In the following results, we present some of the properties that ∆-absolutely continuous functions share.
Hence, |F | is bounded on [a, b].Let K > 0 be one of its bounds.For any partition {x 0 , x 1 , . . ., Consider a partition { y 0 , y 1 , . . ., x ([ y k−1 , y k )), k = 1, 2, . . ., n.Since ∆ x is nondecreasing, the sets I k are empty or they are intervals not necessarily open nor close.Anyway, [a, b] = ∪I k , and so it is enough to show that F has bounded variation on the closure of each I k .We assume the nontrivial case, that is, Proof.Fix x ∈ [a, b] and > 0 and let δ > 0 be given by the definition of ∆ x -absolute continuity of F .Since ∆ x (•) is left-continuous at x, there exists δ > 0 such that if 0 < x − t < δ then, The proof in the case ∆ x is right-continuous at x ∈ [a, b) is analogous, and we omit it.
As a consequence of these two previous propositions, given F , a ∆-absolutely continuous function, there exist two nondecreasing and left-continuous functions, F 1 , F 2 , such that F = F 1 − F 2 .We denote by µ i : ([a, b]) → the Lebesgue-Stieltjes measure defined by F i , i = 1, 2. Recall that Lebesgue-Stieltjes measure are positive measures that are also outer regular, that is, for every E ∈ ([a, b]), we have A natural definition for a signed measure for the function F is given by and so By letting a n tend to a n , we obtain Now, by the first part of the proof, we know that lim Hence, µ F µ x , and since µ x µ, the result follows.
Then F is ∆-absolutely continuous.
Proof.It is enough to consider the case f ≥ 0, as the general case can be expressed as a difference of two functions of this type.
Fix > 0 and x ∈ .Hypothesis (H4, iii) implies that there exists Proof.Lemma 4.13 ensures that the three conditions are sufficient for F to be ∆-absolutely continuous.For the converse, consider µ F to be the Lebesgue-Stieltjes measure defined by F and let z ∈ [a, b] be fixed.Lemma 4.12 and the Radon-Nykodym Theorem guarantee that there exists a measurable function l : In particular, Theorem 4.6 ensures that F ∆ (s) = l(s) for µ-a.a.s ∈ [a, b), and so the result follows.

The relation with Stieltjes derivatives
As commented before, Stieltjes derivatives are, in a first approach, a particular case of ∆-derivatives.However, there can be proven to be equivalent to the displacement derivatives if hypotheses (H1)-(H5) hold.Indeed, we shall prove this equivalence through following results.Therefore, we have that both derivatives are, indeed, equivalent.This shows the interest of studying this type of derivatives.In particular, when looking at differential equations, a wide variety of results exists in several papers such as [21] where the authors showed that Stieltjes differential equations are good for studying equations on time scales and impulsive differential equations, or [10,[17][18][19][20] where we can find different types of existence and uniqueness of solution results.
However, obtaining the corresponding function g can be hard.Indeed, although (3.10) gives an explicit expression of the function, it is defined in terms of the ∆-measure, which depends on the functions (3.3) given by the Radon-Nidokým Theorem.The following result gives a simple way to obtain, under certain hypotheses, the corresponding function g.

A model for smart surface textures
In this section we develop a model for smart surfaces based on a displacement and a derivation similar to that of the diffusion equation.
It is every day more extended to employ biomimetics in order to develop surfaces with extraordinary properties [28].These meta-materials imitate organic tissues with special microstructures which modify their usual behavior.For instance, a cats tongue possesses backwards-facing spines, a disposition which facilitates that particles move towards the interior of the mouth and not in the other direction.This situation is similar to the one on the human respiratory epithelium, with the difference that in this other tissue the effect is due to the active motion of the cells cilia, which move mucus and particles upwards, and not a passive result of the microstructure.
With 3D-printing (or other methods) we may obtain these surfaces for which friction depends on direction, position, pressure, etc.We can measure friction in an indirect but simple way using the definition of work: work is the energy necessary to move and object between two points against a given force field.Thus, on our surface, which, for convenience in the present discussion, we will consider one-dimensional (the higher dimensional case would be analogous), we can define a function W (x, y) which measures the work necessary to move a point mass from x to y.
If we consider now a distribution of particles on the surface subject to random vibrations, this kind of situation may be described as a Brownian motion but, in the case those particles are very small, this model may be approximated by a diffusion process.The derivation of the classical diffusion process -that is, of Fick's second law or, equivalently, the heat equation-can be found in many references -see for instance [12,15,27].By Nernst's law, in the friction-less setting, the mass flowing through the point x during the time interval (t, t + d t) is equal to where u denotes the concentration of mass, ∆u the spatial variation of u, D is the diffusion coefficient and d x, d t are considered to be infinitesimal quantities.With friction, this variation of the mass flow is impeded by the work necessary to move the particles, that is, W .We will assume that the variation of the mass flow is inversely proportional to the spatial variation of this work, that is, On the other hand, the spatial variation of Q can be computed directly as where h is a source term in the case we allow for a continuous inflow of particles.At the same time, the variation in the mass flowing through the sectional volume, being proportional to the concentration of mass, can be computed as  Under the conditions of Theorem (5.2) -that is, (H1)-(H5)-we can apply theorems such as [17,Theorem 3.5] to derive the existence of solution of problem (6.2).

Conclusions
In the work behind we have established a theory of Calculus based on the concept of displacement.We have studied the associated topology and measure and proved some general results regarding their interaction.We have also defined new concepts such as the integral with respect to a path of measures or the displacement derivative, studied their properties and proved a Fundamental Theorem of Calculus that relates them.Finally, we have set up a framework in order to z)| = |∆(x, y)|, where lim z y |∆(x, z)| := sup lim inf n→∞ |∆(x, z n )| : (z n ) n∈ ⊂ X , ∆( y, z n ) n→∞ − −− → 0 .

7 NFig. 2 . 1 . 1 : 3 : 4 :
Fig. 2.1.Graph indicating the time in minutes it takes to go from one place to another in Santiago de Compostela by car (using the least time consuming path) according to Google Maps -good traffic conditions assumed.The points are placed in their actual relative geometric positions, being 1: Faculty of Mathematics (USC), 2: Cathedral, 3: Train station, 4: Bus station.Most of the streets in Santiago are one way, which accounts for the differences in time depending on the direction of the displacement.

Proposition 3 . 2 .
Let ∆ : [a, b] 2 → be a given map and let us denote by D 2 ∆ its partial derivative with respect to its second variable.If D 2 ∆ exists and is continuous on [a, b] 2 , and there exists r > 0 such that D 2 ∆(x, y) ≥ r for all (x, y) ∈ [a, b] 2 , then ∆ satisfies (H3)-(H5).Proof.The assumptions imply that for each x ∈ [a, b], the mapping ∆(x, •) is increasing and continuous, which is more than (H3) and (H5).Now fix z, z ∈ [a, b].For a ≤ x < y ≤ b, the generalized mean value theorem guarantees the existence of ξ ∈ (x, y) such that
Hypothesis (H4) allows us to understand the relationship between the different possible measures on depending on z ∈ [a, b].In particular, given z, z ∈ [a, b], and an interval I ⊂ [a, b], it is clear that µ z (I) ≤ γ(z, z)µ z (I) and, as a consequence of the definition of the Lebesgue-Stieltjes measures, µ z (A) ≤ γ(z, z)µ z (A), for all A ∈ .

Proposition 4 . 3 . 3 . 2 .
Let ([a, b], ≤, ∆) satisfy (H1)-(H5) and H ⊂ (a, b) be such that for a given z ∈ [a, b], there is z ∈ + such that µ * z (H) = z .Then 1.If is any collection of open subintervals of [a, b] that covers H, then there exists a finite disjoint collection {I 1 , I 2 , . . ., I N } of such that N k=1 µ z (I k ) > z If P is a finite subset of [a, b] \ D ∆ and is any collection of open subintervals of [a, b] that covers H \ P, then there exists a finite disjoint collection {I 1 , I 2 , . . ., I N } of such that N k=1 µ z (I k ) >z4.

Proposition 4 . 9 .
Let f : [a, b] → [c, d] be a ∆-absolutely continuous function and let f 2 : [c, d] → satisfy a Lipschitz condition on [c, d].The the composition f 2 • f 1 is ∆-absolutely continuous on [a, b].Proof.Let L > 0 be a Lipschitz constant for f 2 on [c, d].Fix x ∈ [a, b].For each > 0 take δ > 0 in Definition 4.7 with replaced by /L.Now, for an open pairwise disjoint family of subintervals

Proposition 4 . 10 .
Let F : [a, b] → be a ∆-absolutely continuous function.Then F is of bounded variation.Proof.To prove this result we will use the following remark: if for any [α, β] ⊂ (a, b) there exists c > 0 such that the total variation of F on [α, β] is bounded from above by c, then F has bounded variation on [a, b].Indeed, assume that for any [α, β] ⊂ (a, b) there exists c > 0 such that the total variation of F on [α, β] is bounded from above by c.Then for each x ∈ (a, b), and so, our claim holds.Now, to prove that F has bounded variation on [a, b], fix x ∈ [a, b] and take = 1 in the definition of ∆ x -absolute continuity.Then, there exists δ > 0 such that for any family {(a n , b n )} m n=1 of pairwise disjoint open subintervals of [a, b], m n=1

Proposition 4 . 11 .
Now, our previous claim implies that F has bounded variation on each I k , and therefore F has bounded variation on [a, b].Let F : [a, b] → be a ∆-absolutely continuous function.Then F is leftcontinuous everywhere.Moreover, F is continuous where ∆ is continuous.

Lemma 4 . 12 .
Let F : [a, b] → be a ∆-absolutely continuous function.Then for every x ∈ [a, b] we have µ F µ. Proof.Let x ∈ [a, b], > 0 and δ > 0 given by the definition of ∆ x -absolute continuity with replace by /2.Fix an open set V ⊂ (a, b) such that µ x (V ) < δ.Without loss of generality, we can assume that V = n∈ (a n , b n ) for a pairwise disjoint family of open intervals.For each n ∈ , take a n ∈ (a n , b n ).Then, for each m ∈ we have m n=1

Theorem 4 . 14 .
A function F : [a, b] → is ∆-absolutely continuous on [a, b] if and only if the following conditions are fulfilled: (i) there exists F ∆ for µ-a.a.x ∈ [a, b];
z→x h z,x (t)h x,z (t) = h x,z (t), for ∆-a.a.t ∈ [a, b],Thus, (iii) holds, i.e., the map h z is ∆ z -Carathéodory.Now, as it is shown in [10, Lemma 7.2], the composition of a ∆ z -Carathéodory function with a Borel measurable function is ∆-measurable, and so the result follows.Definition 3.9.Let α : ([a, b], ) → ([a, b], ) be a measurable map and z ∈ [a, b].Consider the map µ α : → [0, +∞) given by , noting that (3.6) holds ∆-almost everywhere, it holds µ-almost everywhere and so Now the rest of the proof is analogous to the previous case, and we omit it.
Fix t ∈ [a, b].Since D 2 ∆(t, t) > 0, the function ∆ t is strictly nondecreasing in a neighborhood of t.Thus, t ∈ C ∆ , and so, if we apply Proposition 5.1, we have that D 2 ∆(t, t) = lim (t) exists and equals to D 2 ∆(t, t).The rest of the result now follows.
(x + d x, x + 2 d x) + h(x) d x d t = c∆u(x, t) d x.for some constant c.In the limit d x → 0, ∆u(x, t) W (x, x + d x) ≈ ∂ x,W ∆u(x, t), ∆u(x + d x, t) W (x + d x, x + 2 d x) ≈ ∂ x,W ∆u(x + d x, t).where ∂ x,W denotes the displacement derivative for the displacement W with respect to the variable x.Thus,−D∂ x,W ∆u(x, t) + D∂ x,W ∆u(x + d x, t) + h(x) d x d t = c∆u(x, t) d x, that is, D ∂ x,W ∆u(x + d x, t) − ∂ x,W ∆u(x, t) x ∂x,W u(x, t) + h(x) = c∂ t u(x, t).