Prof Romain Pasquier

Multilevel Governance and the Challenges of Integration and Coordination in Anti-Poverty Policies: the French Experience Between Change and Continuity Paper prepared for the

This paper addresses the issues of coordination and integration of public action in the field of anti-poverty policies. It is based on a recent analysis of the conceptualization and implementation of the "Stratégie nationale de prévention et de lutte contre la pauvreté " (national poverty action strategy SNPLP) from 2018 to 2021 in France. Indeed, this strategy was intended to be emblematic of a desire to profoundly transform the "governance" of poverty in France. From a new political vision focused on the activation of the resources of vulnerable people , the strategy tries to improve the integration of both traditional sectoral policies (social, education, and public health) and multi-scalar systems of actors (central administrations, deconcentrated administrations, local authorities, and associative actors). However, this strategy faces different major obstacles in the policy-making process. It illustrates the tensions between, on the one hand, mechanisms of intersectorality and territorialization that contribute to giving a form of coherence to the strategy and, on the other hand, sectoral mechanisms and territorial fragmentation of initiatives that weaken it, or at least cause it to lose some of its initial transformative power. Our paper will focus on the variables that explain this tension between policy integration and fragmentation. With regard to the first dimension, we will analyse in particular the reference to the social investment theory, the affirmation of presidential leadership, the creation of an ad hoc administration, and the implementation of a territorial contractualization based on monitoring indicators. In the second dimension, we will highlight the impact of several crises on ministerial and sectoral agendas, the plurality of bureaucratic interests intervening in the policy field of "poverty" and the variation of local systems (local authorities and associative networks) involved in the implementation of anti-poverty policies empowerment of some local anti-poverty systems at the intersection of association networks and territorial authorities.

The expansion of a welfare state, which began after World War II, is a key feature of States that have become 'mass producers' of public policies [START_REF] Philipppe | Sociologie de l'Etat et politiques publiques[END_REF]King and Le Galès 2017). This is particularly the case in countries like France that have historically devoted a significant share of their wealth to social policy [START_REF] Robert | Les métamorphoses de la question sociale[END_REF][START_REF] Swan | Sous l'aile protectrice de l'Etat[END_REF][START_REF] Paugam | Repenser la solidarité[END_REF]. For decades, the central State and its specific administrations, the various related agencies, as well as territorial authorities1 and associations have deployed a complex set of social intervention measures (Aquilera and Rouzeau 2020;[START_REF] Renaud | La gouvernance territoriale : une affaire d'État La dimension verticale de la construction de l'action collective dans les territoires[END_REF]Lafore 2020;Pasquier et al. 2013). This accumulation of public policies raises the question of whether they are coherent and to what extent they converge. This paper therefore attempts to take a close look at the questions of coordination and integration of public action in a social policy field that is 'saturated' with sectorial and territorialized measures. We draw from an analysis of the recent establishment of a domain of public action concerning poverty, in particular from 2018 to 2020, involving the formulation and implementation of the French "Stratégie nationale de prévention et de lutte contre la pauvreté" (national poverty action strategy -SNPLP)2 . During the first months of Emmanuel Macron's presidency, the SNPLP was held up as the symbol of a volition to totally transform the 'governance' of poverty in France by interlinking it with a global strategy inspired by social investment [START_REF] Avenel Cyprien | L'investissement social : quelle stratégie pour la France ?[END_REF][START_REF] Bruno | Towards a Worldwide View on the Politics of Social Investment[END_REF] for both traditional sectorial policies (social, medical-social, education, health) and systems featuring multi-scale actors (central administrations, devolved administrations, agencies, local authorities, associations). Yet, unlike a preventative approach mainly aimed at developing human capital and stimulating the resources of vulnerable people, the way that the SNPLP was implemented means that, at the national level, it had to work with what was 'already there'. The impact of the 'yellow vest' crisis followed by the pandemic triggered a return to social protectionism and charity activities in France. Thus the SNPLP constitutes a particularly useful laboratory to investigate the tensions underlying public action between, on the one side, the intersectoral and territorial adaption mechanisms that help make strategy more coherent and pertinent (O'Flynn et al. 2013) and, on the other side, the mechanisms of re-sectoralization [START_REF] Muller | Les politiques publiques entre secteurs et territoires[END_REF]) and the territorial fragmentation of initiatives that weaken a strategy, or at least diminish some of its initial transformative power.

Our paper focuses on the variables that help explain this tension between (sectorial and territorial) integration and (sectorial and territorial) autonomy. Concerning the first dimension, we will in particular analyse the reference to social investment, the affirmation of presidential policy leadership, the creation of ad hoc administrations (inter-ministerial delegation and poverty commissioners), and the promotion of key measures, territorial contracting, and the establishment of monitoring indicators. For the second dimension, we will centre on the impact of the health crisis and the electoral cycle on ministerial and sectoral agendas, the plural nature of 'bureaucratic work' involved in the 'poverty' category (solidarity of rights), freedom of administration by local authorities, their unequal political capacities and their different levels of engagement in action against poverty (départements and municipalities) and, lastly, the

Emergence of a public action field: dispersal, sedimentation and ordering

The oil crisis of the 1970s called into question the prevailing belief that growth was the main source of social development. In parallel, the 'sustainability' of the welfare system no longer seemed definitively guaranteed. The result was an accumulation of forms of social intervention involving the maintenance of assistance and protectionism, combined with a diversification of forms of social intervention, a combination of different types of regulation (sectoral measures, decentralization, socio-territorial strategies, etc.), Europeanization mechanisms translated by a double rhetoric of social cohesion and inclusion, and, in France, the invention of 'active solidarity'.

In this context of accumulation, starting from 2013, poverty became the target of intersectoral 'national mobilization': the new connections made between hitherto dispersed initiatives gave rise to the institutionalization of a new field of public action. Emerging from this background, and marked by ambitions put forward as deeply transformative, the SNPLP (2017-2022) for a while became a fairly consequential vehicle for integration. This shift involved the promotion of social investment, the alignment of government authorities, the enrolment of multiple stakeholders, and the intended, sometimes successful, alignment of numerous measures resulting from more or less autonomous sectoral rationales.

Pluralization of references and sedimentation of measures

Sometimes separately devised and coordinated, sometimes managed in a more unified way, the social-territorial strategies and insertion policies that began to emerge in the 1980s attempted to do away with the fear of exclusion, in particular among the middle and working classes. Centred on social housing neighbourhoods, socio-urban interventions involved interministerial bodies focused on urban policy and the elected representatives of 'urban governments' (Tissot 2007;[START_REF] Renaud | La rénovation urbaine. Démolition et reconstruction de l'État[END_REF]. During the same period, in an attempt to intersect attribution, contracts with beneficiaries, and socio-professional support, inclusion policies mobilized sectoral central administrations (social affairs and, to lesser extent, work and employment) and the départements (administrative divisions). This was combined with charity work done by associations and Centres communaux d'action sociale (municipal social action centres -CCAS) [START_REF] Gilles | Le rôle du département en tant que chef de file dans le domaine de l'insertion[END_REF].

At the same time, the institutional landscape became more diverse following different decentralization phases, which made the départements responsible for social action [START_REF] Robert | La décentralisation de l'action sociale. L'irrésistible ascension du "département-providence[END_REF](Lafore , 2013)). In addition, it became denser and more complex as a result of the more widespread use of public action contracts (Gaudin 2007), the systematic implementation of calls for tender, and the increased role of national and regional agencies [START_REF] Daniel | Les agences, alternatives administratives ou nouvelles bureaucraties techniques[END_REF]. In parallel, with different levels of implementation, a 'territorial administration' model that put the emphasis on 'arrangements' between Prefects and key figures, generated a variety of territorial differences [START_REF] Duran Patrice | L'État et la gestion publique territoriale[END_REF].

At the same time as this diversification of types of intervention and institutional complexification, through the international circulation of ideas, new references (activation, inclusion, social cohesion, etc.) came to join the principles of equality and solidarity of rights characteristic of the French welfare system (Warin 2020). In parallel, the promotion of decompartmentalization and calls for a more transversal approach generated numerous importexports with the sectors of education and training, housing, employment, health, and even culture and sport [START_REF] Garraud | Le chômage dans l'action publique. Le « bricolage » institutionnalisé[END_REF][START_REF] René | Des interventions de plus en plus sociales[END_REF]. In addition, increasing references were made to sustainable development and empowerment, leading to a multiplication of socially just representations.

Thus, with the arrival of the 2000s, the combination of these multiple references and strategic options, the diverse adjustments conceded to the local level, and the mass of measures, rendered the social political landscape disparate and irregular. In this context of accumulation, although the sequencing and sectoral homogeneity of social action were put to the test, the fight against poverty, which had for a long time been a residual object of public action, progressively took shape and became institutionalized [START_REF] Viguier | La cause des pauvres[END_REF]. In this relatively fragmented landscape, the Plans nationaux pour l'inclusion sociale (national plans for social inclusion -PNAI) (2001-2003; 2003-2005; 2006-2008 and 2008-2011) were a first attempt to achieve a degree of consistency. The PNAIs, formulated at the request of European bodies, featured an aggregation of measures and proposed budget consolidation. In addition to this formal gathering together, PNAIs acted as vehicles to disseminate the action principles championed by the EU in the social field, e.g. child poverty action, access to rights, and participation by the people concerned.

Renewal of elites and trajectory for reform

The whole of this period also saw an at least partial renewal of the 'welfare elites'3 , which supported the trajectory for reform in terms of protection and social action. Following the ministerial engagements of Xavier Emmanuelli and Bernard Kouchner in France, this new momentum resulted from the coming together of: efforts made by major associations grouped since 1994 into the Alerte collective and their representatives (François Soulage, Véronique Fayet, Christophe Robert, etc.); emblematic business personalities (Louis Gallois, Louis Schweitzer, etc.) and trade union figures (François Chérèque, Laurent Berger, etc.); and the activities of personalities from unusual backgrounds (Martin Hirsch, Olivier Noblecourt, etc.), and some elected representatives, often local, opting to enter this field of action to strengthen their political position (Michel Dinet, Mathieu Klein, Frédéric Bierry, Brigitte Bourguignon, etc.). The new generation of welfare elites was also fuelled by the mobilization of researchers, some of whom attempted to combine sociological analysis and public action (Bruno Palier, Nicolas Duvoux, Julien Damon, Cyprien Avenel, etc.), while others promoted economic and experimental evaluation in the field of education, professional integration and the fight against poverty (Esther Duflo, François Bourguignon, Yannick L'Horty, Marc Gurgand, etc.) (Arrignon 2021).

Without forming a true 'public policy community', these actors were drivers and disseminators of convergence on numerous subjects, i.e. consideration of social expenditure as an investment and the primacy of prevention; promotion of the notions of cohesion and inclusion leading to the relativization of category-specific policies; the aim of securing more efficient social administrations; improved access to social rights through a simplification of the benefits system; more dynamic local solidarities coupled with solidarity of rights; promotion of intervention and support conceived and implemented locally; dissemination of the notion of pathways; reinforcement of the power to act and participative approaches; development of social engineering and results-based management, etc. Gradually, these elements fed into the hotbed of 'social reform', although some subjects led to differences, if not opposition: use of targeting or reference to universalism; questions concerning benefits amounts and conditions; respective roles of the State and local authorities; encouragement of entrepreneurial approaches or the use of philanthropy in building social action, for example.

Specific tools and social investment perspective

Reflecting demands made by charities, following on from the strategic alignment encouraged by the PNAIs, and constituting a central element of the social reform trajectory promoted by the 'new' welfare elites, the Plan pluriannuel contre la pauvreté et pour l'inclusion sociale (2013-2017), followed by the Stratégie nationale de prévention et de lutte contre la pauvreté (2018-2022) were adopted and implemented, the former under François Hollande's presidency and the latter under Emmanuel Macron.

Led by François Chérèqueformer national figure of the new trade unionismthe structure of the Plan Pauvreté (poverty action plan) comprised a particularly long multi-sectoral list of 61 measures. Advocating consultation, territorialization, participation and the development of committed solidarity (Bouquet, Jaeger 2019), the Plan Pauvreté (2013-2017) moved away from the benefit-focused, protectionist directionand beyond that, from essentially legislative and regulatory enactmentof the historically highly centralized French social action system. In addition, the SNPLP was strongly inspired by a social investment perspective (perspective d'investissement social -peIS). As a vehicle of political integration, peIS densified the backbone of anticipated action, thus bringing together, while adjusting, a whole set of a priori composite measures, from the fields of education and training, health, and social work.

Inspired by the Swedish model, in the United Kingdom this type of social investment took the form of the 'Third Way' driven by Tony Blair, involving strong mobilization to combat child poverty, followed by the 'Big Society' programme pursued by David Cameron's government (Blair, Giddens,2002;[START_REF] Florence | Le bilan des réformes[END_REF]. When taken up by the European authorities, this reference to social investment considers that social expenditure can, in certain conditions, represent economic intervention associated with a positive return. To achieve this, peIS proposes reinforcing human capital with the aim of developing individuals' skills, employability and autonomy. The first step involves investing in childcare, education and training and improving support to enter the work market, while targeting some measures at the most socially vulnerable people and groups [START_REF] Jenson | Lost in translation: The social investment perspective and gender equality[END_REF][START_REF] Chantal | Mettre en perspective la perspective de l'investissement social[END_REF]. A reference to peIS involves building intersectoral public policy strategies and evaluating the results and impacts. For supporters of this approach, identifying that an intervention truly corresponds to a social investment involves measuring its impacts on social and economic development. Our recent research and testimonials gathered from key SNPLP actors concur in considering that, in terms of its initial ambitions and orientations, the SNPLP translated a clear preference of national public authorities for this perspective. The same result emerges from our examination of the measures decided on at its launch and taken up by the Evaluation Committee: in our view, 34 of the 35 initial measures can be considered as relating to peIS.

Illustrating the close involvement of public authorities, the deployment of these two specific tools (the Plan Pauvreté then the SNPLP), and the mobilization of peIS, contribute to make poverty action a field of public action in its own right, thus backing up the statement by Charlotte Halpern and Patrick Le Galès (2011): "All autonomous public action should have its own instruments". In addition, conceived at the top and piloted from the top, this instrumentation underlines 'the nodality of the state' in the field of social policies nevertheless presented as mostly decentralized.

Leadership and daring to modernize

When it was launched, the SNPLP was presented as one of the major projects of President Macron's first term. This is illustrated by the president's initial involvement, the creation of an ad hoc inter-ministerial delegation (the DIPLP) headed by a recognized expert, Olivier Noblecourt, and the establishment of a regional network of Commissioners. Moving away from social protectionism and a rationale of benefits and charity work, the initial targets were early childhood, school dropout, training, and job-hunting strategies. This involved aligning local initiatives with the national framework and involving as many stakeholders as possible through intense cooperation; extending a strategy initially targeting children to become more wideranging; coordination at both national and regional levels; multi-level contracts initially aimed at the départements and then major cities and regions; and a particularly ambitious participation objective.

The combination of a ceremonial launch featuring a presidential announcement on 13 September 2018, the appointment of an inter-ministerial delegate appreciated for both his experience of education and social issues and his sympathetic but combative style, and some smoothly orchestrated communication quickly attracted interest. The testimonials gathered mention a strong national mood during the first period, in particular due to the alignment of government authorities and the involvement of central administrations, social protection bodies and Pôle Emploi (job centres). Despite initial reticence from local authorities put off by the limitations of their initiatives as advocated by the Pacte de Cahors, the initial, well-oiled and promising 'narrative' generated unparalleled inter-institutional mobilization: following in the steps of a dozen 'pilot territories', most départements signed a contract with the central state. The association sector was also keen, with national, regional and local representatives getting involved in the numerous work groups established (15 per region).

To tackle the accumulation and dispersal of objects generated by the growing solidarity of public actionin other words, the dissemination of social objectives in numerous public polices (education, housing, culture, employment, health, etc.) (Ballain 2005), the SNPLP was designed to sustain a preventative policy to combat poverty. To ensure that it remained consistent, the challenge was to successfully promote the doctrinal peIS base without upsetting the various sensitivities and ideologies, and to put forward the key measures that would form the backbone of the SNPLP, i.e. a social diversity bonus to encourage the opening of childcare places, free breakfasts at schools in priority areas, prevention of sudden exits from Aide sociale à l'enfance (childhood social benefits-ASE), shorter guidance periods for beneficiaries of RSA (income support), etc. In order to reinforce this new domain of public action, those driving it also worked to recycle and converge numerous initiatives previously approached from a sectoral angle: halving of first year primary classes, action to combat school dropout and obligatory training for 16-18 year-olds, the 'Logement d'abord' (housing first) programme, establishment of universal social childcare and contact social workers, plan to upgrade social work, etc. All of the above was supported by the creation of numerous, varied tools: scientific council, national and regional conferences, media presentations, a digital collaborative platform, multiple roadmaps, and showcasing of a range of results indicators, etc.

At this stage, adjusting and ordering everything was not important: to maintain the spirit of SNPLP advocates and in the name of an ambition with strong political backing, the focus was on opening up the administrative game. This involved shaking up institutional timetables and avoiding excessively rigid planning operations. Reminiscent of the heyday of the politique de la ville (urban policy) and the creation of the Revenu minimum d'insertion (minimum income allowance -RMI), the launch of the SNPLP therefore shook up the numerous mindsets in place and constituted an attempt to ideologically and strategically reorganize the social policy field, in particular ideas and action relating to prevention and to mobilizing the people concerned.

Integration under pressure: economic shocks and territorial rollout

Public policies are subject to continuity and change processes that need to be analysed to examine the scope of reforms and the integrating capacity of public action measures. In this area, in the 1960s and 1970s, major authors like Charles [START_REF] Charles | The Science of 'Muddling Through[END_REF] and Aaron Wildasky (1975) identified the relative inertia of public action. Advocates of incrementalism considered that, in general, actors 'muddle through', opting to make minimal decisions based on the lowest common denominator in a fragmented, complex system of actors. Pursuing a neoinstitutionalist vein, Paul [START_REF] Paul | Increasing returns, path dependence, and the study of politics[END_REF] insisted on the 'pathway dependence' mechanisms that characterize social policies due to institutional density (procedures, regulations, etc.), the eminently collective nature of the political field, and the complexity of the political arena (interweaving of public and private interests, limited information, influence of electoral cycles).

Thus, the ambitions for integration present throughout the SNPLP's establishment and implementation came up against three types of resistance and/or obstacles: economic shocks that led the government to moderate the cognitive and strategic approach; a collapse of politicaladministrative support that facilitated a return to sectoral agendas; and erratic implementation due to the diverse range of existing local setups and their unequal mobilization in terms of poverty action.

Economic shocks, cognitive moderation and strategic reopening

Public policy studies centred on 'crises' [START_REF] Hassenteufel | Crises économiques et processus de changement dans l'action publique : une approche relationnelle[END_REF] and 'critical situations' [START_REF] Capoccia | The Study of Critical Junctures. Theory, Narrative and Counterfactuals in Historical Institutionalism[END_REF] point out that they can be factors of change, sometimes widescale change. In our case, the succession of two economic shocks: the 'yellow vest' crisis from November 2018, and the Covid-19 crisis from March 2020, on the contrary led to a moderation of the SNPLP doctrine and strategy.

The 'yellow vest' crisis brought to the fore the fear of social downscaling experienced by the working classes (Duvoux, Papuchon 2018), and thus highlighted the deleterious impacts of social-territorial divisions, along with related feelings of disparagement, social injustice and contempt (Rosanvallon 2021). The response, in the form of an increase in the prime d'activité (activity bonus) announced by Emmanuel Macron on 9 December 2018 and leading to the decree of 21 December, marked, if not a return to a kind of social protectionism, at least a reworking of the initial peIS intentions far removed from a benefits-based rationale. In parallel, the government decided to drop the planned reform of the Revenu universel d'activité (universal activity income -RUA), which had been put forward as one of the two key reforms of the SNPLP, along with the Service public de l'insertion et de l'emploi (public service for integration and employment -SPIE). The aim of the reform was to simplify the French social benefits system by merging a number of allowances and paying them automatically to avoid non-take-up. Subject to numerous debates, the reform did not ultimately go through, resulting in a return to the former situation featuring dispersed, non-contribution-based allowances. Veering from postponements to renunciation, this first crisis illustrated the difficulty of changing the game of inter-sectoral integration.

The second economic shock, the Covid-19 pandemic coupled with a social crisis, undermined the establishment of agreements with the départements and some measures like obligatory training for young people. The coordination of the SNPLP was also put to the test: meetings and exchanges between actors from institutions and associations ran out of steam; regional working groups could only meet remotely and their work slowed down; participation by the people concerned was severely limited. More widely, the health crisis upset schedules and called into question the trajectory of the French social action system. Priorities concerning professional integration, youth and housing were reinforced, and support for purchasing power found new legitimacy. Areas of action that had been partially or totally put aside, like the consideration of extreme social exclusion, food aid and shelter, the fight against social isolation in particular in rural locations, the digital divide, and questions relating to mental health became the new focus of poverty action policies.

As part of the ensuing recovery plan, the French government devoted 100 million euros to financing projects to help socially excluded sections of the population deal with the crisis. In particular, this took the form of a support plan launched in November 2020 for national and local associations working on poverty action. Measures to reinforce food aid were also set up, along with an exceptional solidarity support package for those most in difficulty (young people, low-income families, and people on minimum social benefits). In addition, a call for expression of interest (CEI) concerned innovative support targeting people experiencing extreme social exclusion and addictions. These developments saw some outsiders gain or regain a foothold, such as some charities, municipalities through their local social action centres (CCAS), and some citizen initiatives and local networks that, at least for a while, were propelled to centre stage, thus extending the network of actors involved.

Looking back, it is clear that these two shocks contributed to a less asserted use of peIS and prevention. Thus, following on from the initial 35 measures, at least 10 of the 14 measures that emerged appear to be compatible with social protectionism (cf. Figure 2). In sum, the combination of doctrinal moderation, more wide-ranging types of action, and more diverse legitimate actors impaired the coherent, discriminatingand thus integratingcharacter of the SNPLP.

Management shift and the return to sectoral agendas

In addition to the impact of economic shocks, in the space of three years, the politicaladministrative management of the SNPLP changed. In its initial format, the DIPLP (Délégué interministériel à la prevention et à la lutte contre la pauvretéinter-ministerial delegate for poverty action) corresponded to Edgard Pisani's definition of mission administration: "uncomplicated, it has a preference for doing, it is realistic, fluid, it moves towards events; it is involved in life; rather than waiting for initiatives, it solicits them; […] it is dynamic and involves risks. It is based on the shifting thinking of an individual or a team" (Pisani, 1956, p. 325). The SNPLP was constructed by this kind of administration, in other words, a politicaladministrative project organization comprising less than ten people headed by Olivier Noblecourt, whose experience of politics, associations and administration made him highly qualified to act, from local to national level with actors combating poverty. But in January 2020, 15 months after its official launch, Olivier Noblecourt left the SNPLP.

The reasons for his departure, which many viewed as a kind of defection, were numerous, ranging from the loss of one of the leaders when Christelle Dubos was named secretary of state, to the attraction of attempting to win an electoral stronghold (Grenoble), and including his weariness of dealing with bureaucracy, financial realities, and unfruitful decisions. This departure, and the fact that in several weeks the DIPLP's pioneer team abandoned ship, undermined the management of the SNPLP and further eroded the political leadership. Many began to wonder whether the SNPLP was not breaking down, and thus whether this very recent field of action was likely to last.

A new Delegate, Marine Jeantet, was chosen with a less political, more management-based background, aligning the DIPLP more with a central administration direction. The new Delegate employed a less flamboyant style than her predecessor and undertook to represent the SNPLP in parliament before the national evaluation committee and in its relations with the CNLE. In the health crisis context, where financial engagements were confirmed and even consolidated, she ensured the continuity of operations. Operating with a new team and a network of Commissioners 'working the field', she managed to keep the SNPLP afloat. Combined with this cognitive moderation, strategic reopening, and remobilization of historic actors in the fight against poverty, this more management-based orientation of the SNPLP led to a resurgence of sectoral approaches. This can be seen in the evolution of the SNPLP's relations with the different measures launched either by the executive or by legislation. For example, the integration of the 'Housing First' plan into the Strategy was rarely identified by actors, who generally viewed the measure as autonomous. The separate management and rollout of the Stratégie nationale de prévention et de protection de l'enfance (national protection strategy for children, 2019-2022) and the Service public de l'insertion et de l'emploi (public employment service -SPIE) confirmed the weakening of the SNPLP: led by an ad hoc secretary of state, Adrien Taquet, and coordinated by a former DIPLP member, Vincent Reymond, the national protection strategy for children led to contracts being established with départements independently from the contracts system set up as part of the SNPLP, and some of these measures competed with SNPLP measures; although conceived as one of the key areas of the SNPLP, the SPIE was more autonomous since it was led more by the ministry of labour, employment and insertion than by the DIPLP.

Inter-ministerial and transversal objectives then took a back seat with the approach of the 2022 presidential elections. All ministries and divisions entered into a public policy assessment phase that they were responsible for carrying out along with communicating on the results. In this context of a return to sectoral agendas, decentralized state services became more complex due to more numerous programmes that overlapped and competed. Commissioners observed and sometimes regretted this development of badly coordinated ministerial impulses and the profusion of calls for projects, some of which involved contradictory orders. They reported the fatigue experienced by territorial authorities and association actors. As centrifugal approaches began gaining ground, the SNPLP no longer represented a decision-making body with integrated leadership: from this point, it only served as a tracking support and, at best, a support for aligning the scattered initiatives that once again began to accumulate. Initially a transformative instrument working on behalf of a State seeking modernization, the SNPLP had become the management framework of a State returning to a function of adjustment (Epstein 2000).

Rather erratic territorial implementation

The implementation of the SNPLP and its territorial governance (Pasquier et al. 2013) turned out to be rather erratic insofar as, while putting forward an ambition for innovation or even breakthrough, it had to deal with former, well-established arrangements and public action territorial setups that were 'already there'.

In the social domain, the regionalization of poverty action corresponded to the initial ambition to develop and structure a new strategic space capable of circumventing traditional regulations in particular between decentralized state services and département councilswith an inefficient track record of driving peIS-related ambitions. Yet rapidly this intention to circumvent was thwarted by an under-estimation of social jurisdictions decentralized at département level; the presidential intention to pursue more marked departmentalization of public action following the yellow vest crisis; and the ministry of the interior's ambition to implement département-level agreements (Marty, Paux 2021). In an atmosphere of constant tensions between the executive and territorial authorities, the DIPLP finally established a governance model on two levels: territorial strategy would be established and coordinated at regional level via the Commissioners, working groups, and a collaborative platform, while contracts and budget management would be done at département level. Although relatively quick to implement, this institutional design tended to renew and extend the former arrangements between Prefects and the Presidents of département councils.

Unlike the former generation of contracts, current public action contracts are characterized by the fact that the approach proposed by the State is generally expressed less openly. The State, instead of acting as a coordinator, proposing that local protagonists work together in a shared diagnosis framework and support certain specific local strategies, acts rather to incentivize départements to finance and implement five key measures. However, in reality, this 'remote government' (Epstein, 2013) remains relative: the key measures are broken down in a fairly specific manner depending on the territory. In addition, in each contract with the départements, a second part recycles and supports a wide range of measures implemented by local initiatives. This place left to the good will of local authorities is reinforced by the fact that recent contracts signed with the regions and cities are not subject to any specific management by the State. Lastly, despite the ambitions put forward and in the view of all protagonists, the establishment of indicator-based management and evaluation remains extremely tenuous and does not lead to a convergence, even less to a homogenization, of initiatives taken locally in the name of the SNPLP.

Thus, it primarily appears to be a differentiation process featuring highly unequal territorial setups in terms of local management capacity on social issues. Put simply, three main types of setup exist:  Fragmented setups (regions of Ile de France and Occitanie) and/or with little political capacity, and more or less deficient local actors (Martinique), fairly inactive local authorities and a State that has difficulties implementing its role, even though it is encouraged to be assertive and sometimes even directly interventionist.

 Intermediate setups on the way to institutionalization with mobilized local actors, local authorities that are involved (Nord département, Dijon Métropole, Toulouse Métropole) but that experience problems with resources and a State solicited to provide active support, or even to play a role of co-producer and co-leader.

 Stabilized setups with more autonomous political capacity (Brittany region, Hautes Pyrénées département, Rennes Métropole), corresponding to anti-poverty territorial systems for the long term that aim to use resources proposed by the State, while keeping it at a distance.

Depending on more or less deteriorated social-territorial situations and the local historicity of collective action dynamics (Pasquier 2012), this impact of existing networks and unequal distribution of political capacities act to produce a fragmentation of public action.

Integration/differentiation and cohesion/fragmentation

At this point, based on the initial work of Marco Oberti (2000), we can propose a modelling of the mechanisms of integration and differentiation (A and C), and cohesion and fragmentation (B) present in public action on solidarity.

Lines of analysis

The degree of integration or differentiation between a social action regime and a local system depends on A and C. The degree of cohesion or fragmentation within the local social action network depends on B.

In the face of what is often perceived as a very wide range of actors and measures involved in solidarity governance, this model puts the accent on three key dimensions: the persistence of traditional forms of solidarity, related to family, the community, or affinities; the national social action regime historically organized around rights solidarities and now oriented towards a form of active social State; and the diversity of local social action and social intervention networks.

The modelling also draws attention to the possible interactions between these three types. Shown by the A, B and C arrows on the graph above, these interactions concern:

-The relations between national authorities and local authorities: in the case of the SNPLP, we have seen that these relations are largely organized by bilateral contract mechanisms (A).

-Relations between local authorities and organizations that, in the third sector, act at local level (B).

-The relations, within civil society, between federative national organizations and local initiatives (C).

Broadly speaking, we can say that in a regime featuring an accumulation of social policies, this multi-level governance of poverty action is more or less conducive to the integration or differentiation of initiatives depending on the relations forged between national and local authorities (A), but also depending on those that, within civil society and the third sector, connect national dynamics and local initiatives (C).

In this area, the SNPLP was initially subject to governmental framing, but went on to generate fairly strong local autonomy in terms of measures initiated by the départements and thanks to the fact that the agreements drawn up with cities and regions remained very open. Concerning associations, the initial impetus related to innovation and to peIS tended to play into the hands of a multitude of social action outsiders that were positioned on the territories and not part of national charity movements. Yet, having benefited from the 'cohesion' part of the French recovery plan, the latter have successfully returned to take centre stage.

As to the relations at play within local social action networks (B), we have observed that, depending on the territory, they are highly diverse and vacillate between cohesion and fragmentation. They are related to historicity and 'spirit of place'. They therefore depend on the extent to which public powers are mobilized on the social question in general, and that of combating poverty in particular, and the more or less partnership-based relations maintained on this subject between the different levels of territorial authority. They also concern the levels of support and the amount of influence that local authorities wield in the association sector, along with the importance acquired by association partnerships.

Conclusion

This analysis of the SNPLP therefore highlights the complexity of intersectoral, interorganizational and multi-level mechanisms involved in putting together public policy. Following initial strong political leadership, and equipped with a transformative ideational and strategic framework, the strategy was obliged to scale down its initial ambitions as a result of crises, changes in the political agenda, and even more so, the difficulty of shifting the sectoral and inter-institutional anchoring of French social policies. While changes were generated, their durability and scope are questionable in terms of the dependence of social action sectors and sub-sectors on the institutional pathway [START_REF] Giuliano | Phénomènes de path dependence et réformes des systèmes de protection sociale[END_REF]. The 'institutionalized DIY' revealed by the implementation of the SNPLP is made concrete through learning, yet it appears to have acted more to boost the poverty action strategies of already 'efficient' local systems. Ultimately, faced with an accumulation of social policies and the dissemination of social objectives in numerous areas of public action, the potentially integrating character of a coordinated poverty action policy remains to be seen.

Figure 1 .

 1 Figure 1. The institutionalization of poverty action Stage 1 -Sedimentation of measures and emergence of a field of action (1980 -2010)

Figure 2 .

 2 Figure 2. Characterization of SNPLP measures (2018-2021)

Figure 3 :

 3 Figure 3: Institutionalization of the fight against poverty Stage 2 -Special tools and measures concerning exogenous shocks

Figure 4 :

 4 Figure 4: Modelling of multi-level governance in the solidarity field (ARENES team, 2021 after Oberti, 2000) Legend A: Governmental framework versus local political autonomy. B: Nature and intensity of contributions between local authorities and the third sector within the local social action network. C: Level of influence, within civil society, of established national organizations (federations, large groups, foundations, etc.) on territorial actors (companies, associations and local foundations).Lines of analysisThe degree of integration or differentiation between a social action regime and a local system depends on A and C. The degree of cohesion or fragmentation within the local social action network depends on B.

French administrative division with an elected form of local government.

This paper draws from the first results and empirical data collected in the research project 'Gouvernance multiniveaux et stratégie nationale de prévention et de lutte contre la pauvreté (SNPLP). La perspective de l'investissement social à l'épreuve de la complexité politico-administrative française' financed by France Stratégie in 2020-21. Under the scientific coordination of Romain Pasquier and Marc Rouzeau, a research team at the Arènes (UMR 6051) laboratory carried out an original investigation comprising 120 interviews during 2020-21 at national, regional and département scale (5 regions and 15 départements) with the aim of deciphering both the cognitive matrix and the instrumentation proposed by the SNPLP, along with the different ways that it is employed by territorial actor systems.

Note that the notion of elite here refers to the "transforming role that can be played by collective actors structured by shared conceptions. These actors, whom we qualify as programmatic, form a new elite when they succeed in governing a public policy domain" ( Genieys,Hassenteufel, 2012, p.91).


How does psychedelic therapy work?

Dating back to the late 1990s, the growing popularity of psychedelics is not a recent trend. Psychiatrists, psychologists, and neuroscientists are paying increasing attention to these hallucinogenic substances, which seem to hold great promise for treating a wide range of pathologies -including depression, addiction, anxiety and more (see [START_REF] Nichols | Psychedelics[END_REF]Nutt & Carhart-Harris, 2021 for example). According to the results of a growing number of clinical trials, these new substances could far surpass existing treatments, in both speed and efficacy, with negligible side effects (e.g. [START_REF] Andersen | Therapeutic effects of classic serotonergic psychedelics : A systematic review of modern-era clinical studies[END_REF]. But for these substances to be effective, they must not be administered under just any conditions. So-called "psychedelic therapy" usually consists of a single dose taken by the patient after several preparatory sessions, combined with close psychological support during and after administration. Under these controlled conditions, the use of psychedelics would catalyze an intense but brief conscious experience, resulting in beneficial therapeutic effects (Letheby, 2021, p. 11). One of the interesting things about this therapy is that it places at the heart of the treatment process two things that are sometimes seen as conflicting: the biological action of the molecule, and the psychological experience. This "psychedelic experience" is often described by patients as "mystical" or "spiritual", and empirical evidence suggests that the effects of psychedelics may be mediated through this "mystical experience" (see the detailed empirical review done by Letheby in chapter 2 of his book). However, the relationship between the biological effects of these substances, the psychedelic experience, and therapeutic effects is not well understood [START_REF] Nutt | Psychedelics therapeutics : What we know, what we think, and what we need to research[END_REF]. This is precisely the subject of the first book by Chris Letheby, Philosophy of Psychedelics. Through a combination of empirical evidence and philosophical perspectives, the author argues that the central mechanism of psychedelic therapy lies in the alteration of the "sense of self" triggered by a mystical experience. This review of Letheby's book aims to summarize and discuss his position in light of the latest findings in fundamental neuroscience.

The book primarily explores the seemingly crucial role of mystical experience in psychedelic therapy. Letheby uses the definition of "mystical experience" of standard psychometric questionnaires, "as involving a sense of unity and transcendence of individual selfhood, as well as noetic feelings of directly encountering ultimate reality, a transcendence of time and space, a sense of sacredness, ineffability, and paradoxicality, and a deeply felt positive mood." (Letheby, 2022, p. 4) According to Letheby, the crucial nature of the mystical experience in psychedelic therapy raises both epistemic and ethical questions. If the therapy works only by inducing metaphysical hallucinations and what could be conceived as irrational beliefs (epistemic aspect), is it not questionable (ethical aspect)? In other words, as Pollan puts it, "Is psychedelic therapy simply foisting a comforting delusion on the sick and dying? " (2015). This is what Letheby calls the "comforting delusion objection" to psychedelic therapy. The purpose of Letheby's book, then, is twofold: first, to show that this objection fails by taking a naturalistic approach; and second, to propose a working mechanism for this therapy. Letheby's general strategy for dealing with the comforting delusion objection is to first assume that naturalism is true, and to admit that if the epistemic status of psychedelic therapy is doubtful, we should hesitate to prescribe it. For the latter, this means that "realistic" forms of well-being, based on accurate understanding of reality, are preferable over unrealistic ones based on misrepresentation (Letheby, 2021, p. 30). Letheby's thesis is that the epistemic status of psychedelic therapy is unproblematic: "This treatment, despite first appearances, does not work centrally by inducing comforting metaphysical beliefs, but by changing how patients see themselves and relate to their own minds and lives." (Letheby, 2021, p. 3) And in his view, even if there are epistemic risks, there are offset by significant epistemic benefits.

As we said, Letheby starts by assuming that naturalism is true. According to the author, a robust naturalistic worldview as the one he favors generally implies the rejection of supernatural entities and forces, or disembodied minds, and "the acceptance that the universe is populated solely by the kinds of in principle-detectable entities, processes, properties, and forces studied by the natural sciences. (...)" (Letheby, 2021, p. 28). In short, he adopts a metaphysical doctrine that combines a generic physicalism or materialism concerning the mind with a denial of the existence of non-natural entities or properties (such as God, angels, the soul, etc.). Throughout his book, he will maintain this stance, which manifests itself in particular in a keen interest in the existing empirical literature. Letheby's naturalism is thus also methodological: in his view, philosophical conclusions must be based on and integrated with relevant scientific findings (Letheby, 2022, p. 6). According to Letheby, there is nothing essential about psychedelic medicine that cannot be reconciled with this worldview.

Apart from the introduction and conclusion, the author's argument unfolds in eight chapters. Chapters 2 and 3 review the history and phenomenology of psychedelic therapy, but focus primarily on the empirical evidence for its efficacy. Chapters 4 and 5 consider possible mechanisms of psychedelic therapy, and chapters 6 and 7 offer a detailed analysis of one such mechanism. Finally, chapters 8 and 9 set forth the reasons why psychedelic therapies would be agents of epistemic benefit.

In chapter 2, Letheby gives a brief history of psychedelic psychiatry, from Hoffman's discovery of the psychoactive effects of LSD in 1943 to the "renaissance" of psychedelic research in the 1990s. This is followed by a review of the contemporary empirical literature, which is continued in chapter 3. This review focuses on the effects in humans of the classical psychedelics, including LSD, psilocybin, DMT, and mescaline, all of which are thought to act on serotonin 2-a receptors (5-HT2A receptors). Letheby highlights a pattern: the psychological benefits of psychedelics are correlated with the occurrence of a mystical experience. In other words, subjects who have a mystical experience during the drug's action tend to be those who experience lasting benefits. In the literature, the existence of this pattern has led to the formulation of the comforting delusion objection.

In chapters 4 and 5, Letheby turns his attention to the mechanisms of action of psychedelic therapies. He describes the theory of molecular neuroplasticity, according to which the benefits of psychedelic therapies are epiphenomena of changes in synaptic neuroplasticity. He then introduces the alternative idea that the therapeutic benefits of psychedelics derive directly from their ability to induce transcendental visions of another reality. He points out the limitations of these two mechanistic hypotheses and proposes a third: that the benefits of psychedelic therapies are based on aspects of the psychedelic experience that are independent of non-naturalistic metaphysical ideas. More specifically, at the root of the therapeutic effects would be an "experiential factor" consisting of changes in mental representations of the self. Indeed, changes in the ego state would be one of the most salient features of the psychedelic state and mystical experience -which does not necessarily imply non-naturalistic beliefs. In support of his thesis, and following Carhart-Harris [START_REF] Carhart-Harris | The entropic brain : A theory of conscious states informed by neuroimaging research with psychedelic drugs[END_REF], he mentions the idea that the pathologies for which psychedelic therapy is promising (depression, anxiety, addiction) have all been linked to changes in the representation of the self. Letheby reviews other types of evidence that aim to support his idea. One the one hand, two psychological arguments: the good correlation between experiences of psychological insight and clinical benefit, and the increase in mindfulness capacities. On the other hand, a neural mechanistic hypothesis: therapeutic benefits seem to correlate with the modulation of two networks that some scientists consider to support self-representation (default mode and salience networks).

In chapters 6 and 7, Letheby attempts to bridge the explanatory gap between neuroscientific and psychological explanations. His solution is to appeal to a neurocognitive theory of brain function: the predictive processing theory of cognition. This theory describes the brain as an "organ for prediction error minimization, which builds models of the world and uses these to predict sensory inputs." (Letheby, 2022, p. 11) Letheby draws on and refines the REBUS (RElaxed Beliefs Under pSychedelics) model proposed by [START_REF] Carhart-Harris | REBUS and the Anarchic Brain : Toward a Unified Model of the Brain Action of Psychedelics[END_REF]. This model suggests that psychedelics alter consciousness by disrupting the neural substrates of the brain's highlevel predictive models (encoding beliefs about self, space, time, causality, etc.); resulting in reduced confidence in our most abstract and fundamental beliefs. Letheby adds, among other things, that it is certain specific beliefs that are targeted: the self-related beliefs. According to Letheby, this cognitivist framework helps explain psychedelic therapy in terms of two factors -the induction of some plasticity (neural, cognitive, etc.) on the one hand, and the discovery and appropriation of healthier forms of being oneself on the other.

In the final chapters 8 and 9, Letheby details the knowledge that can be acquired through psychedelic therapy: a certain "know-how" (learning "how to let go" -see Letheby, 2021, p. 178); a certain "knoweldge by acquaintance" (of one's own unrealized potential, and of the contingency of the self); as well as a "new knowledge of old facts" (reinterpretation of past facts). He also points to indirect epistemic benefits (e.g. making someone more curious about the world around them and others).

One of the merits of this book is that it is well informed about some of the existing empirical literature. However, while it claims to take into account the "empirical literature", it should be pointed out that this is not just any literature. There is a field that is overlooked in this book, but which is nevertheless a key player in the renaissance of psychedelic research: basic neuroscience and most importantly, preclinical research. For example, Letheby omits all the recent preclinical trials on non-hallucinogenic psychedelics, such as those conducted by [START_REF] Cameron | A non-hallucinogenic psychedelic analogue with therapeutic potential[END_REF]. As part of this research, scientists aim to synthesize non-hallucinogenic psychedelic analogs that have therapeutic benefits without causing acute mind-altering effects. The goal is to produce safer molecules that do not generate a psychedelic or mystical state, reducing the associated risks of such therapies (such as bad trips), and making them more widely accessible. If these molecules prove to be effective in clinical trials, these results would provide a serious counter-argument to Letheby's position that the phenomenological effects are the key to therapeutic success, or, as he said, the "primary cause of the effects on well-being and symptomatology" (2021, p. 207). Although Letheby briefly acknowledges this point, he seems not to be taking the possibility seriously (2021, p. 207). However, there are at least two reasons why he might do so.

First, there are recent case reports of successful treatment with psychedelic drugs in humans in the absence of psychedelic experience (e.g. [START_REF] Rosenblat | Antidepressant Effects of Psilocybin in the Absence of Psychedelic Effects[END_REF]. If such cases multiply, it should lead us to re-examine the centrality of the psychedelic experience in healing. Second, when it comes to neuroscience proper, the author's position is based on a state of the art that sometimes seems simplified compared to the most recent reports. For example, the author assumes that psychedelics work by interacting with 5-HT2A receptors (2021, p. 9, p. 118). But the situation appears to be somewhat more complex: recent literature shows that the therapeutic effects of psilocybin in mouse models of depression do not depend on serotonin 5-HT2A receptors. Moreover, these studies suggest that the hallucinogenic effects of psychedelic compounds can be dissociated from their therapeutic and plasticity-promoting effects (see [START_REF] Hesselgrave | Harnessing psilocybin : Antidepressant-like behavioral and synaptic actions of psilocybin are independent of 5-HT2R activation in mice[END_REF][START_REF] Moliner | Psychedelics promote plasticity by directly binding to BDNF receptor TrkB[END_REF].

Thus, this combination of neuroscientific and clinical findings may call into question the importance of the psychedelic experience in psychedelic therapy. In other words, the role of the "experiential factor" in the efficacy of such treatments would become uncertain if non-hallucinogenic psychedelic molecules that do not produce psychedelic experiences show therapeutic efficacy.

Letheby's broad view of neuroscience probably explains why the main argument for rejecting what he calls "the pure neuroplasticity theory" seems incomplete (2021, p. 65). The author argues that while neuroplastic effects likely contribute to the therapeutic potential of psychedelics, they are not the central mechanism. His argument is simple: "across multiple studies with different populations and different psychiatric conditions, the strongest predictor of lasting therapeutic benefits is not the dosage of the drug, but the occurrence of a mystical-type experience." (Letheby, 2022, p. 7) But this argument needs to be supplemented, because the lack of correlation between dose and effect may well be explained by another factor: the difference in sensitivity between individuals. And this difference could be explained by molecular mechanisms. It may be that the concentration of 5-HT2A receptors or other receptors available per individual, or their binding potential, could be sufficient to predict both the intensity of the psychedelic experience and the therapeutic effects. Overall, his argument does not seem to rule out the possibility that brain mechanisms may be sufficient to explain therapeutic benefits.

Likewise, the three mechanistic models discussed by Letheby in chapters 4 and 5 are not the only ones being considered by scientists today. There are other mechanistic models of psychedelic action at the neural or cognitive level -such as the cortico-striatothalamocortical model, the strong prior model, and the cortico-claustro-cortical model (see [START_REF] Kwan | The neural basis of psychedelic action[END_REF]. It might be interesting for Letheby to include them in his discussion.

In short, one wonders how Letheby's theoretical proposal will develop as research programs on psychedelics progress. Nevertheless, for those interested in psychedelics from a philosophical perspective, this book is a good starting point. In addition to the interesting questions raised, it also provides the reader with a clear, if partial, overview of the complex contemporary debates in the field of psychedelic empirical research. This is both the strength and the weakness of Letheby's proposal: based on a specific empirical corpus, his conceptual framework will have to be re-evaluated in the light of the very next discoveries, especially the results of preclinical research.

  Héloïse Athéa Institut d'Histoire et Philosophie des Sciences et des Techniques (IHPST), CNRS, Paris, FranceInstitut de Biologie Paris Seine, Neuroscience Paris Seine, Paris, France heloise.athea@cnrs.fr

© 2023 https://doi.org/10.1080/09515089.2023.2260422


Louiz Akram
email: louiz.akram@gmail.com

A thesis about Newtonian mechanics rotations and about differential operators

Keywords: Newtonian mechanics, Rotations, derivative, perpendicularity, Nabla, Gradient, Curl, Divergence, differential operators, GPS, reference frame

This work dates from January 2016 as a result of my remarks related to physics made during my high studies. I try in this work to explain the cause behind the inability of Newtonian mechanics to describe correctly many phenomena where the studied object rotates at a very high linear speed. I proved that, in this case, the velocity field is not equiprojective and that the famous formula for changing the reference frame is not correct. I made an application to the case of the GPS system satellites, then I presented a new method for studying a rotating system velocity without needing the conventional steps of changing reference frames. I finished my work by demonstrating the formulas of the main differential operators and I presented them with all the related steps and calculations by using the elementary surfaces. I am eager to discuss the results of this work further with physics and mathematics specialists, and I hope that my formulas will help to simplify the study of many difficult physics phenomena.

Introduction

Even nowadays, there are many electromagnetic phenomena where we will always make mistakes by using the normal mathematical tools in their study. They are in general all the phenomena where the studied object rotates at a high linear speed, especially when its speed exceeds three quarters of the light speed. [START_REF] Savickas | Relations between Newtonian Mechanics, general relativity, and quantum mechanics[END_REF][START_REF] Salesi | Non-newtonian mechanics[END_REF][START_REF] Whitaker | Newton's laws, Euler's laws, and the speed of light[END_REF] Because in this case the study mistake becomes very coarse and impossible to neglect. Furthermore, the scientists are obliged to use the relativity of time as a difficult explanation or statistical physics as a solution to the contradictions found by the conventional change of reference frames. This problematic is described as a historical crisis of Newtonian classical mechanics. [START_REF] Roux | Introduction à la physique statistique et la physique quantique[END_REF] I proved first in the part A of my thesis that in this case the velocity field is not equiprojective and that the famous conventional steps for changing the reference frame are not correct. I presented also a method for studying rotating systems velocity without needing the normal change of reference frames that requires the perpendicularity of a rotating vector and its derivative. [START_REF] Chobotov | Orbital mechanics[END_REF][START_REF] Ardler | 82.45 How the scalar and vector products are derived[END_REF][START_REF] Genta | Dynamics of rotating systems[END_REF] I presented also an application to the case of the GPS system satellites that avoids the relativity of time [8 9,10]. I finished my work by demonstrating geometrically, in the part B, the formulas of the three important differential operators: Gradient (nabla), Divergence and Curl that can be found different in other works. [START_REF] Hyman | Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids[END_REF][START_REF] Kosmann-Schwarzbach | Divergence operators and odd Poisson brackets[END_REF][START_REF] Zubair | Differential electromagnetic equations in fractional space[END_REF] I presented those differential operators with all the related steps and calculations by using the elementary surfaces. However, the results found will make the convinced readers change their vector and matrix calculations especially with the famous navier-stokes equations. [START_REF] Grünbaum | The algebra of differential operators associated to a weight matrix[END_REF][START_REF] Constantin | Navier-stokes equations[END_REF][START_REF] Temam | Navier-Stokes equations: theory and numerical analysis[END_REF] I remind the readers that this work is a revision of personal previous work made since January 2016 entitled "mémoire en physiques". [START_REF] Akram | Mémoire en Physiques[END_REF] Part A: Rotations of Newtonian mechanics:

Introduction:

We know that: ( ) . However, in physics, the infinitesimal variation dx is never null when we study the movement in a physics phenomenon except at rest. Consequently, we shouldn't use f' the derivative of the function f. We should use instead which is the differential divided by the infinitesimal variation. Let's consider the circle of radius r=1 in the figure 1, where b is the length of the circular arc and a is the circular arc chord and dφ is the circular arc infinitesimal angle. Consequently b= dφ and S is the area of the circular segment that is the area of the surface between a and b, where ( ( )).

Remarks:

And we consider that h=sin(dφ) and c=cos(dφ) and c+e=1.

Remark 1:

If dφ is small enough, then

( ) ( ) (1)
and in this case S≈0 therefore a and b become combined which means a≈b. Also, in this case, h≈ dφ therefore by using Pythagoras' theorem we conclude that: consequently: and thus e disappears and cos(dφ)=c≈1. We conclude that:

( ) ( ) (2
) and we remark that in this case ( ) ( ) consequently, trigonometric identities are unusable when sin(dφ)≈ dφ. By Al-Kashi's theorem we prove also that a≈0 which implies that dφ≈0. And this is remarked also with:

( ) and this is absurd since dφ exists during the study of the rotation in a physics phenomenon where we should have normally cos(dφ) < 1. We conclude that the approximation with sin(dφ)≈dφ is unusable. If we use sin(dφ)≈ dφ and ( ) √ (3) by using Pythagoras' theorem in the trigonometric circle, then in this case: √ √ Hence: √

Consequently: √ √

We conclude that: √ ( )

And thus:

We proved that if we use sin(dφ)≈ dφ and ( ) √ , then this implies that which is a contradiction. Hence, the use of this approximation is false and can cause errors in the study of the rotation in a phenomenon.

Remark 2:

Now let's use the limit of ( ) when dφ tends to zero.

We have consequently when dφ is small enough: ( ) .

(4) Consequently, by using Pythagoras' theorem in the trigonometric circle:

( ) √ . (5)
In this case S≠0 even if we notice that a tends to dφ (a≈ dφ) when we use Al-Kashi's theorem in the triangle of the figure 1. S≠0 because a equals approximately dφ but b equals exactly dφ and the formula ( ( ))prevents us to consider that a=b.

This approximation is correct and causes no contradictions. Consequently, we can use it without risking any errors.

All the trigonometric identities are usable in this case since ( ) ( ) and since the triangles of the trigonometric circle in figure 1 stay all valid. Remark 3:

( ) ( ( )) ( ) ( ) (6)
consequently trigonometric identities are not usable for . We conclude that we should fix dφ as the smallest detectable variation in the phenomenon studied rotation which depends on the detection technology in order to avoid mathematical contradictions.

The fixed dφ will respect the formulas: ( ) and ( ) √ .

However, in this case we should use dt as the variable.

During the rotation in Newtonian mechanics, we have a fixed infinitesimal variation which defines the fixed change dφ that happens in different changing durations Δt. For example, in a polar coordinate system (ρ,φ,z=0), we will have:

⃗⃗⃗⃗ ( ) ⃗⃗⃗⃗ ( ) ⃗⃗⃗⃗ ( ) ⃗⃗⃗⃗ ( ) (7)
where: and dt=Δt is the variable duration of each dφ. You will find in this document an example of this method application that deals with the case of GPS systems.

Important:

These approximations imply that:

( ( ) ( )) consequently: ( ) ( )
And thus: ( ) and: ( )

Remark 4:
In order to simplify the results, we will consider that ( ) and ( )

We can also deduce by using these approximations that:

∫ ( ) √ ( ( ) ( )) ( ( ) ( )) (14)
and:

∫ ( ) ( ( ) ( )) √ ( ( ) ( )) (15)
by integrating ( ) and ( ) calculated above.

These two integrals are only approximate sums since dφ is fixed as dφ>0. However Riemann's definition of integrals requires that dφ tends exactly to zero. However, we can also check from above that: We can prove easily that:

∫ ( ) ( ) ( ) and: ∫ ( ) ( ) ( ) ∫
( ) ( ) ( ) (16)
and:

( ) ( ) ( ) (17)
Consequently:

( ) ( ) ( ) ( ) ( ) (18)
and: ( )

( ) ( ) ( ) ( ) (19)
And also:

( ( ) ) ( ( ) ) (20) and: ( ( ) ) ( ( )
) .

(21)

Conclusion 1:

Let's consider that ⃗ ( ) is a rotating vector that belongs to the plane of its rotation φ. Hence:

⃗ ⃗ ( ) ⃗ ⃗ ( ) ⃗ ⃗ ( ) ‖ ⃗ ( )‖ ( ( ( ) ( ) ) √ ( ( ) ( ) )+ ⃗ ( ) √ ⃗ ( ) (22)
Consequently:

⃗ ⃗ ( ) ⃗ ( ) ‖ ⃗ ( )‖ (23)
because dφ exists during the study.

Where:

‖ ⃗ ⃗ ( ) ‖ ‖ ⃗ ( )‖ and: ( ⃗ ⃗ ( ) ⃗ ( )) ^ ( ) (24)

Results and applications:

Result 1:

When using the approximations: ( ) and ( ) √ , if ⃗ ( )is a rotating vector that belongs to the plane of its rotation φ, then

⃗ ⃗ ( )
is not orthogonal to ⃗ ( ).

Consequently, the famous method of reference frames change becomes false and unusable in Newtonian mechanics. Furthermore, the velocity field becomes not equiprojective for solid mechanics.

Result 2:

In cylindrical coordinates (ρ,φ,z):

⃗⃗⃗⃗ ( ) ⃗⃗⃗ ( ) √ ⃗⃗⃗⃗ ( ) (25)
and:

⃗⃗⃗⃗⃗ ( ) ⃗⃗⃗⃗ ( ) √ ⃗⃗⃗ ( ) (26)
Important example:

We will study the case of The GPS system satellites by proving the real time at a given satellite. We will need no time dilation in this proof. Let's consider that the speed V of the satellite is constant V=k 1 , By using the correct approximations above:

⃗⃗⃗⃗⃗⃗⃗ ( ) ( ) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ( ) ( ) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ( ) √ ( ) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (27)
Where M stands for the position of the satellite that has a circular orbit of angle φ.

Since the speed vector given to the satellite is tangent to the circle of the wanted trajectory, we should have:

( ) ( ) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ( ) ( ) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗ (28)
Hence:

( ) ( ) (29)
Consequently:

( ) ( ( )* ( ) (30)
Where: h is the initial altitude of the satellite.

The linear speed V of the GPS satellite is the constant speed that equals the initial speed given to the satellite in order to start orbiting.

Consequently: ⃗ ( ) √ ( ) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (31)
And thus:

√ ( ) ( ( √ ), (32)
We should remark that: ρ(t) increases in the beginning of the GPS satellite lifetime with a very slight change thanks to the initial launching power from earth. φ=0 at the point where the satellite started orbiting after being launched. φ is always increasing and exceeds 2π, consequently φ is always positive and . dφ is the constant smallest variation of the satellite angle that we can detect, and dt is the time needed for that variation. Since we consider in our study that dφ is constant and dt is the variable, we conclude that:

√ ( ) (33)
and thus dt depends on the satellite rotation angle φ.

application:

The real time ΔT needed by the satellite to make the first lap around the earth after it is launched is:

∑ ( √ ( *, √ ∑ ( ( )+ (

A new method to avoid the conventional method of changing reference frames:

We will present a new method to avoid the famous method of changing reference frames that needs the perpendicularity between a rotating vector and its derivative. When a given reference frame with orthonormal direct axes is making any revolution, this revolution can be decomposed to three simple revolutions, and each one of these simple revolutions is around one of the three axes. However each axis changes only by two simple revolutions that are the simple revolutions around the two other axes and not by the revolution around itself. Consequently, let's consider that α is the angle of the simple revolution around the axis , β is the angle of the simple revolution around the axis , and ɣ is the angle of the simple revolution around the axis ⃗ where ( ⃗ )are the three orthonormal direct axes of the reference frame that makes any given revolution. This revolution is composed of the three simple revolutions of angles: α, β and ɣ. Let's study each one of the three axes independently from each other. The simple revolution doesn't influence its axis but the two others. Hence each axis is influenced by two simple revolutions and thus we can use a spherical coordinates system to study each axis.

We will calculate , and ⃗ which are the derivatives of , and ⃗ in an absolute fixed reference frame ( ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ ). This method is useful in order to avoid the difficult projections of , and ⃗ in the absolute fixed reference frame. Furthermore, the three derivatives will be calculated without making any change of reference frames. However, the functions α(t), β(t) and ɣ(t) of the three simple revolutions must be well known before making this study.

Step 1: The study of the vector :

Let's consider that ⃗⃗⃗ , ⃗⃗⃗⃗ and ⃗ ⃗⃗⃗⃗ where: ⃗⃗⃗ , ⃗⃗⃗⃗ and ⃗⃗⃗⃗ are the vectors of a spherical coordinates system. We conclude that: ɣ=θ and α has no effects on . However, we should find the relation between φ and both β and ɣ.

Fig.2 :

The circle of rotation β and the ellipse of rotation φ.

An important advice:

The result of this vector study should preferably be used with the formula: ∫ ⃗⃗⃗⃗ during the energetic study of a system, then simplifications can be made in order to find the correct expression of the kinetic energy variation .

Part B: Differential operators:

The differential operators gradient (nabla), divergence, curl and Laplacian provide information about about a field of scalars or vectors situated immediately in front of the studied point M according to the orientations of the axes of the used coordinates system. Let's prove the formulas of each differential operator geometrically by using the three coordinates systems in an Euclidean space where the field lines are considered continuous vector functions.

The operator gradient (nabla):

Let's consider a function of locations points:

. Where E is the Euclidean space and the function f is differentiable and thus continuous.

Consequently:

( ) makes a scalar field.

In each of the three coordinates system, the operator gradient has this form: ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ .

The Cartesian coordinates System (x,y,z):

( ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ ) ( ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ ) ( ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ ) ⃗⃗⃗⃗⃗⃗ (58)
Consequently:

⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ (59)
The cylindrical coordinates system (ρ,φ,z):

(

And thus:

⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ ( ) ⃗⃗⃗⃗ (73)
Clarifications:

⃗⃗⃗⃗⃗⃗ is an infinitesimal displacement that depends on the used coordinates system. The studied function f must be expressed according to the coordinates system of the used reference frame, then we use the coordinates of M in the final expression. ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ characterizes the variation of f in the space for a given displacement ⃗⃗⃗⃗⃗⃗ . df changes depending on M the studied point of the space, because to each point M corresponds a value f(M), and also, ⃗⃗⃗⃗⃗⃗ is immediately in front of the studied point M according to the orientations of the axes of the used coordinates system . Consequently, ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ is a vector field that depends on the value f(M) at the location of the point M and also on the coordinates system being used. ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ‖ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ‖ ‖ ⃗⃗⃗⃗⃗⃗ ‖ ( ) consequently ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ is located at a rotation angle from ⃗⃗⃗⃗⃗⃗ in the anticlockwise orientation. The level surfaces are the space surfaces where f stays constant. And ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ is perpendicular to its level surfaces.

2.The flow and the operator divergence:

The elementary flow is: ⃗ is a scalar field where: is a vector field dS and dτ are consecutively the elementary surface and volume of the used coordinates system. ⃗ is the unit normal vector to dS. For a closed surface we orientate ⃗ towards outside the surface. Also, dS and dτ change from a coordinates system to an other. Consequently, depends on the coordinates system being used.

We finally conclude that:

The cylindrical coordinates system (ρ,φ,z):

In the cylindrical coordinates system: ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ ⃗ where M is a point of the space. In the figure 3, the elementary volume is: . (88) Also:

(89) and ( ) In the figure 3, we prove by using circular sectors that: (91) And:

( ) (92)
Hence: ⃗⃗⃗⃗

and:

⃗⃗⃗⃗ ⃗⃗⃗⃗ ( ) (94)
where:

Consequently:

( ( )) (96)
because: ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ .

And also: ⃗⃗⃗⃗

and:

⃗⃗⃗⃗ ⃗⃗⃗⃗ (98)
where:

Consequently:

(100) because: ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ .

And also:

⃗⃗⃗⃗ ( ) (101)
and:

⃗⃗⃗⃗ ⃗⃗⃗⃗ ( ) (102)
where:

Consequently:

( ) ( ) (104) because: ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗ . Hence: ( ( ) ( ) ) (105)
We finally conclude that:

( ) ( ) (106)
The spherical coordinates(r,θ,φ):

In the spherical coordinates: ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ and ⃗⃗⃗ is always in the plane of the rotation θ. And the elementary volume is: In the figure 5, the vector that is the radius of the circular arc made by dφ is always in the plane of ⃗⃗⃗⃗ . And by using circular sectors:

( ) . ( 107
)
( ) . ( 108
)
Fig. 6 : The elementary surfaces dS 3 and dS 4 .

In the figure 6:

( ) ( ) (109)
and: And it is a vector field that respects the following Stokes' theorem:

( ) ( ) (110)
⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ Where: dS 12 , is the elementary surface in the plane ( ⃗⃗⃗ ⃗⃗⃗ ),dS 13 is the elementary surface in the plane ( ⃗⃗⃗ ⃗⃗⃗ )and dS 23 is the elementary surface in the plane ( ⃗⃗⃗ ⃗⃗⃗ ), and Also: dC 1 ,dC 2 and dC 3 are respectively their closed boundaries of the corresponding elementary surfaces in the reference frame planes. These boundaries are outlines oriented anticlockwise when we are observing the surfaces from the inside (center) of the reference frame.

We prove that: Where: d ij is the elementary length of the elementary surface sides. And A ij is the field vector of that coincides with d ij . The boundaries are drawn by starting from the studied point M(x,y,z). Consequently, the sides of the elementary surfaces have elementary sides d ij in common.

⃗⃗⃗⃗⃗⃗
The Cartesian coordinates System (x,y,z): In the figure 8 We conclude finally that: ⃗⃗⃗⃗⃗⃗

The cylindrical coordinates system (ρ,φ,z): In the figure 9: (162) and:

(163) and: ( ) (164) And: (165) And also:

(166)

Figure 1 .

 1 Figure 1. The trigonometric circle of radius r=1.

  ( ) and ∫ ( ) are left as a challenge to the readers!!! Remark 5:

Figure 3 .

 3 Figure 3. The elementary volume dτ and the elementary surfaces dS 1 and dS 2 .

Figure 4 .

 4 Figure 4. The elementary similar surfaces dS 3 , dS 4 , and the elementary similar surfaces dS 5 and dS 6 .

Figure 5 .

 5 Figure 5. The six elementary surfaces forming an infinitesimal volume. And the elementary similar surfaces dS 1 and dS 2 .

Figure 7 .

 7 Figure 7. The elementary surfaces dS 5 and dS 6 .

Figure 8 .

 8 Figure 8. The boundaries dC 1 , dC 2 and dC 3 of the cartesian coordinates.

Figure 9 .

 9 Figure 9. The boundaries dC 1 , dC 2 and dC 3 of the cylindrical coordinates.

  ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ )don't change by the variation of r but only θ and φ. ⃗⃗⃗ and ⃗⃗⃗⃗ are always in the rotation plane θ. since ⃗⃗⃗⃗ is always perpendicular to the plane of the rotation θ. Consequently, ⃗⃗⃗⃗ doesn't change by the variation of θ.In order to find:⃗⃗⃗⃗ , ⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗ , we should make the projections of ⃗⃗⃗ , ⃗⃗⃗⃗ and ⃗⃗⃗⃗ in the suitable cylindrical coordinates (ρ,φ,z) with the same plane of rotation φ that contains always ⃗⃗⃗ and the same vector ⃗⃗⃗⃗ of the spherical coordinates. ⃗⃗⃗⃗ is the same in the two coordinates systems.

	Result 3: and also: ⃗⃗⃗⃗			√		( ) ⃗⃗⃗	√	( ) ⃗⃗⃗⃗	⃗⃗⃗⃗
	In spherical coordinates(r,θ,φ): (47)
	⃗⃗⃗		⃗⃗⃗⃗		⃗⃗⃗⃗ and: ⃗⃗⃗⃗	⃗⃗⃗⃗	⃗⃗⃗⃗ and: ⃗⃗⃗⃗	⃗⃗⃗⃗⃗	⃗⃗⃗⃗⃗
	since ( Also:	⃗⃗⃗⃗		⃗⃗⃗ √		⃗⃗⃗⃗	(38)
	and:	⃗⃗⃗⃗		⃗⃗⃗⃗ √		⃗⃗⃗	(39)
	since: And also:
	Hence: ⃗⃗⃗		( ) ⃗⃗⃗	( ) ⃗ and: ⃗⃗⃗⃗	( ) ⃗⃗⃗	( ) ⃗
	and: ⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ where: Also:	⃗⃗⃗⃗	( )	⃗⃗⃗⃗	( ) (	⃗⃗⃗ √	⃗⃗⃗⃗ ) (40)
	And:	⃗⃗⃗⃗	( )	⃗⃗⃗⃗	( ) (	⃗⃗⃗ √	⃗⃗⃗⃗ )	(41)
	with: ⃗⃗⃗	( ) ⃗⃗⃗		( ) ⃗⃗⃗⃗
	and thus:
	⃗⃗⃗⃗				( )	⃗⃗⃗		( ) ⃗⃗⃗⃗	( ) √	⃗⃗⃗⃗
	(42)
	and:	⃗⃗⃗⃗			( ) ⃗⃗⃗		( )	( ) √	⃗⃗⃗⃗	(43)
	and:	⃗⃗⃗⃗⃗		( ) √		⃗⃗⃗	( ) √	⃗⃗⃗⃗	⃗⃗⃗⃗
	(44)
	5. Conclusion 2:
		⃗⃗⃗		(			( ) ) ⃗⃗⃗ (	√	( ), ⃗⃗⃗⃗
					√			( ) ⃗⃗⃗⃗
									(45)
	Where: and: ⃗⃗⃗⃗	⌊ ⌋(the floor of ) ( √		( )+ ⃗⃗⃗ (	(35) ( ) ) ⃗⃗⃗⃗	34)
	and √			√	( ) ⃗⃗⃗⃗		(	)	(36)
	and					.			(37) (46)

⃗⃗⃗⃗⃗ ⃗ ⃗ ⃗ consequently:

  The Cartesian coordinates System (x,y,z): We remark that: dτ=dx.dy.dz and dS 1 =dS 2 =dx.dy and dS 3 =dS 4 =dy.dz and dS 5 =dS 6 =dx.dz.

	And also:		⃗⃗⃗⃗	(74)
	and:	⃗⃗⃗⃗	⃗⃗⃗⃗	(75)
	where:			(76)
	Consequently:		(77)
	because: ⃗⃗⃗⃗	⃗⃗⃗⃗	⃗ .
	And also:		⃗⃗⃗⃗	(78)
	and:	⃗⃗⃗⃗	⃗⃗⃗⃗	(79)
	where:			(80)
	Consequently:		(81)
	because: ⃗⃗⃗⃗	⃗⃗⃗⃗	.
	And also:		⃗⃗⃗⃗	(82)
	and:	⃗⃗⃗⃗	⃗⃗⃗⃗	(83)
	where:			(84)
	Consequently:		(85)
	because: ⃗⃗⃗⃗	⃗⃗⃗⃗	.
	And thus:		(	)

  of the elementary surfaces have elementary sides d ij in common.

	and also:						(151)
	and:						(152)
	with:		and:	and:			(153)
	because the sides And thus:	( (	) )	( (	) )
			(	)	(	)
	(154)
	Consequently:			⃗⃗⃗⃗⃗⃗		(155)
	where: dS 23 =dy.dz					(156)
	And:			⃗⃗⃗⃗⃗⃗			(157)
	where: dS 13 =dx.dz					(158)
	And also:			⃗⃗⃗⃗⃗⃗	⃗	(159)
	where: dS 12 =dx.dy					(160)
		: dS 23 =dy.dz				(140)
	and: dS 13 =dx.dz					(141)
	and dS 12 =dx.dy					(142)
	And:	⃗⃗⃗⃗⃗⃗	⃗⃗⃗ ⃗⃗⃗⃗⃗⃗	⃗⃗⃗ ⃗⃗⃗⃗⃗⃗		⃗⃗⃗ ⃗⃗⃗⃗⃗⃗	⃗⃗⃗
	(143)
	Consequently:
	(144)
	By following the same method we prove that:
	(145)
	And:
	(146)
	Where:						(147)
	and:						(148)
	and also:						(149)
	and:						(150)

The figure 2 shows that the intersection of the rotation β plane and the rotation φ plane is the axis ⃗⃗⃗⃗⃗ of a Cartesian coordinates system. And the orthogonal projection of the circle made by the rotation β inside the plane of ⃗⃗⃗⃗ is an ellipse made by the rotation φ. We consider that O is the center of the ellipse whereas φ is the angle between ⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗ . ⃗⃗⃗⃗⃗ is perpendicular to the plane of the rotation φ which is the plane of ⃗⃗⃗⃗ that is the plane of the two Cartesian vectors ⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗ . ⃗⃗⃗⃗⃗ is in the plane of the rotation β which is the plane of and ⃗⃗⃗⃗⃗ is perpendicular to the axis ⃗⃗⃗⃗⃗ . We can prove that:

( )and ( ) ( ) (48) ⃗⃗⃗⃗⃗ is in the plane of ⃗⃗⃗⃗ with:

And the equation of the ellipse is: where a= (the half of the ellipse major axis), and b= (the half of the ellipse minor axis). Consequently: a=1 and b= sin(θ) And also: ( )and: ( ) Therefore, the ellipse equation becomes:

( ) (51) and this equation doesn't change even if: and: . Since we know exactly the functions β(t) and ɣ(t), we can deduce easily the function φ(t) in order to be able to calculate dφ(t), especially that: and:

and also: and also: .

However we can only use this formula when and .

But we notice that if and ɣ stays fixed, then always φ=β and thus we can study easily by using cylindrical coordinates system. Finally, by using the formula of ⃗⃗⃗ , we conclude that:

Hence:

where: φ 1 (t)=φ(t) in step 1.

Step 2: The study of the vector :

By following the same method, let's consider that ⃗⃗⃗⃗ , ⃗⃗⃗ and ⃗ ⃗⃗⃗⃗ where: ⃗⃗⃗ , ⃗⃗⃗⃗ and ⃗⃗⃗⃗ are the vectors of an other spherical coordinates system. We conclude that: α=θ and β has no effects on . However, we should find the relation between φ and both α and ɣ. And Let's consider that: φ 2 (t)=φ(t) in step 2.

=

The ellipse equation gives:

Consequently, we can deduce easily the function φ 2 (t). And thus by using the formula of ⃗⃗⃗ again, we conclude that:

Step 3: The study of the vector ⃗ :

By following the same method, let's consider that ⃗⃗⃗⃗ , ⃗⃗⃗⃗ and ⃗ ⃗⃗⃗ where: ⃗⃗⃗ , ⃗⃗⃗⃗ and ⃗⃗⃗⃗ are the vectors of an other spherical coordinates system. We conclude that: β=θ and ɣ has no effects on ⃗ . However, we should find the relation between φ and both α and β. And Let's consider that: φ 3 (t)=φ(t) in step 3.

The ellipse equation gives:

Consequently, we can deduce easily the function φ 3 (t). And thus by using the formula of ⃗⃗⃗ again, we conclude that:

In the figure 7, we flatten the surface dS 5 consequently we get a trapezoid shape that has the following height h 1 :

Hence:

We make the same with the surface dS 6 and we get a trapezoid shape that has the following height

And:

We have:

and:

where:

Consequently:

And also:

and:

because: ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ where:

Consequently:

(123) Let's consider that:

And also: ( )

We conclude that:

For: and : Let's consider that:

Consequently:

(130) because: ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ and:

We conclude that:

And thus:

We finally conclude that:

By using the demonstrated approximations:

) And:

) And Also:

) And:

In order to calculate the divergence of in a point of the space M(x,y,z), we should replace by the data of in the expression of the chosen reference frame coordinates system. Then, we integrate the final expression by using the coordinates of the studied part of the space if the integration is possible.

The six vectors ⃗⃗⃗ that are normal to the elementary surfaces are oriented towards outside. Consequently, for a uniform field , when , the field vectors that are in the studied part of the space, in the orientation of the used reference frame axes, have the same orientations of the vectors ⃗⃗⃗ . And thus the field is divergent in the studied part of the space and . In the case when , the field vectors in the studied part of the space have orientations opposite to the vectors ⃗⃗⃗ orientations. Consequently the field is convergent in the studied part of the space and .

In the case when , and the field exists in the studied part of the space, this means that every field vector has its opposite across each elementary surface. In this case , otherwise the field is tangent to all the elementary surfaces and thus it is a rotational field.

The operator curl:

In an orthonormed direct reference frame of axes ( ⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗ ), we can define the operator curl as:

⃗⃗⃗⃗⃗⃗

Consequently:

Where:

And:

Where:

And also:

Where:

Hence:

And:

And also:

We conclude finally that:

The spherical coordinates(r,θ,φ): In the figure 10:

Consequently, by considering that:

We find that:

We remind that the calculations from before gave:

And:

Also:

And also:

We have:

) And also:

Where:

Where:

And also:

Where:

We conclude that:

And: ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗

And also: ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗

We conclude finally that:

),

Clarifications:

During the study, we should replace by the field coordinates in the curl formulas demonstrated above. Then, we integrate according to the studied part of the space when the integration is possible. The operator curl informs about a part of the vicinity of a studied point. This part is the part of the space in front of the studied point in the orientation of the coordinates system being used.

If a coordinate of the curl is positive, then the vector field located at the studied point vicinity that is perpendicular to that coordinate axis is a vortex field. The vortex is anticlockwise around the positive coordinate axis. If that coordinate is negative then the vortex will be oriented clockwise.

If that coordinate is null whereas the field exists, then the field will be uniform in the part of the studied point vicinity concerned by the null coordinate.

The operator Laplacian:

Let's consider a function of locations points:

. Where E is the Euclidean space and the function f is differentiable and thus continuous. The Laplacian of f is:

( ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ). The Cartesian coordinates System (x,y,z): ⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ and:

Where: is a vector field.

Consequently:

The cylindrical coordinates system (ρ,φ,z):

Where: is a vector field.

Consequently:

The spherical coordinates(r,θ,φ):

and:

Where: is a vector field. Consequently:

(

We remind that:

And:

And Also:

And:

Clarifications:

During the study, we should replace by the studied function f in the Laplacian formulas demonstrated above. Then, we integrate according to the studied part of the space when the integration is possible. If Δf=0 then f behaves uniformly in the location that is immediately in front of the studied point M. If Δf>0 then a local minimum of f exists in the location that is immediately in front of the studied point M. If Δf<0 then a local maximum of f exists in the location that is immediately in front of the studied point M. When we are studying the Laplacian of a function f, the gradient vectors cross diagonally the volume made by the elementary surfaces of the divergence.

Remark:

The results found should make the convinced readers change their vector and matrix calculations especially with the famous navier-stokes equations.