On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

Water Resource Pollution and Impacts on the local livelihood: A case study of Beas River in Kullu District, India

The rivers are considered as the life line of any country since they make water available for our domestic, industrial and recreational functions. The quality of river water signifies the health status and hygienic aspects of a particular region, but the quality of these life lines is continuously deteriorating due to discharge of sewage, garbage and industrial effluents into them. Thrust on water demand has increased manifolds due to the increased population, therefore tangible efforts to make the water sources free from pollution is catching attention all across the globe. This paper attempts to highlight the trends in water quality change of River Beas, right from Manali to Larji in India. This is an important river in the state of Himachal Pradesh and caters to the need of water for Manali and Kullu townships, besides other surrounding rural areas. The Manali-Larji Beas river stretch is exposed to the flow of sewage, garbage and muck resulting from various project activities, thereby making it vulnerable to pollution. In addition, the influx of thousands of tourists to these towns also contributes to the pollution load by their recreational and other tourist related activities. Pollution of this river has ultimately affected the livelihood of local population in this region. Hence, water quality monitoring was carried out for the said stretch between January, 2010 and January, 2012 at 15 various locations on quarterly basis, right from the upstream of Manali town and up to downstream of Larji dam. Temperature, color, odor, D.O. , pH, BOD, TSS, TC and FC has been the parameters that were studied. This study gives the broad idea about the characteristics of water at locations in the said river stretch, and suggestions for improving water quality and livelihood of local population in this particular domain.