Paper Title
Theorem on Brand Loyalty in Consumer Dynamics for Moderate Consumption Leading to Well-Being of Society

These are most advanced and new approaches in area of consumer marketing research that its results are proved numerically. Almost forty years ago the subject of Economics was purely theory based but now no one can think about it without involving advanced mathematics. Similarly at present consumer marketing research needs mathematical approaches to prove its results. Using the knowledge of advanced applied mathematics, various explanations in marketing research are possible in an easy and understandable manner which otherwise become a matter of conflict of interests. Mathematically we can always converge our discussions to a numerical value or values and hence it becomes easier to have a logical conclusion(s) by doing its interpretation. Transformative Consumer research(TCR )for personal well-being is an academic marketing research defined as a function of research focused at consumer of a particular segment of society with all of their problems, resources for their well-being. On the other hand this research can be expanded in multiple directions depending upon parameters of consumers and society. The numerical approach in this research process is used to observe the dynamics of consumer behavior in the area of transformative consumer research of any region around the globe. The pattern of the specific region will help in contouring the large diversity of consumer behavior thus enabling the easy and understandable formation of mathematical models(here equations, polynomials, differential calculus, and integral calculus, etc).[7] The focus is on finding the subsets of large interacting forces relative to social values. The recognition of such subsets are the fundamental steps in designing the desired models in TCR This article outlines in a very technical way a specific proof in research which is appropriate to understand the proposed solutions in TCR for a specific segment of society(it is also termed as Singled-valued TCR or STCR) and to obtain the new alternative solutions with available boundary values like brand loyalty, quality, or purchasing power of a consumer, social constraints in context with some specific brand of a product. Key words- Consumer Dynamics Transformative Consumer Research Vector Analysis Vector Dot Product Riemann Integration Vector Valued Function