Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid.
Similarly, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (i.e., a one-dimensional closed curve with no intersections that can be embedded in two-dimensional space), the set of all possible positions of the hour and minute hands taken together is topologically equivalent to the surface of a torus (i.e., a two-dimensional a surface that can be embedded in three-dimensional space), and the set of all possible positions of the hour, minute, and second hands taken together are topologically equivalent to a three-
dimensional object.
Solving of systems of linear equations, which led to linear algebra
Attempts to find formulas for solutions of general polynomial equations of higher degree that resulted in discovery of groups as abstract manifestations of symmetry
Arithmetical investigations of quadratic and higher degree forms and diophantine equations, that directly produced the notions of a ring and ideal.
Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties. This creates a false impression that in algebra axioms had come first and then served as a motivation and as a basis of further study. The true order of historical development was almost exactly the opposite. For example, the hypercomplex numbers of the nineteenth century had kinematic and physical motivations but challenged comprehension. Most theories that are now recognized as parts of algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. An archetypical example of this progressive synthesis can be seen in the history of group theory.
Early group theory
There were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of mathematical tools, such as Fourier series and computer graphics.
Disclaimer: asieformathematics.in doesn't own this Book. We neither create not scan this Book. The Images & Books are copyrighted to their respective Owners. We are providing the PDF of Books which is already available on Internet, Websites and on Social Media like Telegram, Whatsapp, etc. We highly encourage visitors to Buy the Original copy from their Official Sites. If anyway it violates the law or anybody have Copyright issues/ having discrepencies over this post, Please Take our Contact us Page to get touch with us. We will reply as soon as we receive your Mails and immediately remove this content