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Differential Geometry by Mittal and Agarwal It is imperative for geometrical analysis that leads to different geometrical objects. This book presents the basic principles of differential geometry with emphasis on the right approach. Its objective is to fill up in detail, with many illustrations, how certain differential-geometric methods are used in conceptual problems with numerical examples given throughout the text. The book comes in handy for anyone interested in exploring different types of mathematics and mathematical concepts which are related to geometry. It is more of a text book with reference to some specific applications. The book consists of 17 chapters each with its own subject. Here are some of the key insights that are conveyed in the book : · The chapter entitled 'Differentials at a single point' is on differential geometry at a single point. The chapter presents basic properties of differential geometry at points and develops results on geodesics, canonical forms, tangent spaces and more. It also contains the results on the boundary conditions for different types of surfaces including hyperbolic, parabolic and elliptical surfaces. This chapter is followed by 'Spaces of Curves' which covers the results on spaces of plane curves. The chapter also covers the results for plane curves in space. · The third chapter is entitled 'The fundamental form and its applications to geometry'. It contains the fundamental form for plane curvilinear coordinates, canonical forms associated with plane curves, Lie's theorem on motion along a closed curve, geodetic measurements between two points, geodesics in Riemannian spaces and more. Some important sections include : · Section 2 of Chapter 3 deals with the concept of a curve and its local definition (which is equivalent to the global definition)in terms of derivatives at a point. The various differential forms are presented. · Section 3 deals with the fundamental form for plane curvilinear coordinates. The Jacobi matrix is obtained and it is shown how to obtain its 3 × 3 sub-matrices, i.e., the curvature matrix, the torsion matrix and the principal curvature matrix. The second part of this section deals with elliptic charts and their connection with the Jacobian. This is used to give a geometric interpretation for certain differential forms that appear in later sections of this chapter. · Section 4 deals with the concept of canonical forms associated to plane curves defined by polynomials of degree one and degree two in a single variable. The canonical form is presented for plane curves defined by polynomial of degree two in two variables. The section ends with the discussion of the Jacobian matrix for plane curves lying on a surface, and an equation for its determinant (the Riemann curvature tensor). · Section 5 of Chapter 3 presents some complex results on complex curves, including the extension theorem. Some interesting properties are also given. For example, it is proved that if a curve has real parameterization then it is not holomorphic (with respect to its parameterization) on any open set containing the image of its initial point. cfa1e77820
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