Differential Geometry and Its Applications

This book consists of nine chapters; the first chapter deals a brief conceptual study of n-dimensional space, scalar and vector quantities, matrices and determinants and Kronecker delta etc., which are referred to in subsequent chapters. In second chapter, we have diuscussed in detail about the theory of curves in space which deals the conceptual study of space curves, tangent, osculating plane, normal lines and normal plane, rectifying plane, orthogonal triad of fundamental unit vectors t, n, b and fundamental planes, curvature and torsion, screw- curvature, Serret- Frenet formulae, helices (a conic helix, cylindrical helix, circular helix and spherical helix), intrinsic equations (or natural equations) of the curve, fundamental theorems for space curves (Existence and uniqueness theorems), osculating circle (or the circle of curvature), osculating sphere (or the sphere of curvature), Involutes and Evolutes, Bertrand curves and spherical indicatrix. The detail study of a surface and fundamental forms have been discussed in chapter third which covers the detail study of surface, tangent plane and normal plane to the surface, fundamental forms ( first fundamental form or metric, second fundamental form and third fundamental form) and its geometrical interpretation, properties and also the relation between them, arc length and surface area and angle between parametric curves. In chapter four, we have discussed in detail about the study of envelopes and developables which deals the conceptual study of family of surfaces and envelope of family of surfaces, edge of regression, ruled surfaces, developable surfaces and Weingarten map.