Vector is one of the important concepts necessary for the study of Physics, Applied Mathematics and Engineering. This book presents the principal topics in the subject – scalars, vectors, vector algebra, vector differentiation and the differential operator, vector integration and integral theorems – in a clear and simple manner. For each topic, the definitions of important terms, properties and deductions are provided, along with worked-out examples and figures to ensure in-depth understanding and comprehension.
Salient Features
Prasun Kumar Nayak is Assistant Professor at the Department of Mathematics (UG & PG), Midnapore College (Autonomous), Midnapore, West Bengal. He has more than 20 years of teaching experience and has published books such as A Textbook of Mechanics, A Textbook of Tensor Calculus and Differential Geometry, Numerical Analysis, and Linear Algebra. He has also published papers on game theory and inventory theory in national and international journals.
1 Scalars and Vectors 1.1 Scalar 1.2 Vector 1.2.1 Null and Proper Vectors 1.2.2 Position Vector 1.2.3 Unit and Reciprocal Vectors 1.2.4 Rectangular Unit Vectors 1.2.5 Polar and Axial Vectors 1.2.6 Pseudovector 1.2.7 Co-initial Vectors 1.2.8 Parallel Vectors 1.2.9 Equal Vectors 1.2.10 Sliding Vectors 1.2.11 Bound Vectors 1.2.12 Like and Unlike Vectors 1.2.13 Free and Localised Vectors 1.3 Vector Algebra 1.3.1 Addition of Vectors 1.3.2 Subtraction of Vectors 1.3.3 Multiplication of a Vector by a Scalar 1.4 Components of a Vector 1.5 Linear Combination of Vectors 1.5.1 Linearly Dependent 1.5.2 Linearly Independent 1.6 Section Ratio (Point of Division) 1.7 Collinear and Coplanar Vectors 1.7.1 Collinear Vectors 1.7.2 Coplanar Vectors 1.8 Centroids 1.8.1 Centre of Mass Exercises 2 Product of Vectors 2.1 Scalar Product 2.2 Applications to Cartesian Geometry 2.2.1 Distance Between Two Points 2.2.2 Direction Cosines of a Line 2.2.3 Orthogonal Transformation 2.3 Cross Product 2.4 Vector Area 2.4.1 Vector Area of a Parallelogram 2.4.2 Vector Area of a Triangle 2.5 Product of Three Vectors 2.5.1 Scalar Triple Product 2.5.2 Scalar Product of Four Vectors 2.5.3 Vector Triple Product 2.5.4 Vector Product of Four Vectors 2.5.5 Reciprocal System of Vectors Exercises 3 Applications of Vector Algebra 3.1 Vector Equations 3.2 Applications in Geometry 3.2.1 Vector Equation of a Straight Line 3.2.2 Angle Bisector of Two Intersecting Straight Lines 3.3 Plane 3.3.1 Shortest Distance Between Two Skew Lines 3.4 Sphere 3.4.1 Sphere with Given Radius and Centre 3.4.2 Sphere with Given Extremities of a Diameter 3.4.3 Intersection of a Line and a Sphere 3.4.4 Tangent Plane at a Given Point 3.5 Volume of a Tetrahedron 3.5.1 Regular Tetrahedron 3.6 Applications in Mechanics 3.6.1 Concurrent Forces 3.6.2 Work and Power 3.6.3 Rotation of a Rigid Body 3.6.4 Moment of a Force Exercises 4 Vector Differentiation 4.1 Scalar and Vector Fields 4.1.1 Scalar Fields 4.1.2 Vector Fields 4.2 Vector Calculus 4.2.1 Limit and Continuity of Vector Functions 4.2.2 Ordinary Derivatives 4.2.3 Partial Derivatives 4.2.4 Directional Derivative 4.3 Applications in Differential Geometry 4.3.1 Space Curve 4.3.2 Curvature and Torsion of Space Curve Er = Ef (u) 4.3.3 Tangent Plane and Normal to a Surface Er = Er(u, v) 4.4 Curvilinear Coordinates 4.4.1 Transformation of Coordinates 4.4.2 Coordinate Surface and Curves 4.4.3 Unit Vectors in Curvilinear Systems 4.4.4 Arc Length, Surface Area and Volume Element 4.5 Applications in Mechanics 4.5.1 Tangential and Normal Acceleration 4.5.2 Uniform Motion on a Circle 4.5.3 Areal Velocity Exercises 5 Vector Differential Operator 5.1 Differential Operator 5.2 Gradient 5.2.1 Invariance of the Gradient 5.2.2 Gradient of a Vector Function 5.2.3 Gradient in Curvilinear Coordinates 5.3 Divergence 5.3.1 Invariance of Divergence 5.3.2 Divergence in Curvilinear Coordinates 5.4 Curl 5.4.1 Invariance of Curl 5.4.2 Curl in Curvilinear Coordinates 5.5 Laplacian Operator 5.5.1 Laplacian in Curvilinear Coordinates Exercises 6 Vector Integration 6.1 Definitions of Basic Concepts 6.2 Ordinary Integral 6.3 Line Integrals 6.3.1 Tangential Line Integral 6.3.2 Properties of Line Integrals 6.4 Surface Integrals 6.4.1 Normal Surface Integrals 6.5 Volume Integrals Exercises 7 Integral Theorems 7.1 Green’s Theorem in the Plane 7.2 Gauss’s Divergence Theorem 7.3 Stokes’ Theorem Exercises Bibliography Index