دانلود کتاب Vector Calculus

Vector calculus is the foundation stone on which a vast amount of applied mathematics is based. Topics such as fluid dynamics, solid mechanics and electromagnetism depend heavily on the calculus of vector quantities in three dimensions. This book covers the material in a comprehensive but concise manner, combining mathematical rigour with physical insight. There are many diagrams to illustrate the physical meaning of the mathematical concepts, which is essential for a full understanding of the subject. Each chapter concludes with a summary of the most important points, and there are worked examples that cover all of the material. The final chapter introduces some of the most important applications of vector calculus, including mechanics and electromagnetism.1. Vector Algebra .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Vectors and scalars .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Defini tion of a vector and a scalar . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Addition of vectors ................................ 2 1.1.3 Components of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Dot product .................... . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Applications of the dot product . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Cross product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Applications of the cross product. . . . . . . . . . . . . . . . . . . .. 11 1.4 Scalar triple product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14 1.5 Vector triple product ..................................... 16 1.6 Scalar fields and vector fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 2. Line, Surface and Volume Integrals. . . . . . . . . . . . . . . . . . . . . . . .. 21 2.1 Applications and methods of integration. . . . . . . . . . . . . . . . . . . .. 21 2.1.1 Examples of the use of integration .................... 21 2.1.2 Integration by substitution .......................... 22 2.1.3 Integration by parts ................................ 23 2.2 Line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25 2.2.1 Introductory example: work done against a force ....... 25 2.2.2 Evaluation of line integrals .......................... 26 2.2.3 Conservative vector fields. . . . . . . . . . . . . . . . . . . . . . . . . . .. 28 2.2.4 Other forms of line integrals . . . . . . . . . . . . . . . . . . . . . . . .. 30 2.3 Surface integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31 2.3.1 Introductory example: flow through a pipe. . . . . . . . . . . .. 31 2.3.2 Evaluation of surfa-ce integrals ........................ 33 2.3.3 Other forms of surface integrals . . . . . . . . . . . . . . . . . . . . .. 38 2.4 Volume integrals ......................................... 39 2.4.1 Introductory example: mass of an object with variable density . . . . . . . . . . . . . . . .. 39 2.4.2 Evaluation of volume integrals ............ . . . . . . . . . .. 40 3. Gradient, Divergence and Curl ............... . . . . . . . . . . . . . .. 45 3.1 Partial differentiation and Taylor series. . . . . . . . . . . . . . . . . . . . .. 45 3.1.1 Partial differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45 3.1.2 Taylor series in more than one variable. . . . . . . . . . . . . . .. 47 3.2 Gradient of a scalar field ............ . . . . . . . . . . . . . . . . . . .. 48 3.2.1 Gradients, conservative fields and potentials ........... 51 3.2.2 Physical applications of the gradient. . . . . . . . . . . . . . . . .. 52 3.3 Divergence of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53 3.3.1 Physical interpretation of divergence. . . . . . . . . . . . . . . . .. 56 3.3.2 Laplacian of a scalar field ................. . . . . . . . .. 56 3.4 Curl of a vector field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58 3.4.1 Physical interpretation of curl. . . . . . . . . . . . . . . . . . . . . . .. 60 3.4.2 Relation between curl and rotation .... . . . . . . . . . . . . . .. 61 3.4.3 Curl and conservative vector fields ................... 61 4. Suffix Notation and its Applications . . . . . . . . . . . . . . . . . . . . . . .. 65 4.1 Introduction to suffix notation ............................. 65 4.2 The Kronecker delta ij ................................... 68 4.3 The alternating tensor fijk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70 4.4 Relation between fijk and ij ............................. 72 4.5 Grad, div and curl in suffix notation . . . . . . . . . . . . . . . . . . . . . . .. 74 4.6 Combinations of grad, div and curl ................ . . . . . . . .. 76 4.7 Grad, div and curl applied to products of functions . . . . . . . . .. 78 5. Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83 5.1 Divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83 5.1.1 Conservation of mass for a fluid . . . . . . . . . . . . . . . . . . . .. 85 5.1.2 Applications of the divergence theorem. . . . . . . . . . . . . . .. 87 5.1.3 Related theorems linking surface and volume integrals .. 88 5.2 Stokes's theorem ......................................... 91 5.2.1 Applications of Stokes's theorem ..................... 93 5.2.2 Related theorems linking line and surface integrals. . . . .. 95 6. Curvilinear Coordinates 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99 6.1 Orthogonal curvilinear coordinates. . . . . . . . . . . . . . . . . . . . . . . . .. 99 6.2 Grad, div and curl in orthogonal curvilinear coordinate systems 104 6.2.1 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2.2 Divergence ........................................ 105 6.2.3 Curl.............................................. 106 6.3 Cylindrical polar coordinates. . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . 107 6.4 Spherical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7. Cartesian Tensors .......................................... 115 7.1 Coordinate transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.2 Vectors and scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.3 Tensors................................................. 119 7.3.1 The quotient rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.3.2 Symmetric and anti-symmetric tensors . . . . . . . . . . . . . . . . 122 7.3.3 Isotropic tensors ................................... 123 7.4 Physical examples of tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.4.1 Ohm's law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.4.2 The inertia tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8. Applications of Vector Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.1 Heat transfer ............................................ 132 8.2 Electromagnetism........................................ 134 8.2.1 Electrostatics...................................... 135 8.2.2 Electromagnetic waves in a vacuum. . . . . . . . . . . . . . . . . . . 137 8.3 Continuum mechanics and the stress tensor. . . . . . . . . . . . . . . . . . 140 8.4 Solid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.5 Fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.5.1 Equation of motion for a fluid . . . . . . . . . . . . . . . . . . . . . . . . 146 8.5.2 The vorticity equation .............................. 147 8.5.3 Bernoulli's equation ................................ 149 Solutions ................................................... 153 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181