Vector Analysis and Cartesian Tensors

Abstract

Part 1 Rectangular Cartesian co-ordinates and rotation of axes: rectangular Cartesian co-ordinates direction cosines and direction ratios angles between lines through the origin the orthogonal projection of one line on another rotation of axes the summation convention and its use invariance with respect to a rotation of the axes matrix notation. Part 2 Scalar and vector algebra: scalars vectors - basic notions multiplication of a vector by a scalar addition and subtraction of vectors the unit vectors i, j, k scalar products vector products the triple scalar product the triple vector product products of four vectors bound vectors. Part 3 Vector functions of a real variable - differential geometry of curves: vector functions and their geometrical representation differentiation of vectors differentiation rules the tangent to a curve - smooth, piecewise smooth, and simple curves arc length curvature and torsion applications in kinematics. Part 4 Scalar and vector fields: regions functions of several variables definitions of scalar and vector fields gradient of a scalar field properties of gradient the divergence and curl of a vector field the del-operator scalar invariant operators useful identities cylindrical and spherical polar co-ordinates general orthogonal curvilinear co-ordinates vector components in orthogonal curvilinear co-ordinates vector analysis in n-dimensional space. Part 5 Line, surface, and volume integrals: line integral of scalar field line integrals of a vector field repeated integrals double and triple integrals surfaces surface integrals volume integrals. Part 6 Integral theorems: introduction the divergence theorem (Gauss' theorem) Green's theorems Stoke's theorem limit definitions of div F and curl F geometrical and physical significance of divergence and curl. Part 7 Applications in potential theory: connectivity the scalar potential the vector potential Poisson's equation Poisson's equation in vector form Helmholtz's theorem solid angles. Part 8 Cartesian tensors: introduction Cartesian tensors - basic algebra isotropic tensors tensor fields the divergence theorem in tensor field theory. Part 9 Representation theorems for isotropic tensor functions: introduction diagonalization of second order symmetrical tensors invariants of second order symmetrical tensors representation of isotropic vector functions isotropic scalar functions of symmetrical second order tensors representation of an isotropic tensor function. Appendices: determinants the chain rule for Jacobians.