Vector Analysis and Cartesian TensorsAcademic Press, 2014 M05 10 - 266 pages Vector Analysis and Cartesian Tensors, Second Edition focuses on the processes, methodologies, and approaches involved in vector analysis and Cartesian tensors, including volume integrals, coordinates, curves, and vector functions. The publication first elaborates on rectangular Cartesian coordinates and rotation of axes, scalar and vector algebra, and differential geometry of curves. Discussions focus on differentiation rules, vector functions and their geometrical representation, scalar and vector products, multiplication of a vector by a scalar, and angles between lines through the origin. The text then elaborates on scalar and vector fields and line, surface, and volume integrals, including surface, volume, and repeated integrals, general orthogonal curvilinear coordinates, and vector components in orthogonal curvilinear coordinates. The manuscript ponders on representation theorems for isotropic tensor functions, Cartesian tensors, applications in potential theory, and integral theorems. Topics include geometrical and physical significance of divergence and curl, Poisson's equation in vector form, isotropic scalar functions of symmetrical second order tensors, and diagonalization of second-order symmetrical tensors. The publication is a valuable reference for mathematicians and researchers interested in vector analysis and Cartesian tensors. |
Contents
| 1 | |
| 18 | |
Chapter 3 Vector Functions of a Real Variable Differential Geometry of Curves | 49 |
Chapter 4 Scalar and Vector Fields | 72 |
Chapter 5 Line Surface and Volume Integrals | 116 |
Chapter 6 Integral Theorems | 159 |
Chapter 7 Applications in Potential Theory | 186 |
Chapter 8 Cartesian Tensors | 204 |
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Common terms and phrases
axes Ox'y'2 axes Oxyz boundary cartesian coordinate system cartesian tensor closed region Consider continuously differentiable coordinate axes coordinate system curvilinear coordinates cylindrical polar coordinates defined definition denote direction cosines div F divergence theorem double integral eigenvalues Evaluate example EXERCISES expressed field F follows geometrical given grad Q Hence isotropic isotropic function isotropic tensor Laplace's equation line integral magnitude notation origin orthogonal parallel parametric equation partial derivatives perpendicular plane position vector proof prove radius real numbers rectangular cartesian axes rectangular cartesian coordinate region T bounded relative respectively result right-hand rotation satisfied scalar field scalar invariant scalar product second order tensor Show Similarly simple closed curve simple closed surface Solution sphere spherical polar coordinates Stokes's theorem suffixes summation convention tensor gradient transformation matrix triad unit normal unit tangent unit vector variables vector analysis vector field vector product verify x-axis xy-plane z-axis zero