A Course of Mathematics for Engineers and Scientists offers a mathematics course for undergraduate students reading science and engineering at British and Commonwealth Universities and colleges. The aim of this volume is to generalize and develop the ideas and methods of earlier volumes so that the student can appreciate and use the mathematical methods required in the more advanced parts of physics and engineering.
This book begins with elementary ideas of vector algebra which are generalized and developed in two ways. The first is an account of vector analysis and the differential and integral operations and theorems concerning vectors. These ideas find their first generalization in tensor analysis and the transformation of coordinates, including orthogonal curvilinear coordinates. The second development is to matrices, where the properties of arrays of elements, linear equations, and quadratic forms are shown to be the generalizations of elementary algebra and, using 'vector space', of familiar geometrical ideas to n dimensions. The solution of differential equations by series provides a general method for the solution of ordinary and some partial differential equations.
A discussion of the properties of the solutions in the light of the Sturm-Liouville theory introduces the conceptions of eigenvalues and orthogonal functions, forming a link with matrices. A chapter on the special functions gives some of the better known properties of Bessel, Legendre, Laguerre, and Hermite functions, which commonly occur in the solution of boundary and initial value problems.
Preface Chapter I. Vector Analysis Transformation of Coordinates Scalar Fields: Gradient Vector Fields Line and Surface Integrals Applications to Vector Analysis Green's Theorem Discontinuities; Surface Derivatives Uniqueness Theorems and Green's Function Variation with Time Orthogonal Curvilinear Coordinates Suffix Notation and the Summation Convention Cartesian Tensors Chapter II. The Solution of Some Differential Equations Laplace's Equation in Two and Three Dimensions Solution in Series of Ordinary Differential Equations The Behavior of the Solution of a Differential Equation Eigenvalues: Sturm-Liouville Systems Chapter III. Some Special Functions Bessel Functions Legendre Polynomials Other Special Functions Chapter IV. The Differential Equation of Field Lines and Level Surfaces Introduction Field Lines Lagrange's Partial Differential Equation Level Surfaces and Orthogonal Trajectories Chapter V. Matrices Introduction and Notation Matrix Algebra The Rank of a Matrix: Singular Matrices The Reciprocal of a Square Matrix Partitioned Matrices The Solution of Linear Equations Vector Spaces Eigenvalues and Eigenvectors Quadratic Forms Simultaneous Reduction of Quadratic Forms Multiple Eigenvalues Hermitian Matrices Bibliography Answers to the Exercises Index
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- © Pergamon 1964
- 1st January 1964
- eBook ISBN: