## Vector and Tensor Analysis with Applications" … the exposition is clear and the choice of topics is excellent …" Prof. R. E. Williamson, Univ. of Sussex, Brighton, England This concise introduction to a basic branch of applied mathematics is indispensable to mathematicians, physicists and engineers. Eminently readable, it covers the elements of vector and tensor analysis, with applications of the theory to specific physics and engineering problems. It lays particular stress on the applications of the theory to fluid dynamics. The authors begin with a definition of vectors and a discussion of algebraic operations on vectors. The vector concept is then generalized in a natural way, leading to the concept of a tensor. Chapter Three considers algebraic operations on tensors. Next, the authors turn to a systematic study of the differential and integral calculus of vector and tensor functions of space and time. Finally, vector and tensor analysis is considered from both a rudimentary standpoint, and in its fuller ramifications, concluding the volume. The strength of the book lies in the completely worked out problems and solutions at the end of each chapter. In addition, each chapter incorporates abundant exercise material. Intended primarily for advanced undergraduates and graduate students of math, physics and engineering, the work is self-contained and accessible to any student with a good background in calculus. Vector and Tensor Analysis With Applications is one of a series of SELECTED RUSSIAN PUBLICATIONS IN THE MATHEMATICAL SCIENCES, several of which have already been published by Dover. The authors are distinguished Russian mathematicians and specialists in gas dynamics and numerical analysis. Richard A. Silverman, editor of the series as well as editor and translator of this volume, has revised and improved the original edition and added a bibliography." … a concise, clear and comprehensive treatment … " Prof. Henry G. Booker, University of California, San Diego |

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### Contents

DOVER BOOKS ON MATHEMATICS | 1-70 |

THE TENSOR CONCEPT | 1-97 |

TENSOR ALGEBRA | 2-76 |

RUDIMENTS | 4-1 |

RAMIFICATIONS | 5-5 |

BIBLIOGRAPHY | 5-164 |

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### Common terms and phrases

angle antisymmetric arbitrary axis basis e1 basis vectors calculate called Christoffel symbols closed contour closed surface coefficients const contravariant components coordinate curves covariant and contravariant covariant components covariant derivative curl curvilinear coordinates cyclic permutation DIFFERENTIAL EQUATIONS direction electromagnetic equals Example exterior normal fact flux follows formulas Gauss geometric given grad hence implies integral linear magnitude mathematical metric tensor numbers obtain orthogonal orthonormal basis parallelepiped particles perpendicular physical plane potential principal axes Problem Prove pseudotensor quantity radius vector recall rectangular coordinate system rectangular coordinates rectangular coordinates x1 respect right-handed rotation scalar field scalar product second-order tensor Similarly Solution Stokes stress tensor Suppose symmetric system of rectangular tangent tensor of order theorem theory trajectories transformation law unit vector vanishes vector field vector function vector product vectors e1 velocity field volume x3-axis