## Vectors and Tensors in Engineering and PhysicsVectors and Tensors in Engineering and Physics develops the calculus of tensor fields and uses this mathematics to model the physical world. This new edition includes expanded derivations and solutions, and new applications. The book provides equations for predicting: the rotations of gyroscopes and other axisymmetric solids, derived from Euler's equations for the motion of rigid bodies; the temperature decays in quenched forgings, derived from the heat equation; the deformed shapes of twisted rods and bent beams, derived from the Navier equations of elasticity; the flow fields in cylindrical pipes, derived from the Navier-Stokes equations of fluid mechanics; the trajectories of celestial objects, derived from both Newton's and Einstein's theories of gravitation; the electromagnetic fields of stationary and moving charged particles, derived from Maxwell's equations; the stress in the skin when it is stretched, derived from the mechanics of curved membranes; the effects of motion and gravitation upon the times of clocks, derived from the special and general theories of relativity. The book also features over 100 illustrations, complete solutions to over 400 examples and problems, Cartesian components, general components, and components-free notations, lists of notations used by other authors, boxes to highlight key equations, historical notes, and an extensive bibliography. |

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### Contents

1 | |

17 | |

29 | |

GENERAL COMPONENTS | 51 |

TENSOR FIELDS OF ONE VARIABLE | 73 |

TENSOR FIELDS OP MANY VARIABLES | 99 |

APPLICATIONS | 131 |

GENERAL COORDINATES | 177 |

FOURDIMENSIONAL SPACETIME | 207 |

REFERENCES | 225 |

ANSWERS TO PROBLEMS | 231 |

INDEX | 275 |

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### Common terms and phrases

angle angular velocity arbitrary axis base vectors Cartesian components Cartesian coordinate system center of mass Christoffel symbols circular cylindrical constant contravariant coordinate curves covariant covariant derivatives cross section curvature tensor curvilinear coordinates defined deformed denote density derivatives differential equations direction displacement divergence theorem earth eigenvalues eigenvector electric field electromagnetic energy field lines Figure Find flow fluid flux force formulas function geodesic coordinate system gravitational field Green's theorem inertia length line integral linear magnitude matrix membrane metric tensor motion Newton's law obtain orbit orthogonal orthonormal parallel parallelepiped particle plane point mass position vector pressure principal axes Problem radius rectangular relativistic Riemann curvature tensor rigid body rotation tensor scalar potential second-order tensor Show shown in Fig solid solution speed spherical surface tangent temperature tensor fields tensor of order three-dimensional transformation u x v unit vector vector field volume wire zero