Applications of Vector Analysis and Complex Variables in Engineering
This textbook presents the application of mathematical methods and theorems to solve engineering problems, rather than focusing on mathematical proofs. Applications of Vector Analysis and Complex Variables in Engineering explains the mathematica
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Applications of Vector Analysis and Complex Variables in Engineering
Applications of Vector Analysis and Complex Variables in Engineering
Otto D. L. Strack
Applications of Vector Analysis and Complex Variables in Engineering
Otto D. L. Strack CEGE Department University of Minnesota MINNEAPOLIS, MN, USA
ISBN 978-3-030-41167-1 ISBN 978-3-030-41168-8 (eBook) https://doi.org/10.1007/978-3-030-41168-8 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
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Contents xiii
Preface 1
Vectors in Three-Dimensional Space 1.1 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . 1.2 Comparison with Symbolic Notation . . . . . . . . . 1.3 The Dot Product . . . . . . . . . . . . . . . . . . . . 1.3.1 Projection of a Vector onto a Given Direction 1.4 The Cross Product . . . . . . . . . . . . . . . . . . . 1.5 Free Indices and Dummy Indices . . . . . . . . . . . 1.6 Base Vectors . . . . . . . . . . . . . . . . . . . . . . 1.7 The Kronecker Delta . . . . . . . . . . . . . . . . . 1.8 The Levi-Civita Symbol and the Kronecker Delta . .
. . . . . . . . .
1 2 2 3 3 4 7 7 7 8
2
Vector Fields 2.1 Field Lines: Path Lines and Stream Lines . . . . . . . . . . . . . . . 2.1.1 The Lagrangian Description . . . . . . . . . . . . . . . . . . 2.1.2 The Eulerian Description . . . . . . . . . . . . . . . . . . . .
10 10 10 12
3
Fundamental Equations for Fluid Mechanics 3.1 The Euler and Bernoulli Equations . . . . . . . . . . . . . . . . . . . 3.2 The Mass Balance Equation . . . . . . . . . . . . . . . . . . . . . . 3.3 Rotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Bernoulli Equation for Irrotational Flow . . . . . . . . . . .
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