# Applications of Vector Analysis and Complex Variables in Engineering

Authors: Strack, Otto D. L.

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• Application of indicial notation to vectors and tensors, applying this to fluid mechanics to provide physical interpretation
• Derivation of complex variables in general terms, viewing them as a non-Cartesian coordinate transformation. This derivation uses the coordinate transformations presented earlier in the text using indicial notation (covariant and contra variant vector components)
• Applications of the first and second order partial differential equations in engineering. The applications of second order partial differential equations are to fluid mechanics, groundwater flow, heat flow, linear elasticity, and soil mechanics
• Treatment of singularities in elliptic partial differential equations, and discontinuities in hyperbolic partial differential equations
• Quasi-linear partial differential equations of the second order
• Applications of series expansions to obtain solutions to a multitude of elements in problems of groundwater flow. This application is usually given to the students as their final project
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This textbook presents the application of mathematical methods and theorems tosolve engineering problems, rather than focusing on mathematical proofs. Applications of Vector Analysis and Complex Variables in Engineering explains the mathematical principles in a manner suitable for engineering students, who generally think quite differently than students of mathematics. The objective is to emphasize mathematical methods and applications, rather than emphasizing general theorems and principles, for which the reader is referred to the literature.

Vector analysis plays an important role in engineering, and is presented in terms of indicial notation, making use of the Einstein summation convention. This text differs from most texts in that symbolic vector notation is completely avoided, as suggested in the textbooks on tensor algebra and analysis written in German by Duschek and Hochreiner, in the 1960s.

The defining properties of vector fields, the divergence and curl, are introduced in terms of fluid mechanics. The integral theorems of Gauss (the divergence theorem), Stokes, and Green are introduced also in the context of fluid mechanics. The final application of vector analysis consists of the introduction of non-Cartesian coordinate systems with straight axes, the formal definition of vectors and tensors. The stress and strain tensors are defined as an application.

Partial differential equations of the first and second order are discussed. Two-dimensional linear partial differential equations of the second order are covered, emphasizing the three types of equation: hyperbolic, parabolic, and elliptic. The hyperbolic partial differential equations have two real characteristic directions, and writing the equations along these directions simplifies the solution process. The parabolic partial differential equations have two coinciding characteristics; this gives useful information regarding the character of the equation, but does not help in solving problems. The elliptic partial differential equations do not have real characteristics. In contrast to most texts, rather than abandoning the idea of using characteristics, here the complex characteristics are determined, and the differential equations are written along these characteristics. This leads to a generalized complex variable system, introduced by Wirtinger. The vector field is written in terms of a complex velocity, and the divergence and the curl of the vector field is written in complex form, reducing both equations to a single one.

Complex variable methods are applied to elliptical problems in fluid mechanics, and linear elasticity.

The techniques presented for solving parabolic problems are the Laplace transform and separation of variables, illustrated for problems of heat flow and soil mechanics.  Hyperbolic problems of vibrating strings and bars, governed by the wave equation are solved by the method of characteristics as well as by Laplace transform.

The method of characteristics for quasi-linear hyperbolic partial differential equations is illustrated for the case of a failing granular material, such as sand, underneath a strip footing.

The Navier Stokes equations are derived and discussed in the final chapter as an illustration of a highly non-linear set of partial differential equations and the solutions are interpreted by illustrating the role of rotation (curl) in energy transfer of a fluid.

Dr. Strack received his PhD from the Technical University of Delft, the Netherlands, in 1973. He joined the Department of Civil Engineering of the University of Minnesota in 1974, where he is currently a professor. Dr. Strack is the original developer of the Analytic Element Method, which is a popular method in groundwater modeling. He is the author of the textbook Groundwater Mechanics (Prentice-Hall, 1989) and the textbook Analytical Groundwater Mechanics (Cambridge University Press, 2017), and a chapter on basic groundwater flow equations in Wiley’s forthcoming Water Encyclopedia. He has authored 44 refereed papers in major journals, is the third recipient of the Lifetime Achievement Award, granted by the Minnesota Groundwater Association, and is a correspondent (foreign  member) of the Royal Dutch Academy of Sciences. He is the author of the computer programs SLAEM and MLAEM, used for describing groundwater flow.

Professor Strack has taught groundwater flow and mathematics for engineers for over 45 years and has many years of experience as a consultant.

• Vectors in Three-Dimensional Space

Pages 1-9

Strack, Otto D. L.

• Vector Fields

Pages 10-17

Strack, Otto D. L.

• Fundamental Equations for Fluid Mechanics

Pages 18-44

Strack, Otto D. L.

• Integral Theorems

Pages 45-50

Strack, Otto D. L.

• Coordinate Transformations: Definitions of Vectors and Tensors

Pages 51-71

Strack, Otto D. L.

eBook $79.99 price for USA in USD • ISBN 978-3-030-41168-8 • Digitally watermarked, DRM-free • Included format: PDF • ebooks can be used on all reading devices • Immediate eBook download after purchase Hardcover$99.99
price for USA in USD

## Bibliographic Information

Bibliographic Information
Book Title
Applications of Vector Analysis and Complex Variables in Engineering
Authors
2020
Publisher
Springer International Publishing
Springer Nature Switzerland AG
eBook ISBN
978-3-030-41168-8
DOI
10.1007/978-3-030-41168-8
Hardcover ISBN
978-3-030-41167-1
Edition Number
1
Number of Pages
XIV, 216
Number of Illustrations
86 illustrations in colour
Topics