main-content

## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Vectors in Three-Dimensional Space

Abstract
We introduce vector analysis using fluid mechanics as the vehicle for providing physical meaning to the concepts of vectors and the associated definitions and operations.
Otto D. L. Strack

### Chapter 2. Vector Fields

Abstract
The vectors that we deal with in engineering usually are neither constant in space, nor in time. Such vectors are functions of position, i.e., they have different values depending on where we look; the collection of such vectors are called vector fields.
Otto D. L. Strack

### Chapter 3. Fundamental Equations for Fluid Mechanics

Abstract
We consider force equilibrium of an elementary volume of fluid of constant density, dv = dx1dx2dx3, centered at xi.
Otto D. L. Strack

### Chapter 4. Integral Theorems

Abstract
Integral theorems relate integrals over surfaces to integrals over their boundaries, and integrals over volumes to integrals over their bounding surfaces.
Otto D. L. Strack

### Chapter 5. Coordinate Transformations: Definitions of Vectors and Tensors

Abstract
We consider the formal definitions of vectors and tensors, and the equations for their components in a transformed coordinate system. We restrict the analysis to straight coordinates. We first consider Cartesian coordinates and afterward non-Cartesian coordinates.
Otto D. L. Strack

### Chapter 6. Partial Differential Equations of the First Order

Abstract
We consider linear partial differential equations of the first order. A partial differential equation is linear if the dependent variables, i.e., the functions that are being solved for, appear only to the first power and the coefficients, i.e., the factors the derivatives are multiplied by, do not contain the dependent variables.
Otto D. L. Strack

### Chapter 7. Partial Differential Equations of the Second Order

Abstract
We limit the discussion of second-order partial differential equations to linear equations and two-dimensional problems. Although the character of the partial differential equations remains much the same in three dimensions, the problems are much more difficult to solve in three-dimensional space.
Otto D. L. Strack

### Chapter 8. The Elliptic Case: Two Complex Characteristics

Abstract
Elliptic partial differential equations of the second order are relatively common in engineering practice. Examples include irrotational and divergence-free fluid flow, groundwater flow, and linear elasticity.
Otto D. L. Strack

### Chapter 9. Applications of Complex Variables

Abstract
We apply complex variables in this chapter to three topics taken from engineering practice, in the order of presentation: fluid mechanics, groundwater flow, and linear elasticity, as applied to soil mechanics and rock mechanics.
Otto D. L. Strack

### Chapter 10. The Parabolic Case: Two Coinciding Characteristics

Abstract
Parabolic partial differential equations have two coinciding characteristics, which makes them less suitable for the application of complex variables, in contrast to elliptical partial differential equations.
Otto D. L. Strack

### Chapter 11. The Hyperbolic Case: Two Real Characteristics

Abstract
As an illustration of a set of two linear hyperbolic partial differential equations, we consider first longitudinal vibration in a bar, and second transverse vibration in a string. We begin by deriving the equations that govern these two cases of vibration, which are similar in form. We examined the coefficients in these equations in Chapter 7 and established that the system is hyperbolic. We write the differential equations in the characteristic directions, and both determine and examine the solution.
Otto D. L. Strack

### Chapter 12. Hyperbolic Quasi Linear Partial Differential Equations

Abstract
We call a differential equation quasi linear if the partial derivatives of the dependent variables are multiplied by functions of the dependent variables.
Otto D. L. Strack

### Chapter 13. The Navier-Stokes Equations

Abstract
The Navier-Stokes equations are a set of highly non-linear partial differential equations. We present these equations as the final example of partial differential equations, because of their special character and their importance in the field of fluid mechanics. Common forms of these equations are obtained by simplification, for example by setting the viscosity of the fluid equal to zero (open channel flow), or by neglecting the acceleration relative to frictional losses, as in groundwater flow.
Otto D. L. Strack

### Backmatter

Weitere Informationen