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.
General
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