Essential Mathematics for Engineers and Scientists

The textbook covers the topics on applied mathematics which the authors teach the graduate students at the Michigan State University. The book is structured in three main parts containing 13 chapters divided into many sections and subsections.

Part I “Linear Algebra” consists of four chapters. Chapter 1 “Linear Algebra and Finite-Dimensional Vector Spaces” reviews properties of matrices and vectors, spanning and basis, vector spaces and scalar product, orthogonal basis and Gram-Schmidt procedure, and orthogonal matrices and their determinants. Chapter 2 “Linear Transformations” describes change of basis, projection tensors, orthogonal projections, null space, range, and invariant subspaces. Chapter 3 “Application to Systems of Equations” considers solutions of finite-dimension linear algebraic equations, orthogonal complements of the range and null space, and the Fredholm alternative theorem (FAT) about a unique one, infinitely many, or no solutions. The least-squares method, normal equations, and matrix pseudo-inverse are described for the stress-stretch observations in rubber experiments. Chapter 4 “The Spectrum of Eigenvalues” presents the eigenproblem, characteristic equation, real and complex solutions for eigenvalues and eigenvectors, eigenvalue multiplicity, and eigenspaces. Diagonalization procedure for the vibration of mass-spring mechanical networks with friction, dissipation, and resonance is discussed. Rectangular matrices and the singular value decomposition (SVD), connection between FAT and SVD, pseudo-inversion, ill-condition matrices, and application to the filtering noise out of linear systems are described.

Part II “Complex Variables” contains the next four chapters. Chapter 5 “Complex Variables: Basic Concepts” introduces the imaginary and complex numbers overviewing their history in the Italian Renaissance with Cardano’s formulae for cubic equations. Arithmetic of complex numbers, the conjugates, exponential and trigonometric forms, and planar geometry features are described. Cauchy path and contour integration of complex functions in the complex plane is introduced. Non-planar geometrical interpretation of the complex numbers identified with the points on the 3D Riemann sphere, and the point-pair algebra used in computer calculations are studied. Chapter 6 “Analytic Functions of a Complex Variable” starts with properties of power, inversion, linear fraction, exponent, and trigonometric complex functions, location of their roots in the complex plane, the principal branch and value among infinite number of periodic solutions, and their mapping are discussed. The famous Cauchy-Riemann (C-R) equations as necessary conditions for existence of the derivative of an analytic function are defined. Harmonic functions satisfying the Laplace’s equation are introduced. Equations for fluids in motion, incompressible frictionless flow, 2D steady flow, and objects in the flow field are considered. Cartesian, spherical, cylindrical, and other systems of coordinates are described. Schwarz-Christoffel transformation (SCT) and polygon mapping are studied. Chapter 7 “The Cauchy Integral Theorems” covers integration of analytic and non-analytic functions, evaluation of complex contour integrals, Cauchy-Goursat theorem (CGT) for path integrals by the clock-wise (CW) and counter-clock-wise (CCW) closed contours, C-R equations and Green’s line integrals, and closed-path contours around singularities. The Cauchy integral theorem for an analytical function within and on a simple closed contour is proved and applied to evaluation of complex integrals. The fundamental theorem of algebra, Morrera’s theorem, Liouville’s theorem, Lerch’s theorem, harmonic functions, and Laplace transforms with their inverses are also given. Application to fluid mechanics, forces exerted by moving flow, Bernoulli equation, fluid vorticity and circulation, streamlines for the flow, Magnus effect, d’Alembert’s paradox, Kelvin’s theorem, cylinder with blowing, blunt body in crossflow, and other problems in hydrodynamics are considered. Chapter 8 “Series Expansions and Contour Integration” deals with power-series of complex functions, including the Taylor and Laurent series convergency, singularities or poles, and the contour integration defined by the residue (Res). Finding the real integrals via the contour complex integrals, Pickard’s theorem, the residue theorem, Jordan’s lemma, and their applications are described. Branch cuts and mapping similar to Riemann surfaces are discussed for integration of multi-layer functions, including elliptical, polynomials, logs, fractional power functions. Generalized complex Fourier transform in the complex plane, the Plemelj formulae for singularities on the integration contour, and the concept of analytical continuation are also discussed.

Part III “Partial Differential Equations” covers the last five chapters. Chapter 9 “Linear Partial Differential Equations” describes the Laplace partial differential equation (PDE) for the membrane oscillation, its equilibrium and steady states. Similar PDEs are known for the Chaplygin-Karman-Tsien approximation in gas dynamics. The PDEs with specified boundary conditions (BCs) define a boundary-value problem (BVP). The 2D Laplacian operator is presented for the Cartesian and polar coordinates, and in the 3D case for the extended Cartesian, polar, and spherical coordinates. The BCs include Dirichlet, Neumann, and mixed or “Robin” equations for a function, its derivative, and their linear combination, respectively. Solution of the Laplace PDE for a membrane with inhomogeneous BCs can be found by the separation of variables (SoV) technique leading to the ordinary differential equation (ODE) for each variable, with results expressed via the Fourier series. Laplace equations for time-dependent processes can be reduced to the elliptic PDE with equilibrium solution, to the parabolic PDE with diffusion solution, or to the hyperbolic PDE with the wave propagation solution. For the last two types of PDEs the initial conditions (IC) could be given, and the combined initial-BVP (IBVP) can be solved in the SoV approach. The BVP with homogenous BCs, the Helmholtz equation, the Rayleigh quotient, the corresponding eigenproblems are defined with properties of the eigenvalue positivity and eigenfunctions orthogonality. The complete sets of orthogonal functions, approximation of eigenvalues by the Rayleigh quotient, SoV in non-Cartesian coordinates, steady-state limit, transient decay, and other problems are described. Chapter 10 “Linear Ordinary Differential Equations” considers the PDE reduction to the ODE in the SoV approach, properties and solution of an n-order linear homogeneous and non-homogeneous differential equations, Laplace transformation for ODE, and equations in the complex plane with the space or time coordinates of the ODE. Classification of the ordinary, regular or irregular singularities, and the Taylor or Frobenius series for the linear ODEs solutions are described. SoV on the Laplacian in the polar coordinates yields the Bessel equation, in spherical coordinates—the Legendre equations, with the corresponding special functions or polynomial solutions. Recurrence formulae for these functions, integrals of the special functions, and their orthogonality conditions are presented. The second-order ODEs, error and gamma functions, Sturm-Liouville operators, and orthogonal eigenfunctions are formulated. Chapter 11 “Green’s Functions for Ordinary Differential Equations” extends solutions of ODEs with a point localized force or impact defined by the delta ($\delta$) function or Dirac delta function in time or space coordinate. In mechanics it can be a point impulse, in electrical engineering—a point charge, in heat transfer—a point heat, in hydrodynamics—point sources or point sinks. Superposition of the solutions in product with the Green (G) functions serves for problems with multiple unit points or distributed impact. Integral properties of $\delta$ and G functions, the adjoint and FAT for linear differential operators, and applications for different problems are presented. Chapter 12 “Poisson’s Equation and Green’s Functions” considers generalization of the Laplacian to the Poisson equation widely applied in description of diffusion processes in hydrodynamics and viscous flows. The Navier-Stokes momentum-balance and Euler’s equations, Laplace’s and Poisson’s equations are considered with their specifics and solutions based on the SoV technique, eigenproblems for the separated ODEs, and incorporating the G functions. The equations’ extremals are studied with energy and entropy interpretation. A Laplace equation with inhomogeneous BCs can be converted to a Poisson equation with homogeneous BCs. Multidimensional, free-space, finite and semi-infinite domain, and complex analysis for G functions are discussed. The image source and sinks, symmetry and superposition, pull-back and push-forward approaches are given on various examples. Chapter 13 “Combined Green’s Function and Eigenfunction Methods” describes vibration theory and normal modes, harmonic and resonance frequencies, modified G functions for resonance forcing, the Sturm-Liouville (SL) operator formulation, with the Bessel’s equation and other mathematical connections to physical models. The relationships between the SL theory and FAT, complex-contour integration and eigenfunctions expansions for differential operators and G functions are derived and demonstrated on numerous examples. References and Index finalize the book.

Each topic is presented on multiple exercises with detail solutions, and additional exercises are given at the end of each chapter (more than 430 total), some of them could be addressed with help of computational and symbolic manipulation in Mathematica and MATLAB software. Many graphical illustrations are given, and questions-answers dialogues are added sometimes to clarify the essence of problems (for instance, a cartoon and discussion between Batman and Robin). The manual educates students and R&D specialists working in academy and industry for practical usage of the mathematical tools. Support materials are available at the link Resources for Essential Mathematics for Engineers and Scientists | Higher Education from Cambridge. Some additional sources are described in the references (Lipovetsky, 2021, 2022).

Stan Lipovetsky
Minneapolis, MN
[email protected]