Pragmatic Mathematics For Scientists And Engineers

Contents
Preface
About the Authors
1. Infinite Series
	1.1. Applications of series
	1.2. Basic concepts: Infinite sequences and infinite series
	1.3. Basic properties of infinite series
	1.4. Series with positive terms
		1.4.1. Cauchy’s, d’Alembert’s, and Raabe’s tests
		1.4.2. Cauchy’s integral test for convergence
	1.5. Alternating series
	1.6. Absolutely convergent series
	1.7. Series with complex numbers and series of vectors
	1.8. Series of functions
		1.8.1. Power Series
		1.8.2. Taylor series
	1.9. Exercises and problems
2. Complex Numbers
	2.1. Complex numbers in science
	2.2. Definitions
		2.2.1. Basic algebra of complex numbers
	2.3. Complex numbers in z = x + iy form
		2.3.1. Rectangular and polar representations
		2.3.2. Complex conjugate
	2.4. Exponential form
		2.4.1. Products and quotients in exponential form
		2.4.2. Power and roots of complex numbers
		2.4.3. Complex logarithm and complex powers
		2.4.4. The exponential and trigonometric functions
	2.5. Hyperbolic functions
	2.6. Exercises
3. Vectors
	3.1. Introduction
	3.2. Vector algebra
		3.2.1. Addition of vectors
		3.2.2. Subtraction of vectors
		3.2.3. Multiplying a vector by a scalar
	3.3. Base vectors and unit vectors
		3.3.1. Linear independence
		3.3.2. Basis vectors
		3.3.3. The Cartesian coordinates and unit vectors
	3.4. Vector algebra with vector components
	3.5. Scalar and vector products
		3.5.1. Scalar product of two vectors
		3.5.2. Vector product of two vectors
		3.5.3. Scalar and vector triple products
	3.6. Vector calculus
		3.6.1. Differentiation of vectors
		3.6.2. Integration
	3.7. Fields
		3.7.1. Gradient of a scalar field
		3.7.2. Divergence and curl of a vector field
	3.8. Exercises
4. Matrices
	4.1. Introduction
	4.2. Matrices
		4.2.1. Basic operations on matrices
		4.2.2. Types of square matrices
	4.3. Determinants
		4.3.1. Calculating determinants by permutations
		4.3.2. Expansion by minors
		4.3.3. Properties of determinants
	4.4. Linear systems of equations
		4.4.1. Cramer’s rule
		4.4.2. Solutions by the inverse matrix
		4.4.3. Direct elimination methods
		4.4.4. Iterative methods
	4.5. Eigenvalues and eigenvectors
	4.6. Exercises
5. Partial Differentiation
	5.1. Complicated real world
	5.2. The partial derivatives
	5.3. The total differential and total derivative
	5.4. Second partial derivatives
	5.5. Taylor series for functions of two variables
	5.6. Stationary points of a function of two variables
	5.7. Stationary points with constrains
	5.8. Change of variables
	5.9. Implicit functions
	5.10. Exercises
6. Line Integrals and Multiple Integrals
	6.1. Introduction
	6.2. Line integrals: First kind
		6.2.1. Definition
		6.2.2. Reduction to an ordinary definite integral
	6.3. Line integrals: Second kind
		6.3.1. Definition
		6.3.2. Evaluating a second kind of line integral
		6.3.3. Path independence of the line integral
		6.3.4. Integration over a closed path
		6.3.5. Calculation of surface areas using line integrals’ integrals
		6.3.6. Connection between linear integrals of both kinds
	6.4. Application of line integrals
		6.4.1. Mass, center of mass, and rotational inertia of a wire
		6.4.2. Work done by a force
	6.5. Multiple integrals
		6.5.1. Double integrals
		6.5.2. Evaluation of double integrals
		6.5.3. Green’s theorem
		6.5.4. Applications: Mass, center of mass, and moment of inertia
		6.5.5. 2D case
		6.5.6. 3D case
	6.6. Change of variables in multiple integration
		6.6.1. Jacobian
	6.7. Exercises
7. Fourier Series and Transforms
	7.1. Function space and basis vectors
		7.1.1. Orthogonal set of base function
	7.2. Generalized Fourier series
	7.3. Trigonometric Fourier series
		7.3.1. Fourier coefficients by direct integration
		7.3.2. Dirichlet’s conditions
		7.3.3. Case of non-periodic function
		7.3.4. Fourier series in [0, π] interval
		7.3.5. Fourier series in [−L, L] interval
		7.3.6. Complex forms of Fourier
		7.3.7. Applications of Fourier series
	7.4. Fourier transform
		7.4.1. Properties of Fourier transform
		7.4.2. Applications of Fourier transform
	7.5. Exercises
8. First-Order Ordinary Differential Equations
	8.1. A note on classification of ordinary differential equations
		8.1.1. General and particular solutions
	8.2. First-order ordinary differential equations: Basic concepts
	8.3. Separable-variable equations
	8.4. First-order linear ordinary differential equations
		8.4.1. First-order linear homogeneous ordinary differential equation
		8.4.2. First-order linear non-homogeneous ODE: Lagrange’s method
	8.5. Exact equations: Integrating factors
		8.5.1. Exact equations
		8.5.2. Reducible to exact form
	8.6. Some first-order differential equations in physics
		8.6.1. Cooling and heating
		8.6.2. Decay and growth
		8.6.3. Motion along a straight line: Vertical, horizontal, and inclined
		8.6.4. Electric circuits
		8.6.5. Rocket motion
	8.7. Exercises (Math)
	8.8. Physics problems
9. Second-Order Linear Differential Equations
	9.1. General solution of the homogeneous equation
		9.1.1. Wronskian and linear independence
		9.1.2. Reduction of order
	9.2. General solution of the non-homogeneous equation
		9.2.1. Lagrange’s method of variation of parameters
	9.3. Homogeneous equations with constant coefficients
		9.3.1. Two distinct real roots
		9.3.2. Two equal roots
		9.3.3. Complex roots
	9.4. Non-homogeneous equations with constant coefficients
		9.4.1. Lagrange’s method: Variation of constants
		9.4.2. Euler’s method: Undetermined coefficients
	9.5. Series solutions
		9.5.1. Analytic coefficients
		9.5.2. Solutions about regular singular points
	9.6. Periodic series solutions
	9.7. Boundary value problem
	9.8. WKB(J) approximation
		9.8.1. The WKB(J) approximation: An informal treatment
		9.8.2. The next approximation
		9.8.3. General solution
	9.9. Some physics examples
		9.9.1. Oscillations
		9.9.2. R–L–C electric circuits
	9.10. Mathematical problems
	9.11. Physics problems
10. Green’s Function Method
	10.1. Dirac’s delta function
	10.2. Green’s function
		10.2.1. A physics example
		10.2.2. Mathematical approach
	10.3. Green’s function solving for first-order ODE
		10.3.1. Zero initial condition
		10.3.2. Arbitrary initial condition
	10.4. Green’s function for second-order ODE: Initial value problem
		10.4.1. Simple motion driven by a force
		10.4.2. Linear non-homogeneous second-order ODE
	10.5. Green’s function for second-order ODE: Boundary value problem
		10.5.1. Introduction to the boundary value problem
		10.5.2. Case 1: Only trivial solutions for L[y] = 0
		10.5.3. Case 2: Non-trivial solution for L[y] = 0
	10.6. Sturm–Liouville problem and Green’s function
		10.6.1. Sturm–Liouville problem
		10.6.2. Green’s function expansion
	10.7. Exercises
	10.8. Problems
11. Calculus of Variations
	11.1. Introduction
	11.2. From few degrees of freedom to many
	11.3. A necessary condition for an extremum
	11.4. Euler–Lagrange equation
		11.4.1. Special cases of Euler–Lagrange equation
	11.5. Generalizations of the Euler–Lagrange equation
		11.5.1. The Euler–Poisson equation
		11.5.2. The case of two functions
		11.5.3. The case of double integrals
	11.6. Variational problem with constraints
	11.7. A variational problem with variable end points
	11.8. Direct methods for solving variational problems
	11.9. The principle of least action
	11.10. Fermat’s principle of least time
	11.11. Exercises and problems
12. Functions of Complex Variables
	12.1. Introduction
	12.2. Derivatives and the Cauchy–Riemann conditions
	12.3. Integrals
		12.3.1. Cauchy’s integral theorem
		12.3.2. Indefinite integral
		12.3.3. Cauchy integral
	12.4. Series
		12.4.1. Power and Taylor series
		12.4.2. The Laurent series
		12.4.3. Zeros and isolated singular points
	12.5. The residue theorem and its applications
		12.5.1. Evaluating the residue
		12.5.2. Evaluating definite integrals by using residues
	12.6. Exercises and problems
Index