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## Recommended Text

## Glyn James, *Modern Engineering Mathematics*, 4th Ed.

## Glyn James, *Advanced Modern Engineering Mathematics*, 3rd Ed.

The notes of Dr. Tenti are an excellent alternate source of the matrial in this course:

The recommended texts for MATH 211 are James's *Modern Engineering Mathematics*
and *Advanced Modern Engineering Mathematics* series. This sequence of
two books describes most of the mathematics required by electrical and computer
engineering undergraduate students. Ideally these texts will be used in all
three 2nd-year courses: MATH 115, MATH 211, and MATH 212.

The sub-sections which currently appear to be relevant to the above descriptions of the course material and the laboratories are emphasized with bold and and italics, respectively; and those sections where the majority of sub-sections are of interest are also emphasized.

The relevant chapters, sections, and subsections are listed here.

**10 Introduction to Ordinary Differential Equations**

- 10.1 Introduction
- 10.2 Engineering Examples
- 10.2.1 The take-off run of an aircraft
- 10.2.2 Domestic hot-water supply
- 10.2.3 Hydro-electric power generation
**10.2.4 Simple electrical circuits****10.3 The classification of differential equations****10.3.1 Ordinary and partial differential equation****10.3.2 Independent and dependent variables****10.3.3 The order of a differential equation****10.3.4 Linear and nonlinear differential equations****10.3.5 Homogeneous and nonhomogeneous equations****10.4 Solving differential equations**- 10.4.1 Solution by inspection
**10.4.2 General and particular solutions****10.4.3 Boundary and initial conditions****10.4.4 Analytical and numerical solutions**- 10.5 First-order ordinary differential equations
- 10.5.1 A geometrical perspective
**10.5.3 Solutions of separable differential equations**- 10.5.5 Solutions of differential equations of x
^{(1)}(t) = f(x/t) form - 10.5.7 Solutions of exact differential equations
**10.5.9 Solutions of linear differential equations**- 10.5.11 Solutions of Bernoulli differential equations
*10.6 Numerical solution of first-order ordinary differential equations**10.6.1 A simple solution method: Euler's method**10.6.2 Analyzing Euler's method**10.6.3 Using numerical methods to solve engineering problems*- 10.7 Engineering application: analysis of damper performance
**10.8 Linear differential equations****10.8.1 Differential operators****10.8.2 Linear differential equations****10.9 Linear constant-coefficient differential equations****10.9.1 Linear homogeneous constant-coefficient equations****10.9.3 Linear nonhomogeneous constant-coefficient equations****10.10 Engineering applications: second-order linear constant-coefficient differential equations****10.10.1 Free oscillations of elastic systems****10.10.2 Free oscillations of damped elastic systems****10.10.3 Forced oscillations of elastic systems****10.10.4 Oscillations in electrical circuits***10.11 Numerical solutions of second- and higher-order differential equations**10.11.1 Numerical solutions of coupled first-order equations**10.11.2 State-space representation of higher-order systems***10.12 Quantitative analysis of second-order differential equations****10.12.1 Phase-plane plots**

The relevant chapters, sections, and subsections are listed here.

**2 The Laplace Transform**

**2.1 Introduction****2.2 The Laplace Transform****2.2.1 Definition and notation****2.2.2 Transforms of simple functions****2.2.3 Existence of the Laplace transform****2.2.4 Properties of the Laplace transform****2.2.5 Table of Laplace transforms****2.2.7 The inverse Laplace transform****2.2.8 Evaluation of inverse transforms****2.2.9 Inversion using the first shift theorem****2.3 Solutions of Differential Equations****2.3.1 Transforms of derivatives****2.3.2 Transforms of integrals****2.3.3 Ordinary differential equations****2.3.4 Simultaneous differential equations****2.4 Engineering applications: electric circuits and mechanical vibrations****2.4.1 Electrical circuits****2.4.2 Mechanical vibrations****2.5 Step and impulse functions****2.5.1 The Heaviside step function****2.5.2 Laplace transform of the unit step function****2.5.3 The second shift theorem****2.5.4 Inversion using the second shift theorem****2.5.5 Differential equations****2.5.6 Periodic functions****2.5.8 The impulse function****2.5.9 The sifting property****2.5.10 Laplace transforms of impulse functions****2.5.11 Relationship between the Heaviside step and impulse functions**- 2.5.13 Bending of beams
**2.6 Transfer functions****2.6.1 Definitions****2.6.2 Stability****2.6.3 Impulse response****2.6.4 Initial- and final-value theorems****2.6.6 Convolution****2.6.7 System response to an arbitrary input****2.7 Engineering application: frequency response**

**4 Fourier Series**

**4.1 Introduction****4.2 Fourier series expansion****4.2.1 Periodic functions****4.2.2 Fourier's theorem****4.2.3 The Fourier coefficients****4.2.4 Functions of period 2π****4.2.5 Even and odd functions****4.2.7 Even and odd harmonics****4.2.8 Linearity property****4.2.9 Convergence of the Fourier series****4.2.10 Functions of period T****4.3 Functions defined over a finite interval****4.3.1 Full-range series****4.3.2 Half-range cosine and sine series****4.4 Differentiation and integration of Fourier series****4.4.1 Integration of a Fourier series****4.4.2 Differentiation of a Fourier series****4.4.3 Coefficients in terms of jumps at discontinuities****4.5 Engineering application: frequency response and oscillating systems****4.5.1 Response to periodic input**- 4.6 Complex form of Fourier series
- 4.6.1 Complex representation
- 4.6.2 The multiplication theorem and Parseal's theorem
- 4.6.3 Discrete frequency spectra
- 4.6.4 Power spectrum
- 4.7 Orthogonal functions
- 4.7.1 Definitions
- 4.7.2 Generalized Fourier series
- 4.7.3 Convergence of generalized Fourier series
- 4.8 Engineering application: describing functions

**9 Partial-differential Equations**

**9.1 Introduction****9.2 General discussion**- 9.2.1 Wave equation
**9.2.2 Heat-conduction or diffusion equation****9.2.3 Laplace equation**- 9.2.4 Other and related equations
- 9.2.5 Arbitrary functions
- 9.3 Solutions of the wave equation
- 9.3.1 D'Alembert solution and characteristics
**9.3.2 Separated soluations**- 9.3.3 Laplace transform solution
- 9.3.5 Numerical solutions
**9.4 Solutions of the heat-conduction/diffusion equation****9.4.1 Separation method**- 9.4.2 Laplace transform method
- 9.4.4 Numerical solutions
**9.5 Solutions of the Laplace equation****9.5.1 Separated solutions**- 9.5.3 Numerical solutions
- 9.6 Finite elements
- 9.7 General considerations
- 9.7.1 Formal classification
- 9.7.2 Boundary conditions
- 9.8 Engineering application: wave propagation under a moving load
- 9.9 Engineering application: blood-flow model