Mathematics for Electrical Engineering and Computing

Mathematics for Electrical Engineering and Computing

Language: English

Pages: 539


Format: PDF / Kindle (mobi) / ePub

Mathematics for Electrical Engineering and Computing embraces many applications of modern mathematics, such as Boolean Algebra and Sets and Functions, and also teaches both discrete and continuous systems - particularly vital for Digital Signal Processing (DSP). In addition, as most modern engineers are required to study software, material suitable for Software Engineering - set theory, predicate and prepositional calculus, language and graph theory - is fully integrated into the book.

Excessive technical detail and language are avoided, recognising that the real requirement for practising engineers is the need to understand the applications of mathematics in everyday engineering contexts. Emphasis is given to an appreciation of the fundamental concepts behind the mathematics, for problem solving and undertaking critical analysis of results, whether using a calculator or a computer.

The text is backed up by numerous exercises and worked examples throughout, firmly rooted in engineering practice, ensuring that all mathematical theory introduced is directly relevant to real-world engineering. The book includes introductions to advanced topics such as Fourier analysis, vector calculus and random processes, also making this a suitable introductory text for second year undergraduates of electrical, electronic and computer engineering, undertaking engineering mathematics courses.

The book is supported with a number of free online resources. On the companion website readers will find:
* over 60 pages of "Background Mathematics" reinforcing introductory material for revision purposes in advance of your first year course
* plotXpose software (for equation solving, and drawing graphs of simple functions, their derivatives, integrals and Fourier transforms)
* problems and projects (linking directly to the software)

In addition, for lecturers only, features a complete worked solutions manual for the exercises in the book.

Dr Attenborough is a former Senior Lecturer in the School of Electrical, Electronic and Information Engineering at South Bank University. She is currently Technical Director of The Webbery - Internet development company, Co. Donegal, Ireland.

* Fundamental principles of mathematics introduced and applied in engineering practice, reinforced through over 300 examples directly relevant to real-world engineering
* Over 60 pages of basic revision material available to download in advance of embarking on a first year course
* Free website support, featuring complete solutions manual, background mathematics, plotXpose software, and further problems and projects enabling students to build on the concepts introduced, and put the theory into practice

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travelled by the tip of the rod as in Figure 5.2. Figure 5.1 defines the function y = sin(t) and Figure 5.2 defines the function x = cos(t). This definition of the trigonometric function is very similar to that used for the ratios in the triangle, if the hypotenuse is of length 1 unit. The definitions become the same for angles up to a right angle if radians TLFeBOOK Trigonometric functions and waves 89 Figure 5.1 The function y = sin(t ), where t is the distance travelled by the tip of a

Integration 137 We used the trick of replacing (du/dx) dx by du, this can be justified in the following argument. By the definition of the integral as inverse differentiation, if y is differentiated with respect to x and then integrated with respect to x we will get back to y, give or take a constant. This is expressed by dy dx = y + C. dx (7.1) If y is a composite function that can be written in terms of the variable u, then dy du dy = . dx du dx Substituting the chain rule for dy/dx into

0: δy f (t + δt) − δt et+δt − et = = δt δt δt At t = 0 all functions y = a t have value 1 so that the gradient of the chord at t = 0 is δy eδt − 1 = δt δt d t (e ) = et so that at the point dt t = 0 the gradient of the tangent is given by dy/dt = 1. For small δt the gradient of the tangent is approximately equal to the gradient of the chord δy dy ≈ dt δt and therefore We defined e as the number for which 1≈ eδt − 1 δt TLFeBOOK 168 The exponential function Figure 8.3 (a) The graph of y = 3t

find the derivatives of the inverse trigonometric functions in Chapter 5. y = ln(x) where x > 0 ⇔ ey = eln(x) (take the exponential of both sides) ⇔ ey = x (as exp is the inverse function to ln, eln(x) = x) We wish to differentiate both sides with respect to x but the left-hand side is a function of y, so we use the chain rule, setting w = ey , thus, equation ey = x becomes w = x and dw/dy = ey . Differentiating both sides of w = x with respect to x gives dw/dx = 1, where dw dy dw = dx dy dx

on the lower half of the computer screen as in Exercise 1.6(a). (b) Points on the diagonal line and lying in the shaded area represent the set of positions for Exercise 1.6(b). 1.8. Draw arrow diagrams and graphs of the following functions: (a) f (t) = (t − 1)2 t ∈ {0, 1, 2, 3, 4} (b) g(z) = 1/z z ∈ {−1, −0.5, 0.5, 1, 1.5, 2} x x ∈ {−2, −1} (c) y = 2x x ∈ {0, 1, 2, 3} (d) h : t → 3 − t t ∈ {5, 6, 7, 8, 9, 10} 1.9. Given that f : x → 2x − 1, h : x → 3/x (a) Find the following: (iii) (f ◦ f −1 ) :

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