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E-grāmata: Foundation Mathematics for Computer Science: A Visual Approach

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  • Formāts: EPUB+DRM
  • Izdošanas datums: 17-Mar-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • Valoda: eng
  • ISBN-13: 9783030420789
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Foundation Mathematics for Computer Science: A Visual ApproachPalielināt
  • Formāts: EPUB+DRM
  • Izdošanas datums: 17-Mar-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • Valoda: eng
  • ISBN-13: 9783030420789
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In this second edition of Foundation Mathematics for Computer Science, John Vince has reviewed and edited the original book and written new chapters on combinatorics, probability, modular arithmetic and complex numbers. These subjects complement the existing chapters on number systems, algebra, logic, trigonometry, coordinate systems, determinants, vectors, matrices, geometric matrix transforms, differential and integral calculus. During this journey, the author touches upon more esoteric topics such as quaternions, octonions, Grassmann algebra, Barrycentric coordinates, transfinite sets and prime numbers.





John Vince describes a range of mathematical topics to provide a solid foundation for an undergraduate course in computer science, starting with a review of number systems and their relevance to digital computers, and finishing with differential and integral calculus. Readers will find that the authors visual approach will greatly improve their understanding as to why certain mathematical structures exist, together with how they are used in real-world applications.





This second edition includes new, full-colour illustrations to clarify the mathematical descriptions, and in some cases, equations are also coloured to reveal vital algebraic patterns. The numerous worked examples will help consolidate the understanding of abstract mathematical concepts.





Whether you intend to pursue a career in programming, scientific visualisation, artificial intelligence, systems design, or real-time computing, you should find the authors literary style refreshingly lucid and engaging, and prepare you for more advanced texts.
1 Visual Mathematics
1(4)
1.1 Visual Brains Versus Analytic Brains
1(1)
1.2 Learning Mathematics
2(1)
1.3 What Makes Mathematics Difficult?
2(1)
1.4 Does Mathematics Exist Outside Our Brains?
3(1)
1.5 Symbols and Notation
3(2)
2 Numbers
5(30)
2.1 Introduction
5(1)
2.2 Counting
5(1)
2.3 Sets of Numbers
6(1)
2.4 Zero
7(1)
2.5 Negative Numbers
8(2)
2.5.1 The Arithmetic of Positive and Negative Numbers
9(1)
2.6 Observations and Axioms
10(1)
2.6.1 Commutative Law
10(1)
2.6.2 Associative Law
10(1)
2.6.3 Distributive Law
11(1)
2.7 The Base of a Number System
11(8)
2.7.1 Background
11(1)
2.7.2 Octal Numbers
12(1)
2.7.3 Binary Numbers
13(1)
2.7.4 Hexadecimal Numbers
13(4)
2.7.5 Adding Binary Numbers
17(1)
2.7.6 Subtracting Binary Numbers
18(1)
2.8 Types of Numbers
19(6)
2.8.1 Natural Numbers
19(1)
2.8.2 Integers
19(1)
2.8.3 Rational Numbers
20(1)
2.8.4 Irrational Numbers
20(1)
2.8.5 Real Numbers
20(1)
2.8.6 Algebraic and Transcendental Numbers
20(1)
2.8.7 Imaginary Numbers
21(1)
2.8.8 Complex Numbers
22(1)
2.8.9 Quaternions and Octonions
23(1)
2.8.10 Transcendental and Algebraic Numbers
24(1)
2.9 Prime Numbers
25(5)
2.9.1 The Fundamental Theorem of Arithmetic
26(1)
2.9.2 Is 1 a Prime?
27(1)
2.9.3 Prime Number Distribution
27(1)
2.9.4 Infinity of Primes
28(1)
2.9.5 Perfect Numbers
29(1)
2.9.6 Mersenne Numbers
30(1)
2.10 Infinity
30(1)
2.11 Worked Examples
31(2)
2.11.1 Algebraic Expansion
31(1)
2.11.2 Binary Subtraction
31(1)
2.11.3 Complex Numbers
32(1)
2.11.4 Complex Rotation
32(1)
2.11.5 Quaternions
33(1)
References
33(2)
3 Algebra
35(20)
3.1 Introduction
35(1)
3.2 Background
35(1)
3.3 Notation
36(5)
3.3.1 Solving the Roots of a Quadratic Equation
38(3)
3.4 Indices
41(1)
3.4.1 Laws of Indices
42(1)
3.5 Logarithms
42(2)
3.6 Further Notation
44(1)
3.7 Functions
44(6)
3.7.1 Explicit and Implicit Equations
45(1)
3.7.2 Function Notation
45(1)
3.7.3 Intervals
46(1)
3.7.4 Function Domains and Ranges
47(1)
3.7.5 Odd and Even Functions
48(1)
3.7.6 Power Functions
49(1)
3.8 Worked Examples
50(5)
3.8.1 Algebraic Manipulation
50(1)
3.8.2 Solving a Quadratic Equation
51(2)
3.8.3 Factorising
53(2)
4 Logic
55(24)
4.1 Introduction
55(1)
4.2 Background
55(1)
4.3 Truth Tables
56(1)
4.3.1 Logical Connectives
56(1)
4.4 Logical Premises
57(13)
4.4.1 Material Equivalence
57(1)
4.4.2 Implication
58(1)
4.4.3 Negation
59(1)
4.4.4 Conjunction
59(1)
4.4.5 Inclusive Disjunction
59(1)
4.4.6 Exclusive Disjunction
59(1)
4.4.7 Idempotence
60(1)
4.4.8 Commutativity
61(1)
4.4.9 Associativity
62(1)
4.4.10 Distributivity
63(1)
4.4.11 de Morgan's Laws
63(1)
4.4.12 Simplification
64(1)
4.4.13 Excluded Middle
65(1)
4.4.14 Contradiction
65(1)
4.4.15 Double Negation
66(1)
4.4.16 Implication and Equivalence
66(1)
4.4.17 Exportation
66(1)
4.4.18 Contrapositive
66(1)
4.4.19 Reductio Ad Absurdum
67(1)
4.4.20 Modus Ponens
68(1)
4.4.21 Proof by Cases
69(1)
4.5 Set Theory
70(6)
4.5.1 Empty Set
71(1)
4.5.2 Membership and Cardinality of a Set
71(1)
4.5.3 Subsets, Supersets and the Universal Set
72(1)
4.5.4 Set Building
72(1)
4.5.5 Union
73(1)
4.5.6 Intersection
74(1)
4.5.7 Relative Complement
74(1)
4.5.8 Absolute Complement
75(1)
4.5.9 Power Set
76(1)
4.6 Worked Examples
76(3)
4.6.1 Truth Tables
76(1)
4.6.2 Set Building
76(2)
4.6.3 Sets
78(1)
4.6.4 Power Set
78(1)
5 Combinatorics
79(10)
5.1 Introduction
79(1)
5.2 Permutations
79(3)
5.3 Permutations of Multisets
82(1)
5.4 Combinations
83(2)
5.5 Worked Examples
85(4)
5.5.1 Eight-Permutations of a Multiset
85(1)
5.5.2 Eight-Permutations of a Multiset
86(1)
5.5.3 Number of Permutations
87(1)
5.5.4 Number of Five-Card Hands
87(1)
5.5.5 Hand Shakes with 100 People
87(1)
5.5.6 Permutations of MISSISSIPPI
88(1)
6 Probability
89(12)
6.1 Introduction
89(1)
6.2 Definition and Notation
89(6)
6.2.1 Independent Events
91(1)
6.2.2 Dependent Events
91(1)
6.2.3 Mutually Exclusive Events
92(1)
6.2.4 Inclusive Events
93(1)
6.2.5 Probability Using Combinations
93(2)
6.3 Worked Examples
95(6)
6.3.1 Product of Probabilities
95(1)
6.3.2 Book Arrangements
96(1)
6.3.3 Winning a Lottery
96(1)
6.3.4 Rolling Two Dice
96(1)
6.3.5 Two Dice Sum to 7
96(1)
6.3.6 Two Dice Sum to 4
97(1)
6.3.7 Dealing a Red Ace
97(1)
6.3.8 Selecting Four Aces in Succession
97(1)
6.3.9 Selecting Cards
97(1)
6.3.10 Selecting Four Balls from a Bag
98(1)
6.3.11 Forming Teams
98(1)
6.3.12 Dealing Five Cards
99(2)
7 Modular Arithmetic
101(18)
7.1 Introduction
101(1)
7.2 Informal Definition
101(1)
7.3 Notation
102(1)
7.4 Congruence
102(1)
7.5 Negative Numbers
103(1)
7.6 Arithmetic Operations
103(7)
7.6.1 Sums of Numbers
104(1)
7.6.2 Products
105(1)
7.6.3 Multiplying by a Constant
105(1)
7.6.4 Congruent Pairs
106(1)
7.6.5 Multiplicative Inverse
106(2)
7.6.6 Modulo a Prime
108(1)
7.6.7 Fermat's Little Theorem
109(1)
7.7 Applications of Modular Arithmetic
110(5)
7.7.1 ISBN Parity Check
110(3)
7.7.2 ISBN Check Digits
113(2)
7.8 Worked Examples
115(3)
7.8.1 Negative Numbers
115(1)
7.8.2 Sums of Numbers
115(1)
7.8.3 Remainders of Products
116(1)
7.8.4 Multiplicative Inverse
116(1)
7.8.5 Product Table for Modulo 13
117(1)
7.8.6 ISBN Check Digit
117(1)
References
118(1)
8 Trigonometry
119(14)
8.1 Introduction
119(1)
8.2 Background
119(1)
8.3 Units of Angular Measurement
119(1)
8.4 The Trigonometric Ratios
120(3)
8.4.1 Domains and Ranges
123(1)
8.5 Inverse Trigonometric Ratios
123(2)
8.6 Trigonometric Identities
125(1)
8.7 The Sine Rule
126(1)
8.8 The Cosine Rule
126(1)
8.9 Compound-Angle Identities
127(3)
8.9.1 Double-Angle Identities
128(1)
8.9.2 Multiple-Angle Identities
129(1)
8.9.3 Half-Angle Identities
130(1)
8.10 Perimeter Relationships
130(3)
9 Coordinate Systems
133(14)
9.1 Introduction
133(1)
9.2 Background
133(1)
9.3 The Cartesian Plane
134(1)
9.4 Function Graphs
134(1)
9.5 Shape Representation
135(2)
9.5.1 2D Polygons
135(1)
9.5.2 Areas of Shapes
136(1)
9.6 Theorem of Pythagoras in 2D
137(1)
9.7 3D Cartesian Coordinates
137(2)
9.7.1 Theorem of Pythagoras in 3D
138(1)
9.8 Polar Coordinates
139(1)
9.9 Spherical Polar Coordinates
139(1)
9.10 Cylindrical Coordinates
140(1)
9.11 Barycentric Coordinates
141(1)
9.12 Homogeneous Coordinates
142(1)
9.13 Worked Examples
142(3)
9.13.1 Area of a Shape
142(1)
9.13.2 Distance Between Two Points
143(1)
9.13.3 Polar Coordinates
143(1)
9.13.4 Spherical Polar Coordinates
144(1)
9.13.5 Cylindrical Coordinates
144(1)
9.13.6 Barycentric Coordinates
145(1)
Reference
145(2)
10 Determinants
147(18)
10.1 Introduction
147(1)
10.2 Background
147(1)
10.3 Linear Equations with Two Variables
148(4)
10.4 Linear Equations with Three Variables
152(7)
10.4.1 Sarrus's Rule
158(1)
10.5 Mathematical Notation
159(3)
10.5.1 Matrix
159(1)
10.5.2 Order of a Determinant
159(1)
10.5.3 Value of a Determinant
159(2)
10.5.4 Properties of Determinants
161(1)
10.6 Worked Examples
162(3)
10.6.1 Determinant Expansion
162(1)
10.6.2 Complex Determinant
162(1)
10.6.3 Simple Expansion
163(1)
10.6.4 Simultaneous Equations
163(2)
11 Vectors
165(22)
11.1 Introduction
165(1)
11.2 Background
165(1)
11.3 2D Vectors
166(3)
11.3.1 Vector Notation
166(1)
11.3.2 Graphical Representation of Vectors
167(1)
11.3.3 Magnitude of a Vector
168(1)
11.4 3D Vectors
169(12)
11.4.1 Vector Manipulation
170(1)
11.4.2 Scaling a Vector
170(1)
11.4.3 Vector Addition and Subtraction
171(1)
11.4.4 Position Vectors
172(1)
11.4.5 Unit Vectors
173(1)
11.4.6 Cartesian Vectors
173(1)
11.4.7 Products
174(1)
11.4.8 Scalar Product
174(2)
11.4.9 The Vector Product
176(5)
11.4.10 The Right-Hand Rule
181(1)
11.5 Deriving a Unit Normal Vector for a Triangle
181(1)
11.6 Surface Areas
182(2)
11.6.1 Calculating 2D Areas
183(1)
11.7 Worked Examples
184(2)
11.7.1 Position Vector
184(1)
11.7.2 Unit Vector
184(1)
11.7.3 Vector Magnitude
184(1)
11.7.4 Angle Between Two Vectors
185(1)
11.7.5 Vector Product
185(1)
Reference
186(1)
12 Complex Numbers
187(38)
12.1 Introduction
187(1)
12.2 Representing Complex Numbers
187(1)
12.2.1 Complex Numbers
187(1)
12.2.2 Real and Imaginary Parts
188(1)
12.2.3 The Complex Plane
188(1)
12.3 Complex Algebra
188(28)
12.3.1 Algebraic Laws
188(2)
12.3.2 Complex Conjugate
190(2)
12.3.3 Complex Division
192(1)
12.3.4 Powers of i
193(1)
12.3.5 Rotational Qualities of i
194(2)
12.3.6 Modulus and Argument
196(2)
12.3.7 Complex Norm
198(1)
12.3.8 Complex Inverse
199(1)
12.3.9 Complex Exponentials
200(4)
12.3.10 de Moivre's Theorem
204(2)
12.3.11 nth Root of Unity
206(1)
12.3.12 nth Roots of a Complex Number
207(1)
12.3.13 Logarithm of a Complex Number
208(1)
12.3.14 Raising a Complex Number to a Complex Power
209(3)
12.3.15 Visualising Simple Complex Functions
212(3)
12.3.16 The Hyperbolic Functions
215(1)
12.4 Summary
216(1)
12.5 Worked Examples
217(7)
12.5.1 Complex Addition
217(1)
12.5.2 Complex Products
217(1)
12.5.3 Complex Division
217(1)
12.5.4 Complex Rotation
218(1)
12.5.5 Polar Notation
218(1)
12.5.6 Real and Imaginary Parts
219(1)
12.5.7 Magnitude of a Complex Number
219(1)
12.5.8 Complex Norm
220(1)
12.5.9 Complex Inverse
220(1)
12.5.10 de Moivre's Theorem
220(2)
12.5.11 nth Root of Unity
222(1)
12.5.12 Roots of a Complex Number
222(1)
12.5.13 Logarithm of a Complex Number
223(1)
12.5.14 Raising a Number to a Complex Power
223(1)
References
224(1)
13 Matrices
225(34)
13.1 Introduction
225(1)
13.2 Geometric Transforms
225(2)
13.3 Transforms and Matrices
227(3)
13.4 Matrix Notation
230(8)
13.4.1 Matrix Dimension or Order
230(1)
13.4.2 Square Matrix
230(1)
13.4.3 Column Vector
231(1)
13.4.4 Row Vector
231(1)
13.4.5 Null Matrix
231(1)
13.4.6 Unit Matrix
231(1)
13.4.7 Trace
232(1)
13.4.8 Determinant of a Matrix
233(1)
13.4.9 Transpose
233(1)
13.4.10 Symmetric Matrix
234(2)
13.4.11 Antisymmetric Matrix
236(2)
13.5 Matrix Addition and Subtraction
238(1)
13.5.1 Scalar Multiplication
238(1)
13.6 Matrix Products
239(4)
13.6.1 Row and Column Vectors
239(1)
13.6.2 Row Vector and a Matrix
240(1)
13.6.3 Matrix and a Column Vector
241(1)
13.6.4 Square Matrices
241(1)
13.6.5 Rectangular Matrices
242(1)
13.7 Inverse Matrix
243(7)
13.7.1 Inverting a Pair of Matrices
249(1)
13.8 Orthogonal Matrix
250(1)
13.9 Diagonal Matrix
251(1)
13.10 Worked Examples
251(8)
13.10.1 Matrix Inversion
251(1)
13.10.2 Identity Matrix
252(1)
13.10.3 Solving Two Equations Using Matrices
253(1)
13.10.4 Solving Three Equations Using Matrices
254(1)
13.10.5 Solving Two Complex Equations
255(1)
13.10.6 Solving Three Complex Equations
255(1)
13.10.7 Solving Two Complex Equations
256(1)
13.10.8 Solving Three Complex Equations
257(2)
14 Geometric Matrix Transforms
259(30)
14.1 Introduction
259(1)
14.2 Matrix Transforms
259(11)
14.2.1 2D Translation
260(1)
14.2.2 2D Scaling
261(2)
14.2.3 2D Reflections
263(1)
14.2.4 2D Shearing
264(1)
14.2.5 2D Rotation
265(3)
14.2.6 2D Scaling
268(1)
14.2.7 2D Reflection
268(1)
14.2.8 2D Rotation About an Arbitrary Point
269(1)
14.3 3D Transforms
270(6)
14.3.1 3D Translation
270(1)
14.3.2 3D Scaling
271(1)
14.3.3 3D Rotation
271(3)
14.3.4 Rotating About an Axis
274(2)
14.3.5 3D Reflections
276(1)
14.4 Rotating a Point About an Arbitrary Axis
276(3)
14.4.1 Matrices
276(3)
14.5 Determinant of a Transform
279(1)
14.6 Perspective Projection
280(2)
14.7 Worked Examples
282(7)
14.7.1 2D Scale and Translate
282(1)
14.7.2 2D Rotation
283(1)
14.7.3 Determinant of the Rotate Transform
284(1)
14.7.4 Determinant of the Shear Transform
284(1)
14.7.5 Yaw, Pitch and Roll Transforms
285(1)
14.7.6 Rotation About an Arbitrary Axis
285(1)
14.7.7 3D Rotation Transform Matrix
286(1)
14.7.8 Perspective Projection
287(2)
15 Calculus: Derivatives
289(62)
15.1 Introduction
289(1)
15.2 Background
289(1)
15.3 Small Numerical Quantities
290(1)
15.4 Equations and Limits
291(8)
15.4.1 Quadratic Function
291(2)
15.4.2 Cubic Equation
293(1)
15.4.3 Functions and Limits
294(2)
15.4.4 Graphical Interpretation of the Derivative
296(1)
15.4.5 Derivatives and Differentials
297(1)
15.4.6 Integration and Antiderivatives
298(1)
15.5 Function Types
299(1)
15.6 Differentiating Groups of Functions
300(11)
15.6.1 Sums of Functions
300(2)
15.6.2 Function of a Function
302(4)
15.6.3 Function Products
306(3)
15.6.4 Function Quotients
309(2)
15.7 Differentiating Implicit Functions
311(3)
15.8 Differentiating Exponential and Logarithmic Functions
314(4)
15.8.1 Exponential Functions
314(3)
15.8.2 Logarithmic Functions
317(1)
15.9 Differentiating Trigonometric Functions
318(6)
15.9.1 Differentiating tan
318(2)
15.9.2 Differentiating esc
320(1)
15.9.3 Differentiating sec
321(1)
15.9.4 Differentiating cot
322(1)
15.9.5 Differentiating arcsin, arccos and arctan
323(1)
15.9.6 Differentiating arccsc, arcsec and arccot
324(1)
15.10 Differentiating Hyperbolic Functions
324(3)
15.10.1 Differentiating sinh, cosh and tanh
326(1)
15.11 Higher Derivatives
327(1)
15.12 Higher Derivatives of a Polynomial
328(2)
15.13 Identifying a Local Maximum or Minimum
330(2)
15.14 Partial Derivatives
332(6)
15.14.1 Visualising Partial Derivatives
335(1)
15.14.2 Mixed Partial Derivatives
336(2)
15.15 Chain Rule
338(2)
15.16 Total Derivative
340(2)
15.17 Power Series
342(2)
15.18 Worked Examples
344(7)
15.18.1 Antiderivative 1
344(1)
15.18.2 Antiderivative 2
345(1)
15.18.3 Differentiating Sums of Functions
345(1)
15.18.4 Differentiating a Function Product
345(1)
15.18.5 Differentiating an Implicit Function
346(1)
15.18.6 Differentiating a General Implicit Function
346(1)
15.18.7 Local Maximum or Minimum
347(1)
15.18.8 Partial Derivatives
348(1)
15.18.9 Mixed Partial Derivative 1
348(1)
15.18.10 Mixed Partial Derivative 2
349(1)
15.18.11 Total Derivative
349(2)
16 Calculus: Integration
351(44)
16.1 Introduction
351(1)
16.2 Indefinite Integral
351(1)
16.3 Integration Techniques
352(2)
16.3.1 Continuous Functions
352(1)
16.3.2 Difficult Functions
353(1)
16.4 Trigonometric Identities
354(16)
16.4.1 Exponent Notation
356(1)
16.4.2 Completing the Square
357(2)
16.4.3 The Integrand Contains a Derivative
359(1)
16.4.4 Converting the Integrand into a Series of Fractions
360(1)
16.4.5 Integration by Parts
361(5)
16.4.6 Integration by Substitution
366(2)
16.4.7 Partial Fractions
368(2)
16.5 Area Under a Graph
370(1)
16.6 Calculating Areas
370(8)
16.7 Positive and Negative Areas
378(2)
16.8 Area Between Two Functions
380(2)
16.9 Areas with the y-Axis
382(1)
16.10 Area with Parametric Functions
383(2)
16.11 The Riemann Sum
385(1)
16.12 Worked Examples
386(9)
16.12.1 Integrating a Function Containing Its Own Derivative
386(1)
16.12.2 Dividing an Integral into Several Integrals
387(1)
16.12.3 Integrating by Parts 1
388(1)
16.12.4 Integrating by Parts 2
388(2)
16.12.5 Integrating by Substitution 1
390(1)
16.12.6 Integrating by Substitution 2
390(2)
16.12.7 Integrating by Substitution 3
392(1)
16.12.8 Integrating with Partial Fractions
392(3)
Appendix A Limit of (sin θ)/θ 395(4)
Appendix B Integrating cosn θ 399(2)
Index 401
Professor John Vince began working in computer graphics at Middlesex Polytechnic in 1968. His research activities centered on computer animation software and resulted in the PICASO and PRISM animation systems. Whilst at Middlesex, he designed the UKs first MSc course in Computer Graphics and developed a popular program of short courses in computer animation for television designers. In 1986 he joined Rediffusion Simulation as a Research Consultant and worked on the development of real-time computer systems for commercial flight simulators. In 1992 he was appointed Chief Scientist of Thomson Training Simulation Ltd. In 1995 he was appointed Professor of Digital Media at the National Centre for Computer Animation at Bournemouth University and in 1999 he was made Head of Academic Group for Computer Animation. He was awarded a DSc by Brunel University in recognition of his work in computer graphics. He has written and edited over 40 books on computer graphics, computer animation and virtual reality, including the following Springer titles:

Mathematics for Computer Graphics (2014)

Calculus for Computer Graphics (2013)

Matrix Transforms for Computer Games and Animation (2012)

Expanding the Frontiers of Visual Analytics and Visualization (2012)

Quaternions for Computer Graphics (2011)

Rotation Transforms for Computer Graphics (2011)
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