Foundation mathematics for computer science: a visual approach
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| Language: | English |
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[2015]
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 334 Seiten Illustrationen, Diagramme |
| ISBN: | 9783319214368 |
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| 264 | 4 | |c © 2015 | |
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Record in the Search Index
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| adam_text | Contents
1 Visual Mathematics............................................. 1
1.1 Visual Brains Versus Analytic Brains....................... 1
1.2 Learning Mathematics....................................... 2
1.3 What Makes Mathematics Difficult?.......................... 2
1.4 Does Mathematics Exist Outside Our Brains?................. 3
1.5 Symbols and Notation....................................... 3
2 Numbers........................................................ 5
2.1 Introduction............................................... 5
2.2 Sets of Numbers............................................ 5
2.3 Zero....................................................... 7
2.4 Negative Numbers........................................... 7
2.4.1 The Arithmetic of Positive and Negative
Numbers........................................... 8
2.5 Observations and Axioms.................................... 9
2.5.1 Commutative Law................................... 9
2.5.2 Associative Law.................................. 10
2.5.3 Distributive Law................................. 10
2.6 The Base of a Number System............................... 11
2.6.1 Background....................................... 11
2.6.2 Octal Numbers.................................... 12
2.6.3 Binary Numbers................................... 12
2.6.4 Hexadecimal Numbers.............................. 13
2.6.5 Adding Binary Numbers............................ 17
2.6.6 Subtracting Binary Numbers....................... 18
2.7 Types of Numbers........................................ 18
2.7.1 Natural Numbers.................................. 18
2.7.2 Integers..................................... 19
2.7.3 Rational Numbers................................. 19
2.7.4 Irrational Numbers............................... 19
2.7.5 Real Numbers..................................... 20
ix
x Contents
2.7.6 Algebraic and Transcendental Numbers............... 20
2.7.7 Imaginary Numbers.................................. 20
2.7.8 Complex Numbers.................................... 24
2.7.9 Quaternions and Octonions.......................... 27
2.8 Prime Numbers................................................. 29
2.8.1 The Fundamental Theorem of Arithmetic.............. 30
2.8.2 Is 1 a Prime?...................................... 30
2.8.3 Prime Number Distribution.......................... 31
2.8.4 Perfect Numbers.................................... 32
2.8.5 Mersenne Numbers................................... 32
2.8.6 Transcendental and Algebraic Numbers............... 32
2.8.7 Infinity........................................... 33
2.9 Worked Examples............................................... 34
2.9.1 Algebraic Expansion................................ 34
2.9.2 Binary Subtraction................................. 35
2.9.3 Complex Numbers.................................... 35
2.9.4 Complex Rotation................................... 35
2.9.5 Quaternions........................................ 36
3 Algebra.............................................................. 37
3.1 Introduction.................................................. 37
3.2 Background.................................................... 38
3.2.1 Solving the Roots of a Quadratic Equation.......... 39
3.3 Indices....................................................... 43
3.3.1 Laws of Indices.................................... 44
3.4 Logarithms.................................................... 44
3.5 Further Notation.............................................. 46
3.6 Functions..................................................... 46
3.6.1 Explicit and Implicit Equations.................... 47
3.6.2 Function Notation.................................. 47
3.6.3 Intervals.......................................... 48
3.6.4 Function Domains and Ranges........................ 49
3.6.5 Odd and Even Functions............................. 50
3.6.6 Power Functions.................................... 52
3.7 Worked Examples............................................... 52
3.7.1 Algebraic Manipulation............................. 52
3.7.2 Solving a Quadratic Equation....................... 53
3.7.3 Factorising........................................ 55
4 Logic................................................................ 57
4.1 Introduction.................................................. 57
4.2 Truth Tables.................................................. 57
4.2.1 Logical Connectives................................ 58
Contents
XI
4.3 Logical Premises.............................................. 59
4.3.1 Material Equivalence................................ 59
4.3.2 Implication......................................... 60
4.3.3 Negation............................................ 60
4.3.4 Conjunction......................................... 60
4.3.5 Inclusive Disjunction............................... 61
4.3.6 Exclusive Disjunction............................... 61
4.3.7 Idempotence......................................... 62
4.3.8 Commutativity....................................... 62
4.3.9 Associativity....................................... 63
4.3.10 Distributivity...................................... 64
4.3.11 de Morgan’s Laws.................................... 64
4.3.12 Simplification...................................... 65
4.3.13 Excluded Middle..................................... 66
4.3.14 Contradiction....................................... 66
4.3.15 Double Negation..................................... 67
4.3.16 Implication and Equivalence......................... 67
4.3.17 Exportation......................................... 68
4.3.18 Contrapositive...................................... 68
4.3.19 Reductio Ad Absurdum................................ 68
4.3.20 Modus Ponens........................................ 69
4.3.21 Proof by Cases...................................... 70
4.4 Set Theory.................................................... 71
4.4.1 Empty Set........................................... 72
4.4.2 Membership and Cardinality of a Set................. 72
4.4.3 Subsets, Supersets and the Universal Set............ 73
4.4.4 Set Building........................................ 74
4.4.5 Union............................................... 74
4.4.6 Intersection........................................ 75
4.4.7 Relative Complement................................. 75
4.4.8 Absolute Complement................................. 76
4.4.9 Power Set........................................... 77
4.5 Worked Examples............................................... 77
4.5.1 Truth Tables........................................ 77
4.5.2 Set Building........................................ 78
4.5.3 Sets................................................ 79
4.5.4 Power Set........................................... 79
5 Trigonometry...................................................... 81
5.1 Introduction.................................................. 81
5.2 Background................................................ 81
5.3 Units of Angular Measurement................................ 81
5.4 The Trigonometric Ratios...................................... 82
5.4.1 Domains and Ranges.................................. 85
Contents
xii
5.5 Inverse Trigonometric Ratios................................. 85
5.6 Trigonometric Identities..................................... 87
5.7 The Sine Rule................................................ 88
5.8 The Cosine Rule.............................................. 89
5.9 Compound-Angle Identities.................................... 89
5.9.1 Double-Angle Identities............................ 91
5.9.2 Multiple-Angle Identities.......................... 91
5.9.3 Half-Angle Identities.............................. 92
5.10 Perimeter Relationships...................................... 93
6 Coordinate Systems............................................ 95
6.1 Introduction................................................. 95
6.2 Background................................................... 95
6.3 The Cartesian Plane.......................................... 96
6.4 Function Graphs.............................................. 96
6.5 Shape Representation......................................... 97
6.5.1 2D Polygons........................................ 97
6.5.2 Areas of Shapes.................................... 98
6.6 Theorem of Pythagoras in 2D.................................. 99
6.7 3D Cartesian Coordinates..................................... 99
6.7.1 Theorem of Pythagoras in 3D....................... 100
6.8 Polar Coordinates........................................... 101
6.9 Spherical Polar Coordinates................................. 102
6.10 Cylindrical Coordinates..................................... 103
6.11 Barycentric Coordinates..................................... 104
6.12 Homogeneous Coordinates..................................... 104
6.13 Worked Examples............................................. 105
6.13.1 Area of a Shape................................... 105
6.13.2 Distance Between Two Points....................... 105
6.13.3 Polar Coordinates................................. 105
6.13.4 Spherical Polar Coordinates....................... 106
6.13.5 Cylindrical Coordinates........................... 107
6.13.6 Barycentric Coordinates........................... 107
7 Determinants.................................................... 109
7.1 Introduction................................................ 109
7.2 Linear Equations with Two Variables......................... 110
7.3 Linear Equations with Three Variables....................... 113
7.3.1 Sarrus’s Rule..................................... 120
7.4 Mathematical Notation....................................... 120
7.4.1 Matrix............................................ 120
7.4.2 Order of a Determinant............................ 121
7.4.3 Value of a Determinant............................ 121
7.4.4 Properties of Determinants........................ 122
Contents
Xlll
7 5 Worked Examples............................................ 123
7.5.1 Determinant Expansion............................ 123
7.5.2 Complex Determinant.............................. 124
7.5.3 Simple Expansion................................. 124
7.5.4 Simultaneous Equations........................... 125
8 Vectors.......................................................... 127
8.1 Introduction............................................... 127
8.2 2D Vectors................................................. 128
8.2.1 Vector Notation.................................. 128
8.2.2 Graphical Representation of Vectors.............. 129
8.2.3 Magnitude of a Vector............................ 130
8.3 3D Vectors................................................. 131
8.3.1 Vector Manipulation.............................. 132
8.3.2 Scaling a Vector................................. 132
8.3.3 Vector Addition and Subtraction.................. 132
8.3.4 Position Vectors................................. 133
8.3.5 Unit Vectors..................................... 134
8.3.6 Cartesian Vectors................................ 135
8.3.7 Products......................................... 135
8.3.8 Scalar Product................................... 136
8.3.9 The Vector Product............................... 137
8.3.10 The Right-Hand Rule.............................. 142
8.4 Deriving a Unit Normal Vector for a Triangle............... 142
8.5 Surface Areas.............................................. 143
8.5.1 Calculating 2D Areas............................. 144
8.6 Worked Examples............................................ 145
8.6.1 Position Vector.................................. 145
8.6.2 Unit Vector...................................... 145
8.6.3 Vector Magnitude................................. 145
8.6.4 Angle Between Two Vectors........................ 146
8.6.5 Vector Product................................... 146
9 Matrices......................................................... 149
9.1 Introduction............................................... 149
9.2 Geometric Transforms...................................... 149
9.3 Transforms and Matrices................................. 151
9.4 Matrix Notation............................................ 154
9.4.1 Matrix Dimension or Order...................... 154
9.4.2 Square Matrix................................... 154
9.4.3 Column Vector................................... 155
9.4.4 Row Vector..................................... 155
9.4.5 Null Matrix................................... 155
9.4.6 Unit Matrix..................................... 155
XIV
Contents
9.4.7 Trace............................................. 156
9.4.8 Determinant of a Matrix........................... 157
9.4.9 Transpose......................................... 157
9.4.10 Symmetric Matrix.................................. 159
9.4.11 Antisymmetric Matrix.............................. 160
9.5 Matrix Addition and Subtraction.............................. 162
9.5.1 Scalar Multiplication............................. 163
9.6 Matrix Products.............................................. 163
9.6.1 Row and Column Vectors............................ 163
9.6.2 Row Vector and a Matrix .......................... 164
9.6.3 Matrix and a Column Vector........................ 165
9.6.4 Square Matrices................................... 166
9.6.5 Rectangular Matrices.............................. 167
9.7 Inverse Matrix............................................... 167
9.7.1 Inverting a Pair of Matrices...................... 174
9.8 Orthogonal Matrix............................................ 175
9.9 Diagonal Matrix.............................................. 176
9.10 Worked Examples.............................................. 177
9.10.1 Matrix Inversion.................................. 177
9.10.2 Identity Matrix................................... 177
9.10.3 Solving Two Equations Using Matrices.............. 178
9.10.4 Solving Three Equations Using Matrices............ 179
9.10.5 Solving Two Complex Equations..................... 180
9.10.6 Solving Three Complex Equations................... 181
9.10.7 Solving Two Complex Equations..................... 182
9.10.8 Solving Three Complex Equations................... 182
10 Geometric Matrix Transforms....................................... 185
10.1 Introduction................................................. 185
10.2 Matrix Transforms............................................ 185
10.2.1 2D Translation................................... 186
10.2.2 2D Scaling....................................... 187
10.2.3 2D Reflections................................... 189
10.2.4 2D Shearing...................................... 190
10.2.5 2D Rotation...................................... 191
10.2.6 2D Scaling....................................... 194
10.2.7 2D Reflection.................................... 195
10.2.8 2D Rotation About an Arbitrary Point............. 195
10.3 3D Transforms................................................ 196
10.3.1 3D Translation .................................. 196
10.3.2 3D Scaling....................................... 197
10.3.3 3D Rotation...................................... 197
10.3.4 Rotating About an Axis............................ 201
10.3.5 3D Reflections................................... 202
Contents
XV
10.4 Rotating a Point About an Arbitrary Axis......................... 202
10.4.1 Matrices.............................................. 202
10.5 Determinant of a Transform....................................... 206
10.6 Perspective Projection........................................... 207
10.7 Worked Examples.................................................. 209
10.7.1 2D Scale and Translate................................ 209
10.7.2 2D Rotation........................................... 210
10.7.3 Determinant of the Rotate Transform................... 211
10.7.4 Determinant of the Shear Transform.................... 211
10.7.5 Yaw, Pitch and Roll Transforms........................ 211
10.7.6 Rotation About an Arbitrary Axis...................... 212
10.7.7 3D Rotation Transform Matrix.......................... 213
10.7.8 Perspective Projection................................ 213
11 Calculus: Derivatives................................................ 215
11.1 Introduction..................................................... 215
11.2 Small Numerical Quantities....................................... 215
11.3 Equations and Limits............................................. 217
11.3.1 Quadratic Function.................................... 217
11.3.2 Cubic Equation........................................ 218
11.3.3 Functions and Limits.................................. 220
11.3.4 Graphical Interpretation of the Derivative............ 222
11.3.5 Derivatives and Differentials......................... 223
11.3.6 Integration and Antiderivatives....................... 224
11.4 Function Types................................................. 225
11.5 Differentiating Groups of Functions.............................. 226
11.5.1 Sums of Functions..................................... 226
11.5.2 Function of a Function................................ 228
11.5.3 Function Products..................................... 232
11.5.4 Function Quotients.................................... 235
11.6 Differentiating Implicit Functions............................... 237
11.7 Differentiating Exponential and Logarithmic Functions......... 240
11.7.1 Exponential Functions................................. 240
11.7.2 Logarithmic Functions................................. 242
11.8 Differentiating Trigonometric Functions.......................... 244
11.8.1 Differentiating Tan................................... 244
11.8.2 Differentiating Csc................................... 245
11.8.3 Differentiating Sec................................... 246
11.8.4 Differentiating Cot................................... 247
11.8.5 Differentiating Arcsin, Arccos and Arctan............. 248
11.8.6 Differentiating Arccsc, Arcsec and Arccot............. 249
11.9 Differentiating Hyperbohc Functions.............................. 249
11.9.1 Differentiating Sinh, Cosh and Tanh................... 251
XVI
Contents
11.10 Higher Derivatives............................................. 252
11.11 Higher Derivatives of a Polynomial............................. 252
11.12 Identifying a Local Maximum or Minimum......................... 255
11.13 Partial Derivatives............................................ 256
11.13.1 Visualising Partial Derivatives..................... 259
11.13.2 Mixed Partial Derivatives........................... 262
11.14 Chain Rule..................................................... 263
11.15 Total Derivative............................................... 265
11.16 Power Series................................................... 266
11.17 Worked Examples................................................ 271
11.17.1 Antiderivative 1.................................... 271
11.17.2 Antiderivative 2.................................... 272
11.17.3 Differentiating Sums of Functions................... 272
11.17.4 Differentiating a Function Product.................. 272
11.17.5 Differentiating an Implicit Function................ 273
11.17.6 Differentiating a General Implicit Function......... 273
11.17.7 Local Maximum or Minimum............................ 274
11.17.8 Partial Derivatives................................. 275
11.17.9 Mixed Partial Derivative 1.......................... 275
11.17.10 Mixed Partial Derivative 2.......................... 276
11.17.11 Total Derivative.................................... 276
12 Calculus: Integration................................................ 277
12.1 Introduction................................................... 277
12.2 Indefinite Integral............................................ 277
12.3 Integration Techniques......................................... 278
12.3.1 Continuous Functions................................ 278
12.3.2 Difficult Functions................................. 279
12.4 Trigonometric Identities....................................... 280
12.4.1 Exponent Notation................................... 282
12.4.2 Completing the Square............................... 283
12.4.3 The Integrand Contains a Derivative................. 284
12.4.4 Converting the Integrand into a Series
of Fractions........................................ 287
12.4.5 Integration by Parts................................ 287
12.4.6 Integration by Substitution......................... 292
12.4.7 Partial Fractions................................... 294
12.5 Area Under a Graph............................................. 297
12.6 Calculating Areas.............................................. 297
12.7 Positive and Negative Areas.................................... 305
12.8 Area Between Two Functions..................................... 307
12.9 Areas with the Y-Axis.......................................... 309
12.10 Area with Parametric Functions................................. 310
12.11 The Riemann Sum.............................................. 311
Contents
XVII
12.12 Worked Examples.................................................. 313
12.12.1 Integrating a Function Containing Its Own
Derivative........................................... 313
12.12.2 Dividing an Integral into Several Integrals....... 314
12.12.3 Integrating by Parts 1............................... 315
12.12.4 Integrating by Parts 2............................... 316
12.12.5 Integrating by Substitution 1........................ 317
12.12.6 Integrating by Substitution 2........................ 318
12.12.7 Integrating by Substitution 3........................ 319
12.12.8 Integrating with Partial Fractions................... 319
Appendix A: Limit of (Sin O)/0............................................ 323
Appendix B: Integrating Cos n0............................................ 327
Index
329
|
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| id | DE-604.BV042926344 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:12:59Z |
| institution | BVB |
| isbn | 9783319214368 |
| language | English |
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| owner_facet | DE-739 DE-11 |
| physical | xvii, 334 Seiten Illustrationen, Diagramme |
| publishDate | 2015 |
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| publisher | Springer |
| record_format | marc |
| spelling | Vince, John 1941- Verfasser (DE-588)120106604 aut Foundation mathematics for computer science a visual approach John Vince Cham Springer [2015] © 2015 xvii, 334 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Computergrafik (DE-588)4010450-3 gnd rswk-swf Computermathematik (DE-588)4788218-9 gnd rswk-swf Informatik (DE-588)4026894-9 gnd rswk-swf Informatik (DE-588)4026894-9 s Computergrafik (DE-588)4010450-3 s Computermathematik (DE-588)4788218-9 s DE-604 Erscheint auch als 978-3-319-21437-5 Online-Ausgabe (DE-604)BV042730177 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028353526&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vince, John 1941- Foundation mathematics for computer science a visual approach Computergrafik (DE-588)4010450-3 gnd Computermathematik (DE-588)4788218-9 gnd Informatik (DE-588)4026894-9 gnd |
| subject_GND | (DE-588)4010450-3 (DE-588)4788218-9 (DE-588)4026894-9 |
| title | Foundation mathematics for computer science a visual approach |
| title_auth | Foundation mathematics for computer science a visual approach |
| title_exact_search | Foundation mathematics for computer science a visual approach |
| title_full | Foundation mathematics for computer science a visual approach John Vince |
| title_fullStr | Foundation mathematics for computer science a visual approach John Vince |
| title_full_unstemmed | Foundation mathematics for computer science a visual approach John Vince |
| title_short | Foundation mathematics for computer science |
| title_sort | foundation mathematics for computer science a visual approach |
| title_sub | a visual approach |
| topic | Computergrafik (DE-588)4010450-3 gnd Computermathematik (DE-588)4788218-9 gnd Informatik (DE-588)4026894-9 gnd |
| topic_facet | Computergrafik Computermathematik Informatik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028353526&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT vincejohn foundationmathematicsforcomputerscienceavisualapproach |