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Chapter 3 Cartesian Coordinate System

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Chapter 3 Cartesian Coordinate System Physical constant From Wikipedia, the free encyclopedia Contents 1 Amplitude 1 1.1 Definitions of the term ......................................... 1 1.1.1 Peak-to-peak amplitude .................................... 1 1.1.2 Peak amplitude ........................................ 1 1.1.3 Semi-amplitude ........................................ 1 1.1.4 Root mean square amplitude ................................. 1 1.1.5 Ambiguity ........................................... 2 1.1.6 Pulse amplitude ........................................ 3 1.2 Formal representation ......................................... 3 1.3 Units .................................................. 3 1.4 Waveform and envelope ........................................ 3 1.5 Sinusoids ............................................... 4 1.6 See also ................................................ 4 1.7 Notes ................................................. 4 2 Arithmetic progression 5 2.1 Sum .................................................. 5 2.1.1 Derivation .......................................... 6 2.2 Product ................................................ 6 2.3 Standard deviation ........................................... 7 2.4 Intersections .............................................. 7 2.5 Formulas at a Glance ......................................... 7 2.6 See also ................................................ 8 2.7 References .............................................. 8 2.8 External links ............................................. 8 3 Cartesian coordinate system 9 3.1 History ................................................. 11 3.2 Description .............................................. 11 3.2.1 One dimension ........................................ 11 3.2.2 Two dimensions ........................................ 11 3.2.3 Three dimensions ....................................... 12 3.2.4 Higher dimensions ...................................... 13 i ii CONTENTS 3.2.5 Generalizations ........................................ 13 3.3 Notations and conventions ....................................... 14 3.3.1 Quadrants and octants ..................................... 14 3.4 Cartesian formulae for the plane .................................... 14 3.4.1 Distance between two points ................................. 14 3.4.2 Euclidean transformations ................................... 15 3.5 Orientation and handedness ...................................... 18 3.5.1 In two dimensions ....................................... 18 3.5.2 In three dimensions ...................................... 19 3.6 Representing a vector in the standard basis .............................. 21 3.7 Applications .............................................. 21 3.8 See also ................................................ 22 3.9 Notes ................................................. 22 3.10 References ............................................... 22 3.11 Sources ................................................ 22 3.12 Further reading ............................................ 23 3.13 External links ............................................. 23 4 Diophantine equation 24 4.1 Examples ............................................... 25 4.2 Linear Diophantine equations ..................................... 25 4.2.1 One equation ......................................... 25 4.2.2 Chinese remainder theorem .................................. 25 4.2.3 System of linear Diophantine equations ............................ 25 4.3 Diophantine analysis .......................................... 26 4.3.1 Typical questions ....................................... 26 4.3.2 Typical problem ....................................... 27 4.3.3 17th and 18th centuries .................................... 27 4.3.4 Hilbert’s tenth problem .................................... 27 4.3.5 Diophantine geometry .................................... 27 4.3.6 Modern research ....................................... 27 4.3.7 Infinite Diophantine equations ................................ 28 4.4 Exponential Diophantine equations .................................. 28 4.5 Notes ................................................. 28 4.6 References ............................................... 29 4.7 Further reading ............................................ 29 4.8 External links ............................................. 29 5 Equation 30 5.1 Introduction .............................................. 30 5.1.1 Parameters and unknowns .................................. 30 5.1.2 Analogous illustration ..................................... 31 CONTENTS iii 5.1.3 Identities ........................................... 32 5.2 Properties ............................................... 33 5.3 Algebra ................................................ 34 5.3.1 Polynomial equations ..................................... 34 5.3.2 Systems of linear equations .................................. 34 5.4 Geometry ............................................... 36 5.4.1 Analytic geometry ...................................... 36 5.4.2 Cartesian equations ...................................... 36 5.4.3 Parametric equations ..................................... 37 5.5 Number theory ............................................ 38 5.5.1 Diophantine equations .................................... 38 5.5.2 Algebraic and transcendental numbers ............................ 38 5.5.3 Algebraic geometry ...................................... 38 5.6 Differential equations ......................................... 38 5.6.1 Ordinary differential equations ................................ 39 5.6.2 Partial differential equations ................................. 39 5.7 Types of equations .......................................... 39 5.8 See also ................................................ 40 5.9 References ............................................... 40 5.10 External links ............................................. 41 6 Geometric progression 42 6.1 Elementary properties ......................................... 43 6.2 Geometric series ............................................ 44 6.2.1 Derivation ........................................... 44 6.2.2 Related formulas ....................................... 44 6.2.3 Infinite geometric series .................................... 45 6.2.4 Complex numbers ....................................... 47 6.3 Product ................................................ 47 6.4 Relationship to geometry and Euclid’s work .............................. 48 6.5 See also ................................................ 48 6.6 References ............................................... 49 6.7 External links ............................................. 49 7 Geometric series 50 7.1 Common ratio ............................................. 51 7.2 Sum .................................................. 52 7.2.1 Example ............................................ 52 7.2.2 Formula ............................................ 53 7.2.3 Proof of convergence ..................................... 53 7.2.4 Generalized formula ..................................... 54 7.3 Applications .............................................. 55 iv CONTENTS 7.3.1 Repeating decimals ...................................... 55 7.3.2 Archimedes’ quadrature of the parabola ........................... 55 7.3.3 Fractal geometry ....................................... 56 7.3.4 Zeno’s paradoxes ....................................... 57 7.3.5 Euclid ............................................. 58 7.3.6 Economics .......................................... 58 7.3.7 Geometric power series .................................... 58 7.4 See also ................................................ 59 7.4.1 Specific geometric series ................................... 59 7.5 References ............................................... 60 7.5.1 History and philosophy .................................... 60 7.5.2 Economics .......................................... 60 7.5.3 Biology ............................................ 61 7.5.4 Computer science ....................................... 61 7.6 External links ............................................. 61 8 Physical constant 62 8.1 Dimensional and dimensionless physical constants .......................... 62 8.2 How constant are the physical constants? ............................... 63 8.3 Anthropic principle .......................................... 63 8.4 Table of universal constants ...................................... 64 8.5 Table of electromagnetic constants .................................. 64 8.6 Table of atomic and nuclear constants ................................. 64 8.7 Table of physico-chemical constants .................................. 64 8.8 Table of adopted values ........................................ 64 8.9 Natural units .............................................. 64 8.10 See also ................................................ 64 8.11 References ............................................... 64 8.12 External links ............................................. 65 9 Scale factor 66 9.1 See also ................................................ 66 10 Sequence 67 10.1 Examples and notation ........................................ 68 10.1.1 Important examples ...................................... 68 10.1.2 Indexing ............................................ 69 10.1.3 Specifying a sequence by recursion ............................. 70 10.2 Formal definition and basic
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