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Mathematics for Enzyme Reaction Kinetics and Reactor Performance

Mathematics for Enzyme Reaction Kinetics and Reactor Performance

F. Xavier Malcata

ISBN: 978-1-119-49028-9

Jan 2019

1074 pages

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$350.00

Description

Mathematics for Enzyme Reaction Kinetics and Reactor Performance is the first set in a unique 11 volume-collection on Enzyme Reactor Engineering. This two volume-set relates specifically to the wide mathematical background required for systematic and rational simulation of both reaction kinetics and reactor performance; and to fully understand and capitalize on the modelling concepts developed. It accordingly reviews basic and useful concepts of Algebra (first volume), and Calculus and Statistics (second volume). A brief overview of such native algebraic entities as scalars, vectors, matrices and determinants constitutes the starting point of the first volume; the major features of germane functions are then addressed. Vector operations ensue, followed by calculation of determinants. Finally, exact methods for solution of selected algebraic equations – including sets of linear equations, are considered, as well as numerical methods for utilization at large.

The second volume begins with an introduction to basic concepts in calculus, i.e. limits, derivatives, integrals and differential equations; limits, along with continuity, are further expanded afterwards, covering uni- and multivariate cases, as well as classical theorems. After recovering the concept of differential and applying it to generate (regular and partial) derivatives, the most important rules of differentiation of functions, in explicit, implicit and parametric form, are retrieved – together with the nuclear theorems supporting simpler manipulation thereof. The book then tackles strategies to optimize uni- and multivariate functions, before addressing integrals in both indefinite and definite forms. Next, the book touches on the methods of solution of differential equations for practical applications, followed by analytical geometry and vector calculus. Brief coverage of statistics–including continuous probability functions, statistical descriptors and statistical hypothesis testing, brings the second volume to a close.

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Series Preface

Preface

Volume One

1. Basic Concepts of Algebra 2

17.1. Scalars, vectors, matrices and determinants 3

17.2. Function features 8

17.2.1. Series 20

17.2.1.1. Arithmetic series 21

17.2.1.2. Geometric series 24

17.2.1.3. Arithmetic/geometric series 28

17.2.2. Multiplication and division of polynomials 34

17.2.2.1. Product 35

17.2.2.2. Quotient 36

17.2.2.3. Factorization 41

17.2.2.4. Splitting 46

17.2.2.5. Power 57

17.2.3. Trigonometric functions 66

17.2.3.1. Definition and major features 67

17.2.3.2. Angle transformation formulae 73

17.2.3.3. Fundamental theorem of trigonometry 91

17.2.3.4. Inverse functions 99

17.2.4. Hyperbolic functions 100

17.2.4.1. Definition and major features 101

17.2.4.2. Argument transformation formulae 107

17.2.4.3. Euler’s form of complex numbers 112

17.2.4.4. Inverse functions 114

17.3. Vector operations 120

17.3.1. Addition of vectors 123

17.3.2. Multiplication of scalar by vector 125

17.3.3. Scalar multiplication of vectors 128

17.3.4. Vector multiplication of vectors 140

17.4. Matrix operations 147

17.4.1. Addition of matrices 148

17.4.2. Multiplication of scalar by matrix 150

17.4.3. Multiplication of matrices 153

17.4.4. Transposal of matrices 162

17.4.5. Inversion of matrices 165

17.4.5.1. Full matrix 166

17.4.5.2. Block matrix 172

17.4.6. Combined features 175

17.4.6.1. Symmetric matrix 176

17.4.6.2. Positive semidefinite matrix 178

17.5. Tensor operations 181

17.6. Determinants 187

17.6.1. Definition 188

17.6.2. Calculation 195

17.6.2.1. Laplace’s theorem 197

17.6.2.2. Major features 201

17.6.2.3. Tridiagonal matrix 221

17.6.2.4. Block matrix 223

17.6.2.5. Matrix inversion 228

17.6.3. Eigenvalues and -vectors 233

17.6.3.1. Characteristic polynomial 235

17.6.3.2. Cayley & Hamilton’s theorem 239

17.7. Solution of algebraic equations250

17.7.1. Linear systems of equations 251

17.7.1.1. Jacobi’s method 256

17.7.1.2. Explicitation 268

17.7.1.3. Cramer’s rule 269

17.7.1.4. Matrix inversion 273

17.7.2. Quadratic equation 278

17.7.3. Lambert’s W function 283

17.7.4. Numerical approaches 288

 17.7.4.1. Double-initial estimate methods 289

 17.7.4.2. Single-initial estimate methods 307

17.8. Further reading 326

Volume Two

1. Basic Concepts of Calculus 1

18.1. Limits, derivatives, integrals and differential equations 2

18.2. Limits and continuity 3

18.2.1. Univariate limit 4

18.2.1.1. Definition 4

18.2.1.2. Basic calculation 9

18.2.2. Multivariate limit 14

18.2.3. Basic theorems on limits 16

18.2.4. Definition of continuity 26

18.2.5. Basic theorems on continuity 29

18.2.5.1. Bolzano’s theorem 30

18.2.5.2. Weierstrass’ theorem 35

18.3. Differentials, derivatives and partial derivatives 38

18.3.1. Differential 39

18.3.2. Derivative 43

18.3.2.1. Definition 43

18.3.2.2. Rules of differentiation of univariate functions 63

18.3.2.3. Rules of differentiation of multivariate functions 85

18.3.2.4. Implicit differentiation 86

18.3.2.5. Parametric differentiation 89

18.3.2.6. Basic theorems of differential calculus 93

18.3.2.7. Derivative of matrix 116

18.3.2.8. Derivative of determinant 123

18.3.3. Dependence between functions 127

18.3.4. Optimization of univariate continuous functions 131

18.3.4.1. Constraint-free 132

18.3.4.2. Subjected to constraints 135

18.3.5. Optimization of multivariate continuous functions 139

18.3.5.1. Constraint-free 139

18.3.5.2. Subjected to constraints 145

18.4. Integrals 146

18.4.1. Univariate integral 149

18.4.1.1. Indefinite integral 149

18.4.1.2. Definite integral 165

18.4.2. Multivariate integral 185

18.4.2.1. Definition 185

18.4.2.2. Basic theorems 191

18.4.2.3. Change of variables 200

18.4.2.4. Differentiation of integral 204

18.4.3. Optimization of single integral 207

18.4.4. Optimization of set of derivatives 217

18.5. Infinite series and integrals 222

18.5.1. Definition and criteria of convergence 223

18.5.1.1. Comparison test 225

18.5.1.2. Ratio test 226

18.5.1.3. D’Alembert’s test 228

18.5.1.4. Cauchy’s integral test 231

18.5.1.5. Leibnitz’s test 234

18.5.2. Taylor’s series 237

18.5.2.1. Analytical functions 255

18.5.2.2. Euler’s infinite product 293

18.5.3. Gamma function and factorial 304

18.5.3.1. Integral definition and major features 305

18.5.3.2. Euler’s definition 312

18.5.3.3. Stirling’s approximation 319

18.6. Analytical geometry 325

18.6.1. Straight line 326

18.6.2. Simple polygons 329

18.6.3. Conical curves 333

18.6.4. Length of line 340

18.6.5. Curvature of line 353

18.6.6. Area of plane surface 359

18.6.7. Outer area of revolution solid 367

18.6.8. Volume of revolution solid 387

18.7. Transforms 395

18.7.1. Laplace’s transform 396

18.7.1.1. Definition 396

18.7.1.2. Major features 411

18.7.1.3. Inversion 427

18.7.2. Legendre’s transform 438

18.8. Solution of differential equations 447

18.8.1. Ordinary differential equations 448

18.8.1.1. Single first order 449

18.8.1.2. Single second order 456

18.8.1.3. Linear higher order 522

18.8.2. Partial differential equations 535

18.9. Vector calculus 543

18.9.1. Rectangular coordinates 544

18.9.1.1. Definition and representation 544

18.9.1.2. Definition of nabla operator, Ñ 546

18.9.1.3. Algebraic properties of Ñ 553

18.9.1.4. Multiple products involving Ñ 556

18.9.2. Cylindrical coordinates 588

18.9.2.1. Definition and representation 588

18.9.1.2. Redefinition of nabla operator, Ñ 594

18.9.3. Spherical coordinates 601

18.9.3.1. Definition and representation 601

18.9.3.2. Redefinition of nabla operator, Ñ 615

18.9.4. Curvature of three-dimensional surface 636

18.9.5. Three-dimensional integration 646

18.10. Numerical approaches to integration 650

18.10.1. Calculation of definite integrals 651

18.10.1.1. Zero-th order interpolation 653

18.10.1.2. First and second order interpolation 664

18.10.1.3. Composite methods 693

18.10.1.4. Infinite and multidimensional integrals 699

18.10.2. Integration of differential equations 702

18.10.2.1. Single-step methods 705

18.10.2.2. Multiple-step methods 709

18.10.2.3. Multiple-stage methods 718

18.10.2.4. Integral versus differential equation 734

18.11. Further reading

3. Basic Concepts of Statistics 1

19.1. Continuous probability functions2

19.1.1. Basic statistical descriptors 4

19.1.2. Normal distribution 12

19.1.2.1. Derivation 13

19.1.2.2. Justification 20

19.1.2.3. Operational features 27

19.1.2.4. Moment-generating function 31

19.1.2.5. Standard probability density function 48

19.1.2.6. Central limit theorem 52

19.1.2.7. Standard probability cumulative function 64

19.1.3. Other relevant distributions 68

19.1.3.1. Lognormal distribution 69

19.1.3.2. Chi-square distribution 78

19.1.3.3. Student’s t-distribution 94

19.1.3.4. Fisher’s F-distribution 112

19.2. Statistical hypothesis testing 147

19.3. Linear regression 156

19.3.1. Parameter fitting 158

19.3.2. Residual characterization 162

19.3.3. Parameter inference 166

19.3.3.1. Multivariate models 166

19.3.3.2. Univariate models 171

19.3.4. Unbiased estimation 175

19.3.4.1. Multivariate models 175

19.3.4.2. Univariate models 179

19.3.5. Prediction inference 189

19.3.6. Multivariate correction 192

19.4. Further reading 207