Mathematics for Enzyme Reaction Kinetics and Reactor Performance

 
 
Standards Information Network (Verlag)
  • 1. Auflage
  • |
  • erschienen am 30. April 2020
  • |
  • 1072 Seiten
 
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978-1-119-49032-6 (ISBN)
 
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.
1. Auflage
  • Englisch
  • USA
John Wiley & Sons Inc
  • Für Beruf und Forschung
  • 29,22 MB
978-1-119-49032-6 (9781119490326)
weitere Ausgaben werden ermittelt
F. Xavier Malcata, PhD, is Full Professor at in the Department of Chemical Engineering at the University of Porto in Portugal, and Researcher at LEPABE - Laboratory for Process Engineering, Environment, Biotechnology and Energy. He is the author of more than 400 highly cited journal papers, eleven books, four edited books, and fifty chapters in edited books. He has been awarded the Elmer Marth Educator Award by the International Association of Food Protection (USA), and the William V. Cruess Award for excellence in teaching by the Institute of Food Technologists (USA).
About the Author xv

Series Preface xix

Preface xxiii

Volume 1

Part 1 Basic Concepts of Algebra 1

1 Scalars, Vectors, Matrices, and Determinants 3

2 Function Features 7

2.1 Series 17

2.1.1 Arithmetic Series 17

2.1.2 Geometric Series 19

2.1.3 Arithmetic/Geometric Series 22

2.2 Multiplication and Division of Polynomials 26

2.2.1 Product 27

2.2.2 Quotient 28

2.2.3 Factorization 31

2.2.4 Splitting 35

2.2.5 Power 43

2.3 Trigonometric Functions 52

2.3.1 Definition and Major Features 52

2.3.2 Angle Transformation Formulae 57

2.3.3 Fundamental Theorem of Trigonometry 73

2.3.4 Inverse Functions 79

2.4 Hyperbolic Functions 80

2.4.1 Definition and Major Features 80

2.4.2 Argument Transformation Formulae 85

2.4.3 Euler's Form of Complex Numbers 89

2.4.4 Inverse Functions 90

3 Vector Operations 97

3.1 Addition of Vectors 99

3.2 Multiplication of Scalar by Vector 101

3.3 Scalar Multiplication of Vectors 103

3.4 Vector Multiplication of Vectors 111

4 Matrix Operations 119

4.1 Addition of Matrices 120

4.2 Multiplication of Scalar by Matrix 121

4.3 Multiplication of Matrices 124

4.4 Transposal of Matrices 131

4.5 Inversion of Matrices 133

4.5.1 Full Matrix 134

4.5.2 Block Matrix 138

4.6 Combined Features 140

4.6.1 Symmetric Matrix 141

4.6.2 Positive Semidefinite Matrix 142

5 Tensor Operations 145

6 Determinants 151

6.1 Definition 152

6.2 Calculation 157

6.2.1 Laplace's Theorem 159

6.2.2 Major Features 161

6.2.3 Tridiagonal Matrix 177

6.2.4 Block Matrix 179

6.2.5 Matrix Inversion 181

6.3 Eigenvalues and Eigenvectors 185

6.3.1 Characteristic Polynomial 186

6.3.2 Cayley-Hamilton's Theorem 190

7 Solution of Algebraic Equations 199

7.1 Linear Systems of Equations 199

7.1.1 Jacobi's Method 203

7.1.2 Explicitation 212

7.1.3 Cramer's Rule 213

7.1.4 Matrix Inversion 216

7.2 Quadratic Equation 220

7.3 Lambert's W Function 224

7.4 Numerical Approaches 228

7.4.1 Double-initial Estimate Methods 229

7.4.1.1 Bisection 229

7.4.1.2 Linear Interpolation 232

7.4.2 Single-initial Estimate Methods 242

7.4.2.1 Newton and Raphson's Method 242

7.4.2.2 Direct Iteration 250

Further Reading 255

Volume 2

Part 2 Basic Concepts of Calculus 259

8 Limits, Derivatives, Integrals, and Differential Equations 261

9 Limits and Continuity 263

9.1 Univariate Limit 263

9.1.1 Definition 263

9.1.2 Basic Calculation 267

9.2 Multivariate Limit 271

9.3 Basic Theorems on Limits 272

9.4 Definition of Continuity 280

9.5 Basic Theorems on Continuity 282

9.5.1 Bolzano's Theorem 282

9.5.2 Weierstrass' Theorem 286

10 Differentials, Derivatives, and Partial Derivatives 291

10.1 Differential 291

10.2 Derivative 294

10.2.1 Definition 294

10.2.1.1 Total Derivative 295

10.2.1.2 Partial Derivatives 300

10.2.1.3 Directional Derivatives 307

10.2.2 Rules of Differentiation of Univariate Functions 308

10.2.3 Rules of Differentiation of Multivariate Functions 325

10.2.4 Implicit Differentiation 325

10.2.5 Parametric Differentiation 327

10.2.6 Basic Theorems of Differential Calculus 331

10.2.6.1 Rolle's Theorem 331

10.2.6.2 Lagrange's Theorem 332

10.2.6.3 Cauchy's Theorem 334

10.2.6.4 L'Hopital's Rule 337

10.2.7 Derivative of Matrix 349

10.2.8 Derivative of Determinant 356

10.3 Dependence Between Functions 358

10.4 Optimization of Univariate Continuous Functions 362

10.4.1 Constraint-free 362

10.4.2 Subjected to Constraints 364

10.5 Optimization of Multivariate Continuous Functions 367

10.5.1 Constraint-free 367

10.5.2 Subjected to Constraints 371

11 Integrals 373

11.1 Univariate Integral 374

11.1.1 Indefinite Integral 374

11.1.1.1 Definition 374

11.1.1.2 Rules of Integration 377

11.1.2 Definite Integral 386

11.1.2.1 Definition 386

11.1.2.2 Basic Theorems of Integral Calculus 393

11.1.2.3 Reduction Formulae 396

11.2 Multivariate Integral 400

11.2.1 Definition 400

11.2.1.1 Line Integral 400

11.2.1.2 Double Integral 403

11.2.2 Basic Theorems 404

11.2.2.1 Fubini's Theorem 404

11.2.2.2 Green's Theorem 409

11.2.3 Change of Variables 411

11.2.4 Differentiation of Integral 414

11.3 Optimization of Single Integral 416

11.4 Optimization of Set of Derivatives 424

12 Infinite Series and Integrals 429

12.1 Definition and Criteria of Convergence 429

12.1.1 Comparison Test 430

12.1.2 Ratio Test 431

12.1.3 D'Alembert's Test 432

12.1.4 Cauchy's Integral Test 434

12.1.5 Leibnitz's Test 436

12.2 Taylor's Series 437

12.2.1 Analytical Functions 451

12.2.1.1 Exponential Function 451

12.2.1.2 Hyperbolic Functions 458

12.2.1.3 Logarithmic Function 459

12.2.1.4 Trigonometric Functions 463

12.2.1.5 Inverse Trigonometric Functions 466

12.2.1.6 Powers of Binomials 476

12.2.2 Euler's Infinite Product 479

12.3 Gamma Function and Factorial 488

12.3.1 Integral Definition and Major Features 489

12.3.2 Euler's Definition 494

12.3.3 Stirling's Approximation 499

13 Analytical Geometry 505

13.1 Straight Line 505

13.2 Simple Polygons 508

13.3 Conical Curves 510

13.4 Length of Line 516

13.5 Curvature of Line 525

13.6 Area of Plane Surface 530

13.7 Outer Area of Revolution Solid 536

13.8 Volume of Revolution Solid 552

14 Transforms 559

14.1 Laplace's Transform 559

14.1.1 Definition 559

14.1.2 Major Features 571

14.1.3 Inversion 583

14.2 Legendre's Transform 590

15 Solution of Differential Equations 597

15.1 Ordinary Differential Equations 597

15.1.1 First Order 598

15.1.1.1 Nonlinear 598

15.1.1.2 Linear 600

15.1.2 Second Order 602

15.1.2.1 Nonlinear 603

15.1.2.2 Linear 613

15.1.3 Linear Higher Order 650

15.2 Partial Differential Equations 660

16 Vector Calculus 667

16.1 Rectangular Coordinates 667

16.1.1 Definition and Representation 667

16.1.2 Definition of Nabla Operator, 668

16.1.3 Algebraic Properties of 673

16.1.4 Multiple Products Involving 676

16.1.4.1 Calculation of ( . ) 676

16.1.4.2 Calculation of ( . )u 676

16.1.4.3 Calculation of .( u) 677

16.1.4.4 Calculation of .( x u) 679

16.1.4.5 Calculation of .( ) 680

16.1.4.6 Calculation of .(uu) 682

16.1.4.7 Calculation of x ( ) 684

16.1.4.8 Calculation of ( .u) 685

16.1.4.9 Calculation of (u. )u 690

16.1.4.10 Calculation of .( .u) 693

16.2 Cylindrical Coordinates 695

16.2.1 Definition and Representation 695

16.2.2 Redefinition of Nabla Operator, 700

16.3 Spherical Coordinates 705

16.3.1 Definition and Representation 705

16.3.2 Redefinition of Nabla Operator, 715

16.4 Curvature of Three-dimensional Surfaces 729

16.5 Three-dimensional Integration 737

17 Numerical Approaches to Integration 741

17.1 Calculation of Definite Integrals 741

17.1.1 Zeroth Order Interpolation 743

17.1.2 First- and Second-Order Interpolation 750

17.1.2.1 Trapezoidal Rule 751

17.1.2.2 Simpson's Rule 754

17.1.2.3 Higher Order Interpolation 768

17.1.3 Composite Methods 771

17.1.4 Infinite and Multidimensional Integrals 775

17.2 Integration of Differential Equations 777

17.2.1 Single-step Methods 779

17.2.2 Multistep Methods 782

17.2.3 Multistage Methods 790

17.2.3.1 First Order 790

17.2.3.2 Second Order 790

17.2.3.3 General Order 793

17.2.4 Integral Versus Differential Equation 801

Part 3 Basic Concepts of Statistics 807

18 Continuous Probability Functions 809

18.1 Basic Statistical Descriptors 810

18.2 Normal Distribution 815

18.2.1 Derivation 816

18.2.2 Justification 821

18.2.3 Operational Features 826

18.2.4 Moment-generating Function 829

18.2.4.1 Single Variable 829

18.2.4.2 Multiple Variables 835

18.2.5 Standard Probability Density Function 842

18.2.6 Central Limit Theorem 845

18.2.7 Standard Probability Cumulative Function 855

18.3 Other Relevant Distributions 858

18.3.1 Lognormal Distribution 858

18.3.1.1 Probability Density Function 858

18.3.1.2 Mean and Variance 859

18.3.1.3 Probability Cumulative Function 862

18.3.1.4 Mode and Median 863

18.3.2 Chi-square Distribution 865

18.3.2.1 Probability Density Function 865

18.3.2.2 Mean and Variance 869

18.3.2.3 Asymptotic Behavior 870

18.3.2.4 Probability Cumulative Function 872

18.3.2.5 Mode and Median 873

18.3.2.6 Other Features 874

18.3.3 Student's t-distribution 876

18.3.3.1 Probability Density Function 876

18.3.3.2 Mean and Variance 879

18.3.3.3 Asymptotic Behavior 883

18.3.3.4 Probability Cumulative Function 886

18.3.3.5 Mode and Median 887

18.3.4 Fisher's F-distribution 888

18.3.4.1 Probability Density Function 888

18.3.4.2 Mean and Variance 893

18.3.4.3 Asymptotic Behavior 896

18.3.4.4 Probability Cumulative Function 899

18.3.4.5 Mode and Median 902

18.3.4.6 Other Features 903

19 Statistical Hypothesis Testing 915

20 Linear Regression 923

20.1 Parameter Fitting 924

20.2 Residual Characterization 927

20.3 Parameter Inference 931

20.3.1 Multivariate Models 931

20.3.2 Univariate Models 934

20.4 Unbiased Estimation 937

20.4.1 Multivariate Models 937

20.4.2 Univariate Models 940

20.5 Prediction Inference 949

20.6 Multivariate Correction 951

Further Reading 963

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