Mathematics for Enzyme Reaction Kinetics and Reactor Performance, 2 Volume Set

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 xvSeries Preface xixPreface xxiiiVolume 1Part 1 Basic Concepts of Algebra 11 Scalars, Vectors, Matrices, and Determinants 32 Function Features 72.1 Series 172.1.1 Arithmetic Series 172.1.2 Geometric Series 192.1.3 Arithmetic/Geometric Series 222.2 Multiplication and Division of Polynomials 262.2.1 Product 272.2.2 Quotient 282.2.3 Factorization 312.2.4 Splitting 352.2.5 Power 432.3 Trigonometric Functions 522.3.1 Definition and Major Features 522.3.2 Angle Transformation Formulae 572.3.3 Fundamental Theorem of Trigonometry 732.3.4 Inverse Functions 792.4 Hyperbolic Functions 802.4.1 Definition and Major Features 802.4.2 Argument Transformation Formulae 852.4.3 Euler’s Form of Complex Numbers 892.4.4 Inverse Functions 903 Vector Operations 973.1 Addition of Vectors 993.2 Multiplication of Scalar by Vector 1013.3 Scalar Multiplication of Vectors 1033.4 Vector Multiplication of Vectors 1114 Matrix Operations 1194.1 Addition of Matrices 1204.2 Multiplication of Scalar by Matrix 1214.3 Multiplication of Matrices 1244.4 Transposal of Matrices 1314.5 Inversion of Matrices 1334.5.1 Full Matrix 1344.5.2 Block Matrix 1384.6 Combined Features 1404.6.1 Symmetric Matrix 1414.6.2 Positive Semidefinite Matrix 1425 Tensor Operations 1456 Determinants 1516.1 Definition 1526.2 Calculation 1576.2.1 Laplace’s Theorem 1596.2.2 Major Features 1616.2.3 Tridiagonal Matrix 1776.2.4 Block Matrix 1796.2.5 Matrix Inversion 1816.3 Eigenvalues and Eigenvectors 1856.3.1 Characteristic Polynomial 1866.3.2 Cayley–Hamilton’s Theorem 1907 Solution of Algebraic Equations 1997.1 Linear Systems of Equations 1997.1.1 Jacobi’s Method 2037.1.2 Explicitation 2127.1.3 Cramer’s Rule 2137.1.4 Matrix Inversion 2167.2 Quadratic Equation 2207.3 Lambert’s W Function 2247.4 Numerical Approaches 2287.4.1 Double-initial Estimate Methods 2297.4.1.1 Bisection 2297.4.1.2 Linear Interpolation 2327.4.2 Single-initial Estimate Methods 2427.4.2.1 Newton and Raphson’s Method 2427.4.2.2 Direct Iteration 250Further Reading 255Volume 2Part 2 Basic Concepts of Calculus 2598 Limits, Derivatives, Integrals, and Differential Equations 2619 Limits and Continuity 2639.1 Univariate Limit 2639.1.1 Definition 2639.1.2 Basic Calculation 2679.2 Multivariate Limit 2719.3 Basic Theorems on Limits 2729.4 Definition of Continuity 2809.5 Basic Theorems on Continuity 2829.5.1 Bolzano’s Theorem 2829.5.2 Weierstrass’ Theorem 28610 Differentials, Derivatives, and Partial Derivatives 29110.1 Differential 29110.2 Derivative 29410.2.1 Definition 29410.2.1.1 Total Derivative 29510.2.1.2 Partial Derivatives 30010.2.1.3 Directional Derivatives 30710.2.2 Rules of Differentiation of Univariate Functions 30810.2.3 Rules of Differentiation of Multivariate Functions 32510.2.4 Implicit Differentiation 32510.2.5 Parametric Differentiation 32710.2.6 Basic Theorems of Differential Calculus 33110.2.6.1 Rolle’s Theorem 33110.2.6.2 Lagrange’s Theorem 33210.2.6.3 Cauchy’s Theorem 33410.2.6.4 L’Hôpital’s Rule 33710.2.7 Derivative of Matrix 34910.2.8 Derivative of Determinant 35610.3 Dependence Between Functions 35810.4 Optimization of Univariate Continuous Functions 36210.4.1 Constraint-free 36210.4.2 Subjected to Constraints 36410.5 Optimization of Multivariate Continuous Functions 36710.5.1 Constraint-free 36710.5.2 Subjected to Constraints 37111 Integrals 37311.1 Univariate Integral 37411.1.1 Indefinite Integral 37411.1.1.1 Definition 37411.1.1.2 Rules of Integration 37711.1.2 Definite Integral 38611.1.2.1 Definition 38611.1.2.2 Basic Theorems of Integral Calculus 39311.1.2.3 Reduction Formulae 39611.2 Multivariate Integral 40011.2.1 Definition 40011.2.1.1 Line Integral 40011.2.1.2 Double Integral 40311.2.2 Basic Theorems 40411.2.2.1 Fubini’s Theorem 40411.2.2.2 Green’s Theorem 40911.2.3 Change of Variables 41111.2.4 Differentiation of Integral 41411.3 Optimization of Single Integral 41611.4 Optimization of Set of Derivatives 42412 Infinite Series and Integrals 42912.1 Definition and Criteria of Convergence 42912.1.1 Comparison Test 43012.1.2 Ratio Test 43112.1.3 D’Alembert’s Test 43212.1.4 Cauchy’s Integral Test 43412.1.5 Leibnitz’s Test 43612.2 Taylor’s Series 43712.2.1 Analytical Functions 45112.2.1.1 Exponential Function 45112.2.1.2 Hyperbolic Functions 45812.2.1.3 Logarithmic Function 45912.2.1.4 Trigonometric Functions 46312.2.1.5 Inverse Trigonometric Functions 46612.2.1.6 Powers of Binomials 47612.2.2 Euler’s Infinite Product 47912.3 Gamma Function and Factorial 48812.3.1 Integral Definition and Major Features 48912.3.2 Euler’s Definition 49412.3.3 Stirling’s Approximation 49913 Analytical Geometry 50513.1 Straight Line 50513.2 Simple Polygons 50813.3 Conical Curves 51013.4 Length of Line 51613.5 Curvature of Line 52513.6 Area of Plane Surface 53013.7 Outer Area of Revolution Solid 53613.8 Volume of Revolution Solid 55214 Transforms 55914.1 Laplace’s Transform 55914.1.1 Definition 55914.1.2 Major Features 57114.1.3 Inversion 58314.2 Legendre’s Transform 59015 Solution of Differential Equations 59715.1 Ordinary Differential Equations 59715.1.1 First Order 59815.1.1.1 Nonlinear 59815.1.1.2 Linear 60015.1.2 Second Order 60215.1.2.1 Nonlinear 60315.1.2.2 Linear 61315.1.3 Linear Higher Order 65015.2 Partial Differential Equations 66016 Vector Calculus 66716.1 Rectangular Coordinates 66716.1.1 Definition and Representation 66716.1.2 Definition of Nabla Operator, ∇ 66816.1.3 Algebraic Properties of ∇ 67316.1.4 Multiple Products Involving ∇ 67616.1.4.1 Calculation of (∇.∇)ϕ 67616.1.4.2 Calculation of (∇.∇)u 67616.1.4.3 Calculation of ∇.(ϕu) 677           16.1.4.4 Calculation of ∇.(∇ × u) 67916.1.4.5 Calculation of ∇.(ϕ∇ψ) 68016.1.4.6 Calculation of ∇.(uu) 68216.1.4.7 Calculation of ∇ × (∇ ϕ) 68416.1.4.8 Calculation of ∇(∇.u) 68516.1.4.9 Calculation of (u.∇)u 69016.1.4.10 Calculation of ∇.(τ.u) 69316.2 Cylindrical Coordinates 69516.2.1 Definition and Representation 69516.2.2 Redefinition of Nabla Operator, ∇ 70016.3 Spherical Coordinates 70516.3.1 Definition and Representation 70516.3.2 Redefinition of Nabla Operator, ∇ 71516.4 Curvature of Three-dimensional Surfaces 72916.5 Three-dimensional Integration 73717 Numerical Approaches to Integration 74117.1 Calculation of Definite Integrals 74117.1.1 Zeroth Order Interpolation 74317.1.2 First- and Second-Order Interpolation 75017.1.2.1 Trapezoidal Rule 75117.1.2.2 Simpson’s Rule 75417.1.2.3 Higher Order Interpolation 76817.1.3 Composite Methods 77117.1.4 Infinite and Multidimensional Integrals 77517.2 Integration of Differential Equations 77717.2.1 Single-step Methods 77917.2.2 Multistep Methods 78217.2.3 Multistage Methods 79017.2.3.1 First Order 79017.2.3.2 Second Order 79017.2.3.3 General Order 79317.2.4 Integral Versus Differential Equation 801Part 3 Basic Concepts of Statistics 80718 Continuous Probability Functions 80918.1 Basic Statistical Descriptors 81018.2 Normal Distribution 81518.2.1 Derivation 81618.2.2 Justification 82118.2.3 Operational Features 82618.2.4 Moment-generating Function 82918.2.4.1 Single Variable 82918.2.4.2 Multiple Variables 83518.2.5 Standard Probability Density Function 84218.2.6 Central Limit Theorem 84518.2.7 Standard Probability Cumulative Function 85518.3 Other Relevant Distributions 85818.3.1 Lognormal Distribution 85818.3.1.1 Probability Density Function 85818.3.1.2 Mean and Variance 85918.3.1.3 Probability Cumulative Function 86218.3.1.4 Mode and Median 86318.3.2 Chi-square Distribution 86518.3.2.1 Probability Density Function 86518.3.2.2 Mean and Variance 86918.3.2.3 Asymptotic Behavior 87018.3.2.4 Probability Cumulative Function 87218.3.2.5 Mode and Median 87318.3.2.6 Other Features 87418.3.3 Student’s t-distribution 87618.3.3.1 Probability Density Function 87618.3.3.2 Mean and Variance 87918.3.3.3 Asymptotic Behavior 88318.3.3.4 Probability Cumulative Function 88618.3.3.5 Mode and Median 88718.3.4 Fisher’s F-distribution 88818.3.4.1 Probability Density Function 88818.3.4.2 Mean and Variance 89318.3.4.3 Asymptotic Behavior 89618.3.4.4 Probability Cumulative Function 89918.3.4.5 Mode and Median 90218.3.4.6 Other Features 90319 Statistical Hypothesis Testing 91520 Linear Regression 92320.1 Parameter Fitting 92420.2 Residual Characterization 92720.3 Parameter Inference 93120.3.1 Multivariate Models 93120.3.2 Univariate Models 93420.4 Unbiased Estimation 93720.4.1 Multivariate Models 93720.4.2 Univariate Models 94020.5 Prediction Inference 94920.6 Multivariate Correction 951Further Reading 963