Mathematics for Enzyme Reaction Kinetics and Reactor Performance: (Enzyme Reaction Engineering)

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.

Table of Contents:
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’Hôpital’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 (∇.∇)ϕ 676 16.1.4.2 Calculation of (∇.∇)u 676 16.1.4.3 Calculation of ∇.(ϕu) 677            16.1.4.4 Calculation of ∇.(∇ × u) 679 16.1.4.5 Calculation of ∇.(ϕ∇ψ) 680 16.1.4.6 Calculation of ∇.(uu) 682 16.1.4.7 Calculation of ∇ × (∇ ϕ) 684 16.1.4.8 Calculation of ∇(∇.u) 685 16.1.4.9 Calculation of (u.∇)u 690 16.1.4.10 Calculation of ∇.(τ.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

About the Author :
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).