Оглавление
F. Xavier Malcata. Mathematics for Enzyme Reaction Kinetics and Reactor Performance
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
Enzyme Reactor Engineering: Forthcoming Titles
Mathematics for Enzyme Reaction Kinetics and Reactor Performance
About the Author
Series Preface
Preface
1 Basic Concepts of Algebra
1 Scalars, Vectors, Matrices, and Determinants
2 Function Features
2.1 Series
2.1.1 Arithmetic Series
2.1.2 Geometric Series
2.1.3 Arithmetic/Geometric Series
2.2 Multiplication and Division of Polynomials
2.2.1 Product
2.2.2 Quotient
2.2.3 Factorization
2.2.4 Splitting
2.2.5 Power
2.3 Trigonometric Functions
2.3.1 Definition and Major Features
2.3.2 Angle Transformation Formulae
2.3.3 Fundamental Theorem of Trigonometry
2.3.4 Inverse Functions
2.4 Hyperbolic Functions
2.4.1 Definition and Major Features
2.4.2 Argument Transformation Formulae
2.4.3 Euler’s Form of Complex Numbers
2.4.4 Inverse Functions
3 Vector Operations
3.1 Addition of Vectors
3.2 Multiplication of Scalar by Vector
3.3 Scalar Multiplication of Vectors
3.4 Vector Multiplication of Vectors
4 Matrix Operations
4.1 Addition of Matrices
4.2 Multiplication of Scalar by Matrix
4.3 Multiplication of Matrices
4.4 Transposal of Matrices
4.5 Inversion of Matrices
4.5.1 Full Matrix
4.5.2 Block Matrix
4.6 Combined Features
4.6.1 Symmetric Matrix
4.6.2 Positive Semidefinite Matrix
5 Tensor Operations
6 Determinants
6.1 Definition
6.2 Calculation
6.2.1 Laplace’s Theorem
6.2.2 Major Features
6.2.3 Tridiagonal Matrix
6.2.4 Block Matrix
6.2.5 Matrix Inversion
6.3 Eigenvalues and Eigenvectors
6.3.1 Characteristic Polynomial
6.3.2 Cayley and Hamilton's Theorem
7 Solution of Algebraic Equations
7.1 Linear Systems of Equations
7.1.1 Jacobi’s Method
7.1.2 Explicitation
7.1.3 Cramer’s Rule
7.1.4 Matrix Inversion
7.2 Quadratic Equation
7.3 Lambert’s W Function
7.4 Numerical Approaches
7.4.1 Double‐initial Estimate Methods
7.4.1.1 Bisection
7.4.1.2 Linear Interpolation
7.4.2 Single‐initial Estimate Methods
7.4.2.1 Newton and Raphson’s Method
7.4.2.2 Direct Iteration
Further Reading
Enzyme Reactor Engineering: Forthcoming Titles
Mathematics for Enzyme Reaction Kinetics and Reactor Performance
About the Author
Series Preface
Preface
2 Basic Concepts of Calculus
8 Limits, Derivatives, Integrals, and Differential Equations
9 Limits and Continuity
9.1 Univariate Limit. 9.1.1 Definition
9.1.2 Basic Calculation
9.2 Multivariate Limit
9.3 Basic Theorems on Limits
9.4 Definition of Continuity
9.5 Basic Theorems on Continuity
9.5.1 Bolzano’s Theorem
9.5.2 Weierstrass’ Theorem
10 Differentials, Derivatives, and Partial Derivatives
10.1 Differential
10.2 Derivative. 10.2.1 Definition
10.2.1.1 Total Derivative
10.2.1.2 Partial Derivatives
10.2.1.3 Directional Derivatives
10.2.2 Rules of Differentiation of Univariate Functions
10.2.3 Rules of Differentiation of Multivariate Functions
10.2.4 Implicit Differentiation
10.2.5 Parametric Differentiation
10.2.6 Basic Theorems of Differential Calculus. 10.2.6.1 Rolle’s Theorem
10.2.6.2 Lagrange’s Theorem
10.2.6.3 Cauchy’s Theorem
10.2.6.4 L’Hôpital’s Rule
10.2.7 Derivative of Matrix
10.2.8 Derivative of Determinant
10.3 Dependence Between Functions
10.4 Optimization of Univariate Continuous Functions
10.4.1 Constraint‐free
10.4.2 Subjected to Constraints
10.5 Optimization of Multivariate Continuous Functions. 10.5.1 Constraint‐free
10.5.2 Subjected to Constraints
11 Integrals
11.1 Univariate Integral. 11.1.1 Indefinite Integral. 11.1.1.1 Definition
11.1.1.2 Rules of Integration
11.1.2 Definite Integral. 11.1.2.1 Definition
11.1.2.2 Basic Theorems of Integral Calculus. 11.1.2.2.1 Mean Value Theorem for Integrals
11.1.2.2.2 First Fundamental Theorem of Integral Calculus
11.1.2.2.3 Second Fundamental Theorem of Integral Calculus
11.1.2.3 Reduction Formulae
11.2 Multivariate Integral. 11.2.1 Definition. 11.2.1.1 Line Integral
11.2.1.2 Double Integral
11.2.2 Basic Theorems. 11.2.2.1 Fubini’s Theorem
11.2.2.2 Green’s Theorem
11.2.3 Change of Variables
11.2.4 Differentiation of Integral
11.3 Optimization of Single Integral
11.4 Optimization of Set of Derivatives
12 Infinite Series and Integrals
12.1 Definition and Criteria of Convergence
12.1.1 Comparison Test
12.1.2 Ratio Test
12.1.3 D’Alembert’s Test
12.1.4 Cauchy’s Integral Test
12.1.5 Leibnitz’s Test
12.2 Taylor’s Series
12.2.1 Analytical Functions. 12.2.1.1 Exponential Function
12.2.1.2 Hyperbolic Functions
12.2.1.3 Logarithmic Function
12.2.1.4 Trigonometric Functions
12.2.1.5 Inverse Trigonometric Functions
12.2.1.6 Powers of Binomials
12.2.2 Euler’s Infinite Product
12.3 Gamma Function and Factorial
12.3.1 Integral Definition and Major Features
12.3.2 Euler’s Definition
12.3.3 Stirling’s Approximation
13 Analytical Geometry
13.1 Straight Line
13.2 Simple Polygons
13.3 Conical Curves
13.4 Length of Line
13.5 Curvature of Line
13.6 Area of Plane Surface
13.7 Outer Area of Revolution Solid
13.8 Volume of Revolution Solid
14 Transforms
14.1 Laplace’s Transform. 14.1.1 Definition
14.1.2 Major Features
14.1.3 Inversion
14.2 Legendre’s Transform
15 Solution of Differential Equations
15.1 Ordinary Differential Equations
15.1.1 First Order
15.1.1.1 Nonlinear
15.1.1.2 Linear
15.1.2 Second Order
15.1.2.1 Nonlinear. 15.1.2.1.1 Dependent Variable‐free
15.1.2.1.2 Independent Variable‐free
15.1.2.1.3 Hartman and Grobman’s Theorem
15.1.2.2 Linear
15.1.2.2.1 Frobenius’ Method of Solution
15.1.2.2.2 Bessel’s Equation
15.1.2.2.3 MacLaurin’s Method of Solution
15.1.2.2.4 Independent Solutions
15.1.3 Linear Higher Order
15.2 Partial Differential Equations
16 Vector Calculus
16.1 Rectangular Coordinates. 16.1.1 Definition and Representation
16.1.2 Definition of Nabla Operator, ∇
16.1.3 Algebraic Properties of ∇
16.1.4 Multiple Products Involving ∇ 16.1.4.1 Calculation of (∇ ⋅ ∇)ϕ
16.1.4.2 Calculation of (∇⋅∇)u
16.1.4.3 Calculation of ∇⋅(ϕu)
16.1.4.4 Calculation of ∇⋅(∇ × u)
16.1.4.5 Calculation of ∇⋅(ϕ∇ψ)
16.1.4.6 Calculation of ∇⋅(uu)
16.1.4.7 Calculation of ∇ × (∇ ϕ)
16.1.4.8 Calculation of ∇(∇⋅u)
16.1.4.9 Calculation of (u⋅∇)u
16.1.4.10 Calculation of ∇⋅(τ⋅u)
16.2 Cylindrical Coordinates. 16.2.1 Definition and Representation
16.2.2 Redefinition of Nabla Operator, ∇
16.3 Spherical Coordinates. 16.3.1 Definition and Representation
16.3.2 Redefinition of Nabla Operator, ∇
16.4 Curvature of Three‐dimensional Surfaces
16.5 Three‐dimensional Integration
17 Numerical Approaches to Integration
17.1 Calculation of Definite Integrals
17.1.1 Zeroth Order Interpolation
17.1.2 First‐ and Second‐Order Interpolation
17.1.2.1 Trapezoidal Rule
17.1.2.2 Simpson’s Rule
17.1.2.3 Higher Order Interpolation
17.1.3 Composite Methods
17.1.4 Infinite and Multidimensional Integrals
17.2 Integration of Differential Equations
17.2.1 Single‐step Methods
17.2.2 Multistep Methods
17.2.3 Multistage Methods
17.2.3.1 First Order
17.2.3.2 Second Order
17.2.3.3 General Order
17.2.4 Integral Versus Differential Equation
3 Basic Concepts of Statistics
18. Continuous Probability Functions
18.1 Basic Statistical Descriptors
18.2 Normal Distribution
18.2.1 Derivation
18.2.2 Justification
18.2.3 Operational Features
18.2.4 Moment‐generating Function. 18.2.4.1 Single Variable
18.2.4.2 Multiple Variables
18.2.5 Standard Probability Density Function
18.2.6 Central Limit Theorem
18.2.7 Standard Probability Cumulative Function
18.3 Other Relevant Distributions
18.3.1 Lognormal Distribution. 18.3.1.1 Probability Density Function
18.3.1.2 Mean and Variance
18.3.1.3 Probability Cumulative Function
18.3.1.4 Mode and Median
18.3.2 Chi‐square Distribution. 18.3.2.1 Probability Density Function
18.3.2.2 Mean and Variance
18.3.2.3 Asymptotic Behavior
18.3.2.4 Probability Cumulative Function
18.3.2.5 Mode and Median
18.3.2.6 Other Features
18.3.3 Student’s t‐distribution. 18.3.3.1 Probability Density Function
18.3.3.2 Mean and Variance
18.3.3.3 Asymptotic Behavior
18.3.3.4 Probability Cumulative Function
18.3.3.5 Mode and Median
18.3.4 Fisher’s F‐distribution. 18.3.4.1 Probability Density Function
18.3.4.2 Mean and Variance
18.3.4.3 Asymptotic Behavior
18.3.4.4 Probability Cumulative Function
18.3.4.5 Mode and Median
18.3.4.6 Other Features
19 Statistical Hypothesis Testing
20 Linear Regression
20.1 Parameter Fitting
20.2 Residual Characterization
20.3 Parameter Inference. 20.3.1 Multivariate Models
20.3.2 Univariate Models
20.4 Unbiased Estimation. 20.4.1 Multivariate Models
20.4.2 Univariate Models
20.5 Prediction Inference
20.6 Multivariate Correction
Further Reading
Index. a
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