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Applying Maths in the Chemical & Biomolecular Sciences
Applying Maths in the Chemical and Biomolecular Sciences.
1 Numbers, Equations, Operators, and Algorithms.
1 Numbers to Algorithms.
Questions 1 - 7
5 Trig, hyperbolic and inverse functions, waves, polar coordinates, factorials
Questions 8 - 16
9 Sophisticated Counting.
Questions 17 - 34
10 Modulo arithmetic,
\(\delta\)
functions, types of series & estimating quantities
Questions 35 - 39
Solutions Q1 - 7
Solutions Q8 - 16
Solutions Q17 - 34
Solutions Q35 - 39
2. Complex numbers.
1 Real, imaginary, conjugate and modulus
Questions 1 - 9
3 De Moivre’s theorem and integer powers of complex numbers
Questions 10 -13
Questions 14 - 29
Solutions Q1 - 27
3. Differentiation.
1 - 3.8 Differentiation, gradients & basic functions
3.9 Trig functions, logs, power, reciprocals and integrals
Questions 1-11
4 Product rule and function of function or chain rule.
Questions 12-42
6 Limits, l’Hopital’s rule, Maximum, Minimum and Calculus of Variations
Questions 43 - 73
9 Numerically finding the roots of an equation
Questions 74 - 85
10 Minimizing or maximizing with constraints: Lagrange Undetermined Multipliers
Questions 86 - 92
11 Partial differentiation
Questions 93-114
12 Differentiation of vectors
Solutions Q 1-11
Solutions Q 12-42
Solutions Q 43 - 73
Solutions Q 74 - 85
Solutions Q 86 - 94
Solutions Q 93-114
4 Integration.
1 Integration basics
Questions 1-14
3 Integration by substitution and ‘by parts’. Differentiation and the Leibniz formula. Feynman’s ‘Trick’
Questions 15-30
6 Integration and parametric equations
Questions 31 - 48
8 Calculating an Average.
Questions 49 - 72
9 The Variational Method in Quantum Mechanics
Questions 73 - 77
10 Multiple integrals, Change of variables: Jacobians
Questions 78 - 86
12 Calculating the energy of a Chemical Bond using Molecular Orbitals
13 Line integrals
Questions 87 - 96
Solutions Q1 - 14
Solutions Q15 - 30
Solutions Q31 - 48
Solutions Q49 - 72
Solutions Q73 - 96
5. Summations, Series and expansion of Functions.
1 Series, averages, waves, partition functions, DNA melting, atom entropy
Questions 1 - 13
6 Maclaurin and Taylor series expansions. Paramagnetic spins. Euler-Maclaurin formula.
Questions 14 - 34
Questions 35 - 44
8 Perturbation Theory
Questions 45 - 48
9 Quantum superposition and wavepackets.
Questions 49 - 52
Solutions Q 1 - 7
Solutions Q8 - 13
Solutions Q 14 - 26
Solutions Q27 - 34
Solutions Q33 - 44
Solutions Q45 - 52
6. Vectors.
1 Vector basics
Questions 1 - 24
6 Projections and components
Questions 25 - 32
8 Axes need not be right-angled or of equal length
Questions 33 - 38
12 Basis sets with more than three dimensions
Questions 39 - 46
15 Cross product or vector product
Questions 47 - 52
18 Torsion or dihedral angles
Questions 53 - 48
21 Torque and angular momentum
Questions 59 - 61
Solutions Q1 - 24
Solutions Q25 - 32
Solutions Q33 - 38
Solutions Q39 - 46
Solutions Q47 - 52
Solutions Q53 - 61
7. Matrices
1 Determinants
Questions 1 - 10
4 Matrices
Questions 11 - 16
6 Molecular Group Theory
Questions 17 - 30
7 Rotation matrices: moving molecules. Coordinate transform as rotation.
Questions 31 - 34
9 Matrices in optics and designing laser cavities
Questions 35 - 39
11 Polarizing optics
Questions 40 - 44
12 Solving equations using matrices. Eigenvectors and Eigenvalues.
Questions 45 - 50
13 Rate equations and Chemical Kinetics
Questions 51 - 54
14 Molecular vibrations and pendulums
Questions 55 - 58
15 Moments of Inertia
Questions 59 - 62
16 Principal axes and Moments of Inertia
Question 61 - 62
Solutions Q1 - 10
Solutions Q11 - 16
Solutions Q17 - 30
Solutions Q31 - 34
Solutions Q35 - 39
Solutions Q40 - 44
Solutions Q45 - 50
Solutions Q51 - 54
Solutions Q55 - 58
Solutions Q59 - 61
8. Matrices in Quantum Mechanics
1 Matrices in Quantum Mechanics
Questions 1 - 7
4 Basis sets and bra-ket algebra
Questions 8 - 12
5 Continuous basis sets
Solutions Q1 - 7
Solutions Q8 - 12
9. Fourier Series and Transforms.
1 Fourier series, Gibbs phenomenon, generalised series.
Questions 1 - 6
5 Fourier Transforms
Questions 7 - 15
7 Convolution and Autocorrelation
Questions 16 - 21
9 Discrete Fourier (DFT) and Fast Fourier transforms (FFT)
11 The Hadamard Transform: Encoding and Decoding
Solutions Q 1 - 6
Solutions Q 7 - 15
Solutions Q 16 - 21
12 The Fourier transform in two dimensions: images and x-ray Diffraction
14 Computed Tomography
10. Differential Equations.
1 Basics, Initial and Boundary Value problems, Steady state, Phase portrait, Chemical kinetics
Questions 1 - 23
9 First order equations & Integrating factors. Second order equations, Newton’s laws and equations of motion.
12 The ‘D’ operator. Solving linear differential equations with constant coefficients
13 Simultaneous equations
Questions 24 - 37
14 Linear equations with variable coefficients
15 Partial Differential Equations, PDE. 1D & 2D diffusion, Particle in a box, Lateral flow tests, Diffusion and Reaction, Cable equation, Algal Blooms
16 PDE continued. Diffusion, Wave and Schroedinger eqns. Solution by the Fourier method. Transient grating.
Solutions Q1 - 21
Solutions Q22 - 37
11. Numerical Methods
1 Numerical Methods
Questions 1 - 7
4 Numerical solution of differential equations
Questions 8 - 12
5 Coupled equations
Questions 13 - 16
6 The phase plane, nullclines, stable points and separatrix. The pendulum, Euler-Cromer eqns, SIR model of disease, bacterial growth.
Questions 17 - 20
8 Reaction schemes with feedback. Predator -Prey (Lotka-Volterra) & Nerve impulses (Fitzhugh-Nagumo) equations.
Questions 21 - 34
9 Boundary value problems.
Questions 35 - 40
Solutions Q1 - 7
Solutions Q 8 - 12
Solutions Q13 - 16
Solutions Q17 - 20
Solutions Q21 - 34
Solutions Q35 - 40
12. Monte Carlo Methods
1 Monte - Carlo integration & importance sampling
Questions 1 - 6
2 Solving rate equations. The Gillespie method
Questions 7 - 10
3 Monte Carlo Simulations
4 Energy transfer. Autocatalytic reaction and spreading of fires
Questions 11 - 17
6 The Metropolis algorithm
Solutions Q1 - 6
Solutions Q7 - 10
Solutions Q11 - 17
13. Data Analysis
1 Characterizing experimental data. Accuracy, precision, mean and standard deviation
Questions 1 - 3
6 Modelling data. Least squares, chi squared, residuals, ANNOVA
Questions 4 - 9
7 Modelling data is simpler using matrices
Questions 10 - 13
10 Non-linear least squares. Least absolute deviation. Principal component analysis
Solutions Q1 -11
14 SI Units, Scientific Constants, unit conversion, Glossary, Python
1 SI Units, Unit conversions, Scientific Constants
5 Appendix: Some basic Python instructions with a few examples.
15 References and index
1 References
2 Index
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