Mathematical Methods for Molecular Science

✍ Scribed by John E. Straub

• Publisher: University Science Books
• Year: 2022
• Tongue: English
• Leaves: 542
• Edition: 1
• Category: Library

✦ Synopsis

This brilliant new text by John Straub (Boston University) is designed to bridge the “mathematics knowledge gap” between what is commonly known by students after completing a year of introductory calculus, and what is required for success in the physical sciences and in physical chemistry courses. Key concepts from the introductory calculus sequence are reviewed and carefully selected topics in multivariate calculus, probability and statistics, ordinary differential equations, and linear algebra are explored. Additional chapters cover advanced topics, including partial differential equations, Fourier analysis, and group theory. Engaging narratives, fully worked examples, hundreds of colorful visualizations, and ample end-of-chapter problems with complete answers combine to make this stunning new text an excellent choice for a one-semester course on mathematical methods, as a supplement for courses in physical chemistry, or as a self-study guide. Ancillaries for adopting faculty include in-class worksheets, sample exams, and an answer manual.

✦ Table of Contents

Introduction
Functions and coordinate systems
Survey of common functions of continuous variables
Exploring coordinate systems and their utility
End-of-chapter problems
Complex numbers and logarithms
Complex numbers and the complex plane
Special properties of logarithms
Visualizing complex functions of complex variables
Application of logarithms and the logarithmic scale
Logarithms and Stirling's approximation
Connecting complex numbers and logarithms
End-of-chapter problems
Differentiation in one and many dimensions
Differentiating functions of one variable
Partial derivatives of functions of many variables
Infinitesimal change and the total differential
Euler's theorem for homogeneous functions
Geometric interpretation of the total differential
End-of-chapter problems
Scalars, vectors, and vector algebra
Fundamental properties of scalars and vectors
Multiplication of vectors
Building orthogonal vectors using Gram-Schmidt orthogonalization
End-of-chapter problems
Scalar and vector operators
Scalar operators
Vector operators and the gradient and divergence
The force and the potential energy
A survey of potential energy landscapes
Explicit forms of vector operations
Deriving explicit forms for vector operations
End-of-chapter problems
Extremizing functions of many variables
Extremizing functions of one and many variables
The method of Lagrange undetermined multipliers
Variational calculation of the energy of a one electron atom
Extremizing the multiplicity subject to constraints
End-of-chapter problems
Integration in one and many dimensions
Integrating functions of one variable
Integrating functions of many variables
An alternative to integration by parts for exponential integrals
Evaluating the definite integral of a gaussian function
An alternative to integration by parts for gaussian integrals
Properties of delta functions
End-of-chapter problems
Sequences, series, and expansions
Series, convergence, and limits
Power series
Expanding functions as Maclaurin and Taylor series
Taylor series expansions of potential energy functions
Useful approximations to functions based on power series
Self-similarity and fractal structures
End-of-chapter problems
Fundamentals of probability and statistics
Probability distributions of discrete variables
Probability distributions of continuous variables
Probability distributions in the physical sciences
Connecting the gaussian and binomial probability distributions
Uniform distributions of independent random variables
Gaussian distributions of independent random variables
Three definitions of Pythagorean means
Propagation of error through total differentials and Taylor series
End-of-chapter problems
Ordinary differential equations
First order ordinary differential equations
Applications of first order differential equations
Functions derived from exact differentials and integrating factors
End-of-chapter problems
More ordinary differential equations
Second order ordinary differential equations
Applications of second order differential equations
Power series solutions to differential equations
Quantum theory of a particle in a box
Classical theory of motion of a harmonic oscillator
Classical theory of a damped harmonic oscillator
Power series solutions to special equations in quantum theory
End-of-chapter problems
Partial differential equations
The classical heat equation
The classical diffusion equation
The classical wave equation
Survey of partial differential equations in the physical sciences
End-of-chapter problems
Fourier series, Fourier transforms, and harmonic analysis
Fourier series
Fourier transforms
Orthogonal vectors and orthogonal functions
Building orthogonal polunomials using Gram-Schmidt orthgonalization
Deriving Fourier integral transforms from complex Fouier series
End-of-chapter problems
Matrices and matrix algebra
Vectors, matrices, and determinants
Basic properties of matrix algebra
Solving coupled linear equations using Cramer's rule
Applications of determinants in Hückel theory
Solving coupled linear equations using Gaussian elimination
Finding the matrix inverse using Gauss-Jordan elimination
End-of-chapter problems
Eigenvalues and eigenvectors
Matrix eigenvalues and eigenvectors
Matrix methods for coupled differential equations
Scalar operators and eigenfunctions
End-of-chapter problems
Geometric transforms and molecular symmetry
Eigenvectors, geometric transforms, and symmetry
Matrix transformations and molecular symmetry
Point groups and the symmetry decision tree
End-of-chapter problems
Supplements
Notes on notation
Formulas from geometry
Formulas from trigonometry
Table of power series
Table of indefinite integrals
Error function table
Complementary error function table
Table of Fourier transform pairs
Answers to end-of-chapter problems
Bibliography
Index
Colophon