Numbers, variables, and units / 1: |
Concepts / 1.1: |
Real numbers / 1.2: |
Factorization, factors, and factorials / 1.3: |
Decimal representation of numbers / 1.4: |
Variables / 1.5: |
The algebra of real numbers / 1.6: |
Complex numbers / 1.7: |
Units / 1.8: |
Exercises / 1.9: |
Algebraic functions / 2: |
Graphical representation of functions / 2.1: |
Factorization and simplification of expressions / 2.3: |
Inverse functions / 2.4: |
Polynomials / 2.5: |
Rational functions / 2.6: |
Partial fractions / 2.7: |
Solution of simultaneous equations / 2.8: |
Transcendental functions / 2.9: |
Trigonometric functions / 3.1: |
Inverse trigonometric functions / 3.3: |
Trigonometric relations / 3.4: |
Polar coordinates / 3.5: |
The exponential function / 3.6: |
The logarithmic function / 3.7: |
Values of exponential and logarithmic functions / 3.8: |
Hyperbolic functions / 3.9: |
Differentiation / 3.10: |
The process of differentiation / 4.1: |
Continuity / 4.3: |
Limits / 4.4: |
Differentiation from first principles / 4.5: |
Differentiation by rule / 4.6: |
Implicit functions / 4.7: |
Logarithmic differentiation / 4.8: |
Successive differentiation / 4.9: |
Stationary points / 4.10: |
Linear and angular motion / 4.11: |
The differential / 4.12: |
Integration / 4.13: |
The indefinite integral / 5.1: |
The definite integral / 5.3: |
The integral calculus / 5.4: |
Uses of the integral calculus / 5.5: |
Static properties of matter / 5.6: |
Dynamics / 5.7: |
Pressure-volume work / 5.8: |
Methods of integration / 5.9: |
The use of trigonometric relations / 6.1: |
The method of substitution / 6.3: |
Integration by parts / 6.4: |
Reduction formulas / 6.5: |
Rational integrands. The method of partial fractions / 6.6: |
Parametric differentiation of integrals / 6.7: |
Sequences and series / 6.8: |
Sequences / 7.1: |
Finite series / 7.3: |
Infinite series / 7.4: |
Tests of convergence / 7.5: |
MacLaurin and Taylor series / 7.6: |
Approximate values and limits / 7.7: |
Operations with power series / 7.8: |
Algebra of complex numbers / 7.9: |
Graphical representation / 8.3: |
Complex functions / 8.4: |
Euler's formula / 8.5: |
Periodicity / 8.6: |
Evaluation of integrals / 8.7: |
Functions of several variables / 8.8: |
Partial differentiation / 9.1: |
The total differential / 9.4: |
Some differential properties / 9.6: |
Exact differentials / 9.7: |
Line integrals / 9.8: |
Multiple integrals / 9.9: |
The double integral / 9.10: |
Change of variables / 9.11: |
Functions in 3 dimensions / 9.12: |
Spherical polar coordinates / 10.1: |
Functions of position / 10.3: |
Volume integrals / 10.4: |
The Laplacian operator / 10.5: |
Other coordinate systems / 10.6: |
First-order differential equations / 10.7: |
Solution of a differential equation / 11.1: |
Separable equations / 11.3: |
Separable equations in chemical kinetics / 11.4: |
First-order linear equations / 11.5: |
An example of linear equations in chemical kinetics / 11.6: |
Electric circuits / 11.7: |
Second-order differential equations. Constant coefficients / 11.8: |
Homogeneous linear equations / 12.1: |
The general solution / 12.3: |
Particular solutions / 12.4: |
The harmonic oscillator / 12.5: |
The particle in a one-dimensional box / 12.6: |
The particle in a ring / 12.7: |
Inhomogeneous linear equations / 12.8: |
Forced oscillations / 12.9: |
Second-order differential equations. Some special functions / 12.10: |
The power-series method / 13.1: |
The Frobenius method / 13.3: |
The Legendre equation / 13.4: |
The Hermite equation / 13.5: |
The Laguerre equation / 13.6: |
Bessel functions / 13.7: |
Partial differential equations / 13.8: |
General solutions / 14.1: |
Separation of variables / 14.3: |
The particle in a rectangular box / 14.4: |
The particle in a circular box / 14.5: |
The hydrogen atom / 14.6: |
The vibrating string / 14.7: |
Orthogonal expansions. Fourier analysis / 14.8: |
Orthogonal expansions / 15.1: |
Two expansions in Legendre polynomials / 15.3: |
Fourier series / 15.4: |
Fourier transforms / 15.5: |
Vectors / 15.7: |
Vector algebra / 16.1: |
Components of vectors / 16.3: |
Scalar differentiation of a vector / 16.4: |
The scalar (dot) product / 16.5: |
The vector (cross) product / 16.6: |
Scalar and vector fields / 16.7: |
The gradient of a scalar field / 16.8: |
Divergence and curl of a vector field / 16.9: |
Vector spaces / 16.10: |
Determinants / 16.11: |
Determinants of order 3 / 17.1: |
The general case / 17.3: |
The solution of linear equations / 17.4: |
Properties of determinants / 17.5: |
Reduction to triangular form / 17.6: |
Alternating functions / 17.7: |
Matrices and linear transformations / 17.8: |
Some special matrices / 18.1: |
Matrix algebra / 18.3: |
The inverse matrix / 18.4: |
Linear transformations / 18.5: |
Orthogonal matrices and orthogonal transformations / 18.6: |
Symmetry operations / 18.7: |
The matrix eigenvalue problem / 18.8: |
The eigenvalue problem / 19.1: |
Properties of the eigenvectors / 19.3: |
Matrix diagonalization / 19.4: |
Quadratic forms / 19.5: |
Complex matrices / 19.6: |
Numerical methods / 19.7: |
Errors / 20.1: |
Solution of ordinary equations / 20.3: |
Interpolation / 20.4: |
Numerical integration / 20.5: |
Methods in linear algebra / 20.6: |
Gauss elimination for the solution of linear equations / 20.7: |
Gauss-Jordan elimination for the inverse of a matrix / 20.8: |
Systems of differential equations / 20.9: |
Probability and statistics / 20.11: |
Descriptive statistics / 21.1: |
Frequency and probability / 21.3: |
Combinations of probabilities / 21.4: |
The binomial distribution / 21.5: |
Permutations and combinations / 21.6: |
Continuous distributions / 21.7: |
The Gaussian distribution / 21.8: |
More than one variable / 21.9: |
Least squares / 21.10: |
Sample statistics / 21.11: |
Standard integrals / 21.12: |
Solutions to exercises |
Index |
Numbers, variables, and units / 1: |
Concepts / 1.1: |
Real numbers / 1.2: |