Twenty books, one notebook.

Determinants

Defined through permutations rather than cofactor recursion, so every property (multilinearity, alternation, \(\det AB=\det A\det B\)) falls out of one formula:

Cofactor expansion and Cramer's rule follow. Read this carefully: the permutation viewpoint returns in exterior algebra and volume forms.

Linear spaces

Axioms, linear dependence, bases, dimension, subspaces, sums and direct sums. Key identity:

Isomorphism theorem: every \(n\)-dimensional space over \(K\) is isomorphic to \(K^n\), so coordinates are always available but never canonical.

Systems of linear equations

Rank of a matrix as the maximal number of independent rows (equals columns). The Kronecker–Capelli theorem: \(Ax=b\) is solvable iff \(\operatorname{rank}A=\operatorname{rank}[A\mid b]\). The homogeneous solution space has dimension \(n-r\), and the general solution is a particular solution plus that space.

Linear functions and operators

Linear forms, linear operators and their matrices in a basis, kernel and image, rank–nullity \(\dim\ker A+\dim\operatorname{im}A=n\). Invariant subspaces, eigenvectors, and the characteristic polynomial \(\det(A-\lambda I)\). Operators with \(n\) distinct eigenvalues are diagonalizable.

Coordinate transformations

Under a change of basis with matrix \(C\), an operator's matrix becomes \(A'=C^{-1}AC\) and a bilinear form's matrix becomes \(B'=C^{\mathsf T}BC\). Knowing which law applies is the whole difference between operators and forms. Trace, determinant and the characteristic polynomial are similarity invariants.

Canonical form of an operator

The space splits into generalized eigenspaces (root subspaces); on each, \(A-\lambda I\) is nilpotent and decomposes into cyclic blocks. Result: the Jordan normal form. Shilov also develops \(\lambda\)-matrices and elementary divisors, the algebraic route to the same answer, plus the minimal polynomial (Jordan form is diagonal iff the minimal polynomial has simple roots).

Bilinear and quadratic forms

Reduction of a quadratic form to a sum of squares (Lagrange's method, Jacobi's method via leading minors). Sylvester's law of inertia: the counts of positive, negative and zero coefficients are invariants. Sylvester's criterion: a symmetric \(A\) is positive definite iff all leading principal minors are positive.

Euclidean and unitary spaces

Inner products, Cauchy–Schwarz, Gram–Schmidt, orthogonal complements, and the Gram determinant as squared volume. Adjoint operators. The spectral theorem: a self-adjoint operator has real eigenvalues and an orthonormal eigenbasis. Unitary and normal operators. Extremal properties of eigenvalues via the Rayleigh quotient and the Courant minimax principle:

Simultaneous diagonalization of two quadratic forms when one is positive definite.

Algebras, representations, categories

Closing chapters on finite-dimensional algebras, their representations, and a short introduction to categories of vector spaces. Optional on a first pass; read once you start group or representation theory.

Vectors, bases, summation convention

Einstein convention: a repeated upper and lower index is summed, \(x=x^ie_i\). The reciprocal (dual) basis is defined by

Components with upper indices are contravariant, with lower indices covariant. In an orthonormal basis the distinction disappears, which is why many courses never mention it.

Change of basis

If \(e_{i'}=A^i_{i'}e_i\), then contravariant components transform with the inverse matrix and covariant ones with the direct matrix:

This bookkeeping is the core skill of the book; drill it until it is automatic.

The metric tensor

\(g_{ij}=e_i\cdot e_j\) encodes lengths and angles in any basis: \(x\cdot y=g_{ij}x^iy^j\). Its inverse \(g^{ij}\) raises indices, \(x^i=g^{ij}x_j\), and \(g_{ij}\) lowers them. \(\det(g_{ij})\) is the squared volume of the basis parallelepiped.

Linear transformations as rank-2 tensors

An operator has components \(a^i_j\) and transforms as a mixed tensor of type (1,1). Eigenvalues come from \(\det(a^i_j-\lambda\delta^i_j)=0\); the trace \(a^i_i\) and determinant are invariants because they are full contractions.

Symmetric tensors and quadrics

A symmetric bilinear form \(a_{ij}\) can be brought to principal axes by an orthogonal change of basis. Geometric payoff: classify second-order curves and surfaces (ellipsoids, hyperboloids, paraboloids) by the signs of the eigenvalues.

General tensors and tensor algebra

Tensors of type \((p,q)\), addition, tensor product, contraction, symmetrization and alternation. The Levi-Civita symbol and the cross product \( (a\times b)_i=\varepsilon_{ijk}a^jb^k \) (in an orthonormal basis), and the triple product as a determinant.

Sets, relations, mappings

Injective, surjective, bijective maps, composition, inverses. Equivalence relations and the partition into equivalence classes: the single most reused idea in the book (cosets, quotient rings, \(\mathbb Z/n\mathbb Z\)).

Groups

Axioms, subgroups, cyclic groups, permutation groups. Cosets and Lagrange's theorem \(|G|=[G:H]\,|H|\). Normal subgroups, quotient groups, homomorphisms and the first isomorphism theorem:

Rings and integral domains

Rings, subrings, ideals, quotient rings, ring homomorphisms. Integral domains (no zero divisors) and the construction of the field of fractions, which is how \(\mathbb Q\) is built from \(\mathbb Z\).

Polynomial rings and factorization

Division algorithm and Euclidean algorithm in \(K[x]\), greatest common divisors, irreducible polynomials, unique factorization. \(K[x]/(p)\) is a field exactly when \(p\) is irreducible.

Fields and extensions

Prime fields, algebraic elements, minimal polynomials, degree of an extension and the tower law \([L:K]=[L:M][M:K]\). Applications to ruler-and-compass constructions: constructible numbers live in extensions of degree \(2^k\), which rules out trisecting an arbitrary angle and doubling the cube.

Toward Galois theory

Automorphisms of field extensions and the correspondence between subgroups and intermediate fields, presented as an introduction. Clark's book completes this story.

Set theory

Brisk foundations: maps, equivalence relations, cardinality, Zorn's lemma. Skim if you have read Landin.

Group theory

Group actions, orbits and stabilizers (orbit-stabilizer \(|Gx|=[G:G_x]\)), class equation, \(p\)-groups have nontrivial center. Sylow theorems: for \(|G|=p^am\) with \(p\nmid m\), Sylow \(p\)-subgroups exist, are conjugate, and their number satisfies

Composition series, Jordan–Hölder, solvable groups, and the structure theorem for finitely generated abelian groups.

Field theory

Algebraic extensions, splitting fields, algebraic closure, separability, normality. Finite fields: for each prime power \(p^n\) there is exactly one field \(\mathbb F_{p^n}\) up to isomorphism, the splitting field of \(x^{p^n}-x\).

Galois theory

For a finite Galois extension \(L/K\), there is an inclusion-reversing bijection between subgroups \(H\le\operatorname{Gal}(L/K)\) and intermediate fields \(L^H\), with \([L:L^H]=|H|\). Consequences: a polynomial is solvable by radicals iff its Galois group is solvable; the general quintic is not, since \(S_5\) is not solvable. Gauss's theorem: a regular \(n\)-gon is constructible iff \(n=2^kp_1\cdots p_r\) with distinct Fermat primes \(p_i\).

Ring theory

Euclidean domains ⊂ PIDs ⊂ UFDs, Gauss's lemma, \(R[x]\) is a UFD when \(R\) is. Noetherian rings and the Hilbert basis theorem.

History and the foundational crisis

Frege's logicism, Russell's paradox \(R=\{x: x\notin x\}\), and Hilbert's program to prove mathematics consistent by finitary means.

Propositional and predicate logic

Syntax vs semantics: formulas, proofs (\(\vdash\)) and truth in a structure (\(\models\)). Quantifiers, models, and why first-order logic is the default framework.

Completeness and model theory

Gödel's completeness theorem (1930): \(\Gamma\models\varphi\iff\Gamma\vdash\varphi\). Compactness: a set of sentences has a model iff every finite subset does. Löwenheim–Skolem: a countable theory with an infinite model has a countable model, and models of every larger size. Non-standard models of arithmetic exist.

Computability

Turing machines, recursive functions, Church's thesis, and the undecidability of the halting problem by diagonalization.

Incompleteness

Gödel's first theorem: any consistent, effectively axiomatized theory containing basic arithmetic has a true-but-unprovable sentence. Second theorem: such a theory cannot prove its own consistency, which ends Hilbert's program in its original form.

Set theory and independence

ZF axioms, the axiom of choice, and the continuum hypothesis. Gödel showed CH is consistent with ZFC (constructible universe); Cohen showed its negation is too (forcing). So CH is independent.

Theory of sets

Functions, images and preimages (note \(f^{-1}\) preserves unions, intersections and complements; \(f\) does not), countability, Cartesian products.

Metric spaces

Metrics, open balls, open and closed sets, neighborhoods. Key equivalence: \(f\) is continuous iff the preimage of every open set is open. That sentence is what makes topology possible.

Topological spaces

Topologies, bases, subspace, product and quotient topologies, closure, interior, boundary, Hausdorff spaces, homeomorphisms. A homeomorphism is a continuous bijection with continuous inverse.

Connectedness

A space is connected if it is not a union of two disjoint nonempty open sets. Continuous images of connected sets are connected, which gives the intermediate value theorem. Path-connected implies connected; the topologist's sine curve shows the converse fails.

Compactness

Every open cover has a finite subcover. Heine–Borel in \(\mathbb R^n\): compact iff closed and bounded. Continuous image of compact is compact, so a continuous real function on a compact set attains its max. In metric spaces, compact iff sequentially compact. Compact subsets of Hausdorff spaces are closed.

Homotopy and the fundamental group

Homotopic paths, \(\pi_1(X,x_0)\) as loops modulo homotopy under concatenation. \(\pi_1(\mathbb R^n)=0\), and

via winding number. This gives the Brouwer fixed point theorem in dimension 2.

Sets, functions, and the move to abstraction

Why mathematicians started talking about sets and maps, and how abstraction lets one proof cover many cases.

Algebra: groups and symmetry

Arithmetic mod \(n\), groups as the mathematics of symmetry, subgroups, isomorphism. Preview for Landin and Clark.

Axiomatics and counting

Axiomatic method, consistency, and infinite cardinals: \(\mathbb Q\) is countable, \(\mathbb R\) is not (Cantor's diagonal argument). Preview for Crossley.

Topology

Rubber-sheet geometry, the Möbius band, knots, topological invariants like the Euler characteristic \(V-E+F=2\) for polyhedra, and an intuitive tour of algebraic topology. Preview for Mendelson.

Linear algebra and hyperspace

Vector spaces, linear maps, and thinking in \(n\) dimensions. Preview for Shilov.

Analysis and probability

Limits, continuity and the real numbers made rigorous; probability as measure. Preview for Wilcox and Myers.

Limits of the Riemann integral

The Dirichlet function \(\mathbf 1_{\mathbb Q}\) on \([0,1]\) is not Riemann integrable, yet it is a pointwise limit of integrable functions. Riemann theory does not commute well with limits.

Lebesgue measure

Outer measure \(m^*(E)=\inf\sum\ell(I_k)\) over interval covers, Carathéodory measurability, the σ-algebra of measurable sets, Borel sets, sets of measure zero (every countable set), and the existence of non-measurable sets.

Measurable functions and the integral

Measurable functions, simple functions, the integral defined as a supremum over simple functions below \(f\). Riemann integrable implies Lebesgue integrable with the same value; a bounded function is Riemann integrable iff it is continuous almost everywhere.

Convergence theorems

Monotone convergence, Fatou's lemma, and the dominated convergence theorem:

DCT is the workhorse; know its hypotheses exactly.

The space \(L^2\)

Square-integrable functions modulo a.e. equality form a Hilbert space with \(\langle f,g\rangle=\int f\bar g\). Riesz–Fischer: \(L^2\) is complete.

Fourier series

Orthonormal system \(\{e^{inx}/\sqrt{2\pi}\}\), Bessel's inequality, completeness of the trigonometric system, and Parseval:

Riemann–Lebesgue lemma (\(c_n\to0\)), the Dirichlet kernel and pointwise convergence for piecewise smooth \(f\), and Fejér's theorem: Cesàro means of the series of a continuous periodic function converge uniformly.

Functions as vectors

Inner product \(\langle f,g\rangle=\int_a^b f g\,w\,dx\), orthogonality, projection onto an orthonormal set as best least-squares approximation. Fourier coefficients are just coordinates.

Fourier series

On \([-L,L]\):

Even and odd extensions, half-range sine and cosine series, complex form.

Convergence

For piecewise smooth \(f\), the series converges to the average of one-sided limits at jumps. Gibbs phenomenon: about 9% overshoot of the jump near a discontinuity, which does not shrink. Coefficient decay reflects smoothness: jumps give \(O(1/n)\), a continuous kink gives \(O(1/n^2)\).

Boundary value problems

Separation of variables for the heat equation \(u_t=ku_{xx}\), the vibrating string \(u_{tt}=c^2u_{xx}\), and Laplace's equation in rectangles and disks. Each mode evolves independently; the initial data's Fourier coefficients set the amplitudes.

Sturm–Liouville theory

Problems \((p y')'+(q+\lambda w)y=0\) with suitable boundary conditions have real eigenvalues \(\lambda_1<\lambda_2<\dots\to\infty\) and eigenfunctions orthogonal with weight \(w\). This is the spectral theorem for a differential operator.

Bessel functions and Legendre polynomials

Bessel's equation from separation in polar coordinates (vibrating drum), with modes \(J_n(\alpha_{nk}r/a)\). Legendre's equation from spherical coordinates, with orthogonal polynomials \(P_n\) on \([-1,1]\), Rodrigues' formula \(P_n=\frac1{2^nn!}\frac{d^n}{dx^n}(x^2-1)^n\).

Fourier integrals

Letting \(L\to\infty\) turns the series into the Fourier transform, for problems on unbounded domains.

Test functions and distributions

\(\mathcal D\): smooth functions with compact support, with a strong notion of convergence. A distribution \(T\in\mathcal D'\) is a continuous linear functional \(\phi\mapsto\langle T,\phi\rangle\). Every locally integrable \(f\) defines one by \(\langle f,\phi\rangle=\int f\phi\), and

Derivatives

Defined by moving the derivative onto the test function: \(\langle T',\phi\rangle=-\langle T,\phi'\rangle\). So every distribution is infinitely differentiable, the Heaviside step satisfies \(H'=\delta\), and differentiation commutes with limits of distributions.

Convolution

\(T*S\) under support conditions; \(\delta\) is the identity: \(\delta*T=T\), and \(\delta'*T=T'\). Differential operators become convolutions.

Fourier transform and tempered distributions

Schwartz space \(\mathcal S\) of rapidly decreasing functions is invariant under the Fourier transform, so its dual \(\mathcal S'\) is too. Results: \(\hat\delta=1\), \(\hat1=2\pi\delta\) (with the convention \(\hat f(\omega)=\int f e^{-i\omega t}dt\)), and \(\widehat{T'}=i\omega\hat T\).

Laplace transform and periodic distributions

Laplace transforms of distributions supported on \(t\ge0\), used for initial value problems. Periodic distributions have Fourier series; the Dirac comb \(\sum\delta(t-nT)\) is the basic example and underlies sampling theory.

Applications to systems

Linear time-invariant systems as convolution operators: output \(y=h*x\) with impulse response \(h\). Fundamental solutions of differential equations (\(LE=\delta\)), and passivity conditions on system transfer functions.

Preliminaries and first-order linear equations

Complex-valued functions, \(e^{(a+ib)x}\). The equation \(y'+a(x)y=b(x)\) solved with an integrating factor \(e^{\int a}\).

Linear equations with constant coefficients

\(L(y)=y^{(n)}+a_1y^{(n-1)}+\dots+a_ny\). Characteristic polynomial \(p(r)\); root \(r\) of multiplicity \(m\) gives \(e^{rx},xe^{rx},\dots,x^{m-1}e^{rx}\). The Wronskian tests independence. Nonhomogeneous equations by annihilators or variation of parameters.

Linear equations with variable coefficients

Existence of a basis of \(n\) solutions, Abel's formula \(W(x)=W(x_0)\exp\!\big(-\!\int_{x_0}^x a_1\big)\), reduction of order, variation of parameters. Legendre's equation solved by power series at an ordinary point.

Regular singular points

Euler equations \(x^2y''+axy'+by=0\), then the Frobenius method: try \(y=x^r\sum c_kx^k\), solve the indicial equation; roots differing by an integer can force a logarithm. Bessel's equation as the main example.

Existence and uniqueness: first order

For \(y'=f(x,y)\) with \(f\) continuous and Lipschitz in \(y\), Picard iteration

converges uniformly to the unique solution on a small interval. Without Lipschitz, uniqueness can fail: \(y'=3y^{2/3}\), \(y(0)=0\).

Existence and uniqueness: systems

The same proof in vector form for \(\mathbf y'=\mathbf f(x,\mathbf y)\), which covers \(n\)th-order equations. Linear systems have solutions on the whole interval where coefficients are continuous.

Calculus of finite differences

Difference operator \(\Delta y_k=y_{k+1}-y_k\), shift \(Ey_k=y_{k+1}\), so \(\Delta=E-1\). Factorial powers \(k^{(n)}=k(k-1)\cdots(k-n+1)\) satisfy \(\Delta k^{(n)}=nk^{(n-1)}\), the discrete power rule. Summation is anti-differencing.

First-order linear equations

\(y_{k+1}=ay_k+b\) has solution

Converges to the equilibrium iff \(|a|<1\); oscillates if \(a<0\). Application: the cobweb model of supply and demand.

Higher-order linear equations

Constant coefficients: try \(y_k=\lambda^k\), solve the characteristic equation. Repeated root gives \(k\lambda^k\); complex roots \(re^{\pm i\theta}\) give \(r^k\cos k\theta\), \(r^k\sin k\theta\) (damped oscillation if \(r<1\)). Particular solutions by undetermined coefficients.

Stability

Equilibrium is stable iff every characteristic root has \(|\lambda|<1\). For \(\lambda^2+a_1\lambda+a_2=0\) this is

Applications

Samuelson's multiplier-accelerator: \(Y_t=c(1+v)Y_{t-1}-cvY_{t-2}+G\), producing growth, decay or business cycles depending on \(c,v\). Harrod–Domar growth, stochastic learning models in psychology, and population and diffusion models in sociology.

The one-dimensional wave equation

d'Alembert's solution of \(u_{tt}=c^2u_{xx}\):

Characteristics \(x\pm ct=\)const, domain of dependence, finite speed of propagation, reflections at boundaries.

Classification and characteristics

Second-order equations \(Au_{xx}+2Bu_{xy}+Cu_{yy}+\dots\) are hyperbolic, parabolic or elliptic as \(B^2-AC\) is \(>0\), \(=0\), \(<0\). Which data give a well-posed problem depends on type.

Separation of variables and Fourier series

Product solutions, eigenvalue problems, convergence of the resulting series, and verifying that the series actually solves the PDE.

Laplace's equation and the maximum principle

Harmonic functions attain their max and min on the boundary, which gives uniqueness for the Dirichlet problem. Poisson's integral formula for the disk.

Eigenfunction expansions

Sturm–Liouville problems and their multidimensional analogs, completeness, and expansions in Bessel and Legendre functions.

Green's functions

Solve \(Lu=f\) as \(u(x)=\int G(x,\xi)f(\xi)\,d\xi\), where \(G\) solves the equation with a point source and the homogeneous boundary conditions.

Complex variables and conformal mapping

Analytic functions, Cauchy's theorem and integral formula, residues. Real and imaginary parts of analytic functions are harmonic, so conformal maps carry solutions of Laplace's equation between domains.

Fourier and Laplace transforms

Transforms turn derivatives into multiplication; inversion by contour integration. Heat equation on the line gives the Gaussian kernel \(u=\frac1{\sqrt{4\pi kt}}e^{-x^2/4kt}*f\).

Curves, surfaces and vector fields

Integral curves of vector fields and first integrals: the geometry that first-order PDEs are really about.

First-order linear and quasilinear equations

For \(a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u)\), solutions are surfaces made of characteristic curves

The Cauchy problem is solvable near an initial curve that is not characteristic. Shock formation in nonlinear equations like \(u_t+uu_x=0\).

Fully nonlinear first-order equations

Monge cones, characteristic strips, and complete integrals for \(F(x,y,u,p,q)=0\). Connection to Hamilton–Jacobi equations.

Cauchy–Kovalevskaya and classification

Analytic Cauchy problems have unique local analytic solutions. Second-order classification, canonical forms, and characteristic surfaces in higher dimensions.

Laplace, wave and heat equations

Harmonic functions and the mean value property, the wave equation in 1, 2 and 3 dimensions (Huygens' principle holds in odd dimensions ≥ 3), and the heat equation with its maximum principle and smoothing.

Introduction and classification

Classification, canonical forms, the Cauchy–Kovalevskaya theorem, and Hadamard's example showing analytic solvability is not enough: for Laplace's equation, small Cauchy data can produce huge solutions, so the problem is ill-posed.

Hyperbolic equations

The Cauchy problem for the wave equation, energy integrals and uniqueness, domains of dependence. Kirchhoff's formula in 3D and Poisson's in 2D (method of descent).

Elliptic equations

Harmonic functions, Dirichlet and Neumann problems, Green's function, single- and double-layer potentials. Reducing boundary problems to integral equations, then the Fredholm alternative:

Parabolic equations

The heat equation, maximum principle, uniqueness, and the Cauchy problem solved by the heat kernel.

Derivations and the wave equation

Physical derivations of the classical equations, then the wave equation in one and several dimensions, energy, and Huygens' principle.

Fourier series and integrals

Convergence theory and transform methods applied to boundary and initial value problems.

The heat equation

Maximum principle, the heat kernel on \(\mathbb R^n\), and problems on bounded domains.

Laplace's equation and potential theory

Mean value property, Harnack's inequality, Green's functions, Newtonian potentials, and \(\Delta\big(\tfrac{-1}{4\pi|x|}\big)=\delta\) in \(\mathbb R^3\).

Integral equations

Fredholm equations of the second kind, the Fredholm alternative, symmetric kernels and the Hilbert–Schmidt theorem: a symmetric compact kernel has an eigenfunction expansion \(K(x,y)=\sum\lambda_n\phi_n(x)\phi_n(y)\) in the \(L^2\) sense. Volterra equations always have unique solutions.

Variational methods and eigenvalues

Dirichlet's principle, weak solutions, and Rayleigh–Ritz approximation of eigenvalues by minimizing the Rayleigh quotient.

Classical PDE review

Laplace, heat and wave equations, separation of variables, Green's functions, characteristics. Moves quickly if you have read Weinberger.

Hilbert spaces and distributions

\(L^2\), orthogonal projection, Riesz representation, generalized functions and weak derivatives.

Sobolev spaces and weak solutions

\(H^1_0(\Omega)\): functions vanishing on the boundary with square-integrable weak gradient. The weak form of \(-\Delta u=f\) is

Lax–Milgram: a bounded coercive bilinear form gives a unique weak solution.

Operator and spectral theory

Unbounded operators, self-adjointness, compact operators, and the spectral theorem in infinite dimensions. Eigenfunction expansions become theorems about self-adjoint operators with compact resolvent.

Numerical methods

Finite differences and finite elements. FEM is the weak form restricted to a finite-dimensional subspace (Galerkin), so error analysis comes straight from the Hilbert space theory.

Preliminaries

Fourier series, integral transforms, and an introduction to integral equations as tools used throughout.

First-order equations and classification

Characteristics, the Cauchy problem, the Cauchy–Kovalevskaya theorem, and reduction of second-order equations to canonical form.

The heat equation

Initial and boundary value problems, the heat kernel, maximum principle, and integral-equation methods for boundary problems.

Laplace's equation and potential theory

Green's functions, layer potentials, and Fredholm theory applied to the Dirichlet and Neumann problems.

The wave and Helmholtz equations

Time-harmonic waves \(u(x,t)=\mathrm{Re}\{v(x)e^{-i\omega t}\}\) reduce the wave equation to \(\Delta v+k^2v=0\). Radiation conditions, scattering by obstacles, and the setup of inverse scattering problems.

Nonlinear algebraic and operator equations

Newton's method \(x_{k+1}=x_k-J(x_k)^{-1}F(x_k)\) with quadratic convergence near a simple root. Contraction mapping theorem: if \(\|T x-Ty\|\le q\|x-y\|\), \(q<1\), on a complete space, then \(T\) has a unique fixed point and iteration converges geometrically. Brouwer and Schauder fixed point theorems for existence without uniqueness.

Nonlinear difference equations

Fixed points and their stability via \(|f'(x^*)|<1\); period doubling in maps like the logistic map \(x_{k+1}=rx_k(1-x_k)\).

Nonlinear ordinary differential equations

Phase plane, linearization at equilibria (node, saddle, focus, center), Lyapunov functions for stability, limit cycles and the Poincaré–Bendixson theorem, perturbation and averaging methods.

Nonlinear integral and functional equations

Hammerstein and Urysohn integral equations solved by iteration or fixed-point arguments; equations where the unknown is a function of transformed arguments.

Nonlinear partial differential equations

Examples such as Burgers' equation (solvable by the Cole–Hopf transform into the heat equation) and the role of similarity solutions.

Riccati equation

\(y'=P(x)+Q(x)y+R(x)y^2\). The substitution \(y=-u'/(Ru)\) turns it into a linear second-order equation. Knowing one solution reduces it to a linear first-order one.

Elliptic functions and the pendulum

\(\theta''+\frac gL\sin\theta=0\) is solved by Jacobi elliptic functions. The period with amplitude \(\theta_0\) is

which reduces to \(2\pi\sqrt{L/g}\) for small swings.

Painlevé transcendents

Second-order equations whose only movable singularities are poles reduce to known functions or to six new ones, the Painlevé transcendents. Example: \(y''=6y^2+x\) (P-I).

Nonlinear oscillations and the phase plane

Duffing \(x''+\delta x'+\alpha x+\beta x^3=\gamma\cos\omega t\) (amplitude-dependent frequency, jump phenomena) and van der Pol \(x''-\mu(1-x^2)x'+x=0\) (a stable limit cycle). Singular points classified by linearization.

Population models

Logistic growth \(N'=rN(1-N/K)\) and Volterra's predator-prey system with closed orbits around the coexistence equilibrium.

Nonlinear integral equations and numerics

Volterra-type nonlinear integral equations, existence by successive approximation, and numerical integration of nonlinear ODEs.