Analysis

The value a function approaches as the variable approaches some value.

A function is continuous at a point if the limit equals the function value at that point.

The instantaneous rate of change of a function, or slope of the tangent line.

Differentiation of composite functions: derivative of outer times derivative of inner.

Derivative formulas for trigonometric functions.

Derivative formulas for exponential and logarithmic functions.

Derivatives of derivatives. The second derivative relates to curvature.

The reverse of differentiation, finding the original function from its derivative.

Represents the area between a function and the x-axis, defined as a limit.

The reverse of the chain rule, simplifying complex integrals.

The reverse of the product rule, for integrating products of functions.

A series expansion that represents a function as an infinite polynomial.

Differentiating a multivariable function with respect to one variable while holding others constant.

Integration over multiple variables, used to compute volume, mass, etc.

Studies phenomena where qualitative properties of dynamical systems change abruptly with parameter variation. Includes saddle-node, Hopf bifurcations.

Studies irregular behavior in deterministic systems that is extremely sensitive to initial conditions. Lorenz equations and logistic map are classic examples.

Exponents measuring the rate at which nearby trajectories separate. Positive Lyapunov exponents characterize chaos.

Chaotic attractors with fractal structure. Lorenz and Hénon attractors are classic examples, exhibiting self-similar structure.

Dynamics of conservative systems described by Hamiltonian functions. Preserves symplectic structure in phase space, encompassing both integrable and chaotic systems.

Studies long-term average behavior of dynamical systems using measure theory. The equality of time and space averages is the key concept.

An invariant manifold tangent to neutral directions (zero eigenvalues) near a fixed point. Essential for bifurcation analysis and dimension reduction.

A technique reducing continuous dynamical systems to discrete maps. Essential for analyzing periodic orbits and studying chaos.

Expansion of periodic functions as series of trigonometric functions. Convergence conditions, Gibbs phenomenon, and Parseval's theorem are key topics.

Transforms functions to frequency domain. Schwartz space, Plancherel theorem, and inverse transform are essential.

Continuous linear functionals on test function spaces. Allows differentiation of singular objects like Dirac delta.

A family of localized oscillating functions. Complements limitations of Fourier analysis in time-frequency analysis.

Eigenfunctions of the Laplacian on the sphere. Form an orthonormal basis for functions on the sphere.

Integral operators with singular kernels. Hilbert transform and Calderón-Zygmund theory are central.

Generalizes Fourier analysis to locally compact abelian groups. Pontryagin duality is the key theorem.

Theory decomposing functions by frequency bands. Essential for characterizing function spaces and harmonic analysis.