change
45 theorems carry this thread across the degree.
The chain rule
The derivative of a composition is the product of the derivatives.
Rolle's theorem
A differentiable function equal at two points has a stationary point between them.
The mean value theorem
A differentiable function attains its average rate of change at some interior point.
The extreme value theorem
A continuous function on a closed interval attains a maximum and a minimum.
The fundamental theorem of calculus
Differentiation and integration are inverse operations.
Taylor's theorem with remainder
A smooth function equals its Taylor polynomial plus a controllable error term.
L'Hôpital's rule
Indeterminate limits of ratios can be resolved by differentiating numerator and denominator.
The algebra of limits
Limits respect sums, products, and quotients.
The squeeze theorem
A sequence trapped between two convergents to L also converges to L.
The monotone convergence theorem
A bounded monotone sequence converges.
The Bolzano–Weierstrass theorem
Every bounded sequence has a convergent subsequence.
Completeness of the reals
A sequence of reals converges if and only if it is Cauchy.
The ratio test
A series converges absolutely when the limiting ratio of terms is below one.
The alternating series test
An alternating series with terms decreasing to zero converges.
The intermediate value theorem
A continuous function takes every value between two of its values.
The Heine–Borel theorem
A subset of R^n is compact iff it is closed and bounded.
Uniform continuity on compact sets
Continuous on a compact set implies uniformly continuous.
The Riemann integrability criterion
A bounded function is integrable iff upper and lower sums can be made arbitrarily close.
Uniform limits of continuous functions
A uniform limit of continuous functions is continuous.
The Weierstrass M-test
A dominated series of functions converges uniformly.
Change of variables and the Jacobian
How volumes transform under a smooth change of coordinates.
The method of Lagrange multipliers
Constrained extrema occur where gradients align.
Green's theorem
A planar circulation integral equals a double integral of curl.
Stokes' theorem
Circulation around a boundary equals the flux of curl through the surface.
The divergence theorem
Flux through a closed surface equals the integral of divergence inside.
The Picard–Lindelöf theorem
A Lipschitz ODE has a unique local solution.
Superposition for linear ODEs
Solutions of a linear equation form a vector space.
The Wronskian and independence
A non-vanishing Wronskian certifies linear independence of solutions.
Variation of parameters
A particular solution from the homogeneous solutions.
Linear stability and the phase plane
Eigenvalues of the linearisation classify equilibria.
The Cauchy–Riemann equations
Complex differentiability is equivalent to a pair of PDEs.
Cauchy's integral theorem
The integral of a holomorphic function round a closed loop is zero.
Cauchy's integral formula
A holomorphic function's values are determined by its boundary values.
Liouville's theorem
A bounded entire function is constant.
The fundamental theorem of algebra
Every non-constant polynomial over C has a root.
The residue theorem
A contour integral equals the sum of enclosed residues.
Separation of variables
Reducing a PDE to ODEs via product solutions.
d'Alembert's solution
The 1D wave equation solved by travelling waves.
The maximum principle
A harmonic function attains its extrema on the boundary.
Convergence of Fourier series
When and how a periodic function equals its Fourier series.
The method of characteristics
First-order PDEs solved along characteristic curves.
Convergence of fixed-point iteration
Contraction guarantees convergence of iterative schemes.
Convergence of Newton's method
Quadratic convergence near a simple root.
Lagrange interpolation and its error
The unique interpolating polynomial and its remainder.
Gaussian quadrature
Optimal node placement integrates high-degree polynomials exactly.