Integration by parts is the product rule read backwards. It moves the derivative from one factor to the other, at the cost of a boundary term.

Two disjoint cycles move different points. Each cycle does its work without touching the points of the other cycle. So the order in which you do them does not matter.

An exhaustion function is like a height function on a bowl. Each region below a given height is a compact piece, and the pieces fill up the whole space as the height grows. Asking the function to be plurisubharmonic means that it is subharmonic on every complex line, a complex form of convexity. On $\C^n$ the function $\abs{z}^2$ is a plurisubharmonic exhaustion function: its sublevel sets are balls. On the unit disc, $-\log(1 - \abs{z}^2)$ is one: it is subharmonic and blows up at the circle.

In a polygon or a solid in space, two corners are usually not joined by an edge. In four or more dimensions this can change: there are four-dimensional solids where every pair of corners is joined by an edge. Such shapes are called neighborly, because every corner is a neighbour of every other. In higher dimensions the same happens with triples, quadruples and so on of corners.

Sometimes a formula is written down only for nice inputs, and the resulting operator is not closed. One then tries to enlarge the domain by taking limits, keeping the rule that limits of inputs go with limits of outputs. This works when the same input limit can never come with two different output limits. For example, differentiation on polynomials, in the space $C[0, 1]$ with the supremum norm, is closable, and its closure is differentiation on continuously differentiable functions.

Take a rubber sheet, glue two opposite sides to make a tube, then glue the two ends of the tube with a flip. In ordinary space the flip forces the tube to pass through its own wall, which gives the familiar glass bottle picture. The result has no inside and no outside: a walker on it can come back mirror-reversed.

A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of faces at each vertex. There are exactly five: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.

The adjoint equation swaps the roles of the two variables of the kernel, like transposing a matrix. For complex kernels one also takes complex conjugates, as in the conjugate transpose of a matrix. The original equation and its adjoint have solvability properties that mirror each other. For example, the adjoint of $\varphi(x) = f(x) + \lambda \int_0^1 x \, \varphi(s) \, ds$ is $\psi(x) = g(x) + \bar\lambda \int_0^1 s \, \psi(s) \, ds$.

In a linear model with well-behaved noise, the least squares fit is the most precise estimator among those that are linear in the data and correct on average. "Best linear unbiased estimator", or BLUE, is the short name for this property. It is the main reason why least squares is the default method for fitting linear models.

Every smooth manifold, however it was defined, can be realised as a smooth shape inside an ordinary Euclidean space. Twice its dimension is always enough room. So abstract manifolds are no more general than the curved surfaces and solids that sit inside some $\R^N$.

These identities are the algebra rules of vector calculus. They let you expand derivatives of products and simplify repeated derivatives, the way the ordinary product rule does in one variable.

In one variable, a power series converges on a disc, and the disc is fixed by one number, the radius of convergence. In several variables, the region of convergence can have many shapes. The theorem says which shapes can occur. They are symmetric under rotating each coordinate, they are "filled in" towards the origin, and they become convex after taking logarithms of absolute values.

Some problems amplify small errors in the data enormously, so that the naive answer is useless. Undoing a blur in a photograph is an example: tiny noise gets magnified into wild patterns. Regularization adds a mild preference, such as "the answer should not be too large or too rough", which tames this amplification. The strength of the preference is tuned to the amount of noise. Too little regularization leaves the noise in, and too much smooths away the real details.

The sum $\frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \cdots$ becomes larger than each number that you choose, if you take enough terms. It grows very slowly: the terms for the primes up to $10^9$ still give a sum less than $4$.

Spaces can be suspended, but not desuspended. In the stable homotopy category every object can also be desuspended, so shifting up and down are inverse operations. Maps between objects form abelian groups, much as linear maps between vector spaces do. For two finite CW complexes, the maps between their suspension spectra are exactly the stable homotopy classes of maps between them. For example, the maps from the sphere spectrum to itself form the group $\Z$, given by degree.

The adjoint of a linear map between finite-dimensional inner product spaces is a linear map in the opposite direction. It moves the map from one side of the inner product to the other.

Take a linear map with constant gradient $\xi$ on a region, and wiggle it inside while keeping its boundary values. Quasiconvexity says that the wiggled map always has at least as much energy as the linear one. So, on average, an affine map is the cheapest way to achieve given affine boundary values. Convex functions have this property by Jensen's inequality, because the average of $\nabla \varphi$ over the region is zero. The condition is hard to check in general, because it is not local in $\xi$: it involves all test maps $\varphi$.

A Steiner system is a way to group points into blocks of equal size so that any choice of $t$ points determines exactly one block. With $t = 2$, two points always lie together in exactly one block, like two points on exactly one line. The Fano plane is a Steiner system $S(2, 3, 7)$: $7$ points, $7$ blocks of size $3$, and every pair of points in one block.

If events grow step by step, their probabilities grow toward the probability of the limit event. For example, the chance that a fair coin shows heads at least once in the first $n$ tosses is $1 - 2^{-n}$. It tends to the chance $1$ of heads at some toss.

A loss function says how bad a guess is when the truth is known. Guessing $7$ when the truth is $5$ costs $(7 - 5)^2 = 4$ under squared error loss, and $\abs{7 - 5} = 2$ under absolute error loss. Different losses express different priorities: squared error punishes large mistakes heavily, while a $0$-$1$ loss only cares whether the answer is right. The choice of loss changes which estimator is best.