For numbers, $e^x e^y = e^{x + y}$. For matrices, and in Lie groups generally, this rule fails because the order of multiplication matters. The formula says how to repair it: the exponent of the product is $X + Y$ plus correction terms built only from commutators $[X, Y] = XY - YX$ and commutators of commutators.

A rearrangement of a series is the series of the same terms taken in a new order, given by a bijection of the indices. Rearrangements keep the sum of an absolutely convergent series. The node rearrangement of a series gives the precise definition.

A function in the plane that only depends on the distance to the origin has a Fourier transform with the same symmetry. Its two-dimensional Fourier transform then reduces to a one-dimensional integral over the radius, with the Bessel function $J_0$ as kernel. That integral is the Hankel transform of order $0$, which is why Hankel transforms appear in every problem with circular symmetry.

Think of $u(x, t)$ as the temperature at the place $x$ at the time $t$ in a body where heat simply spreads out. A caloric function is any temperature profile that obeys the law of heat flow. A harmonic function that does not depend on time is caloric, since it describes a steady temperature. A simple example in one space variable is $u(x, t) = x^2 + 2t$: here $\partial_t u = 2$ and $\partial_x^2 u = 2$. Another is $u(x, t) = e^{-t} \sin x$, a wave of heat that fades as time goes on.

Start with the average of a function over $[0, 1]$. Then record how the average over the left half differs from the average over the right half. Then do the same inside each half, inside each quarter, and so on. Each step uses one function that is $+1$ on one half of a dyadic interval and $-1$ on the other half, scaled to have length $1$ in $L^2$. These numbers rebuild the function step by step, with finer and finer detail.

The secant method draws a line through two points of the graph and takes its crossing with the axis. Muller's method draws a parabola through three points and takes its crossing nearest to the last point. A parabola can miss the real axis, and then the method moves into the complex numbers, which lets it find complex roots.

A Hamiltonian path is a tour that visits every city exactly once, without needing to come back home. On a chessboard, a knight's tour that visits every square once is a Hamiltonian path in the knight's move graph. The path graph $1 - 2 - 3 - 4$ is itself a Hamiltonian path, while a star with three arms has none.

A Rees matrix semigroup is like a grid of copies of a group. The rows are labelled by $I$ and the columns by $\Lambda$. When you multiply two elements, you keep the row of the first and the column of the second. You multiply the group parts with a twist taken from the matrix $P$. If the group is trivial, a product just keeps the row of the left factor and the column of the right factor.

Given two functions $f$ and $g$ from $X$ to $Y$, glue $f(x)$ to $g(x)$ in $Y$ for every $x$, and keep gluing until the identifications are consistent. The result is a quotient set $Q$ of $Y$, and $q$ sends each element of $Y$ to its glued class. For example, let $X = \{*\}$, $Y = [0, 1]$, $f(*) = 0$ and $g(*) = 1$. The coequalizer glues the two ends of the interval, and with the quotient topology it gives a circle.

Remove the root of a binary tree, and you get a left tree and a right tree. The sizes of the two trees add up to one less than the size of the original tree, and this gives the Catalan recurrence.

At a point of a surface, the bending changes with the direction you look in. Euler's theorem says that this change follows a very simple rule. Once you know the strongest and the weakest bending, and their directions, you know the bending in every other direction.

The invertible $2 \times 2$ matrices form a group, and both the product and the inverse are given by formulas in the entries. So the group is also a geometric object, the set where the determinant is nonzero. An algebraic group is any group of this kind, where the elements form a variety and the group operations are polynomial maps. Other examples are the numbers under addition, the nonzero numbers under multiplication, and the points of an elliptic curve.

Suppose you measured the gold content of rock at a few drill holes and want a guess for a new spot. Nearby holes should count more than far ones, and holes that cluster together should not be counted twice. Kriging chooses the weights to make the expected squared error of the guess as small as possible. At a drill hole itself, kriging returns the measured value, so it interpolates the data. It also gives the size of the expected error, which is larger far from the data.

Throw points at random over the plane, with no point influencing any other. Around each point draw a shape, such as a disc whose radius is also chosen at random. The Boolean model is the union of all these shapes, a random pattern of overlapping blobs. If all discs have radius $r$ and there are on average $\lambda$ centres per unit area, then each fixed point of the plane is uncovered with probability $e^{-\lambda \pi r^2}$. With $\lambda \pi r^2 = 1$, about $63\%$ of the plane is covered.

Think of a ball resting in a valley of a landscape that slowly changes shape. Most of the time the ball moves smoothly as the valley shifts. But if the valley flattens out and disappears, the ball suddenly rolls to another valley far away. Catastrophe theory describes the few typical ways this can happen. In the cusp catastrophe, for example, two valleys exist for some settings and the jump between them depends on the direction from which you approach.

The de Rham operator $d + d^*$ sends forms of even degree to forms of odd degree. Its solutions are the harmonic forms, which by Hodge theory count the holes of the manifold in each dimension. So its index is the alternating count of holes, the Euler characteristic. This is the simplest case where the index theorem recovers a classical theorem, here Gauss–Bonnet.

The circle $x^2 + y^2 = 1$ has many points, but only some have both coordinates rational, such as $(3/5, 4/5)$. These are its rational points over $\Q$. Each one corresponds to a Pythagorean triple, here $3^2 + 4^2 = 5^2$. The circle $x^2 + y^2 = 3$ has real points but no rational points at all. Which curves and surfaces have rational points, and how many, is a deep question.

A linear form on a vector space is a linear map from the space to its field of scalars.

Think of $\mathcal{C}$ as a base, and above each point $c$ of the base sits a whole category $F(c)$, called the fibre. The Grothendieck construction puts all the fibres side by side and adds arrows that move from one fibre to another, following arrows of the base. An object of $\int F$ is a pair: a base object $c$ and an object $x$ of the fibre over $c$. For a small example, if $F$ is constant with value a category $\mathcal{D}$, then $\int F$ is the product category $\mathcal{C} \times \mathcal{D}$. If every fibre is a set with only identity arrows, the construction gives the category of elements of $F$.

In many physical problems one quantity is of main interest, but another one comes with it, such as the flow of heat together with the temperature. The standard finite element method computes the temperature and gets the heat flow by differentiating, which loses accuracy. A mixed method computes both at the same time, as separate unknowns linked by the equations. For the Poisson equation $-\Delta u = f$, one introduces the flux $\sigma = \nabla u$ and solves for the pair $(\sigma, u)$ with $\nabla \cdot \sigma = -f$.