At a constrained extremum, the derivative of the objective is a combination of the derivatives of the constraints. In infinite dimensions there may be infinitely many constraints, so the coefficients form a continuous linear functional on the constraint space instead of a list of numbers.

The vector projection of a vector $a$ onto a nonzero vector $b$ is the multiple of $b$ that is nearest to $a$. It is the shadow of $a$ on the line of $b$, under light that falls perpendicular to that line.

The first root of unity after $1$, going counterclockwise around the circle, is a primitive one. Its powers walk around the circle step by step and visit every $n$th root of unity before they return to $1$.

The same group can be written with very different generators and rules. For example, $\langle a, b \mid a^2, b^3, a b a^{-1} b^{-1} \rangle$ and $\langle c \mid c^6 \rangle$ both define the cyclic group of order $6$, with $c = ab$. The isomorphism problem asks whether a computer can always tell when two descriptions define the same group. In general it cannot: no program can even decide whether a given description defines the group with one element.

In infinite dimensions a sequence can tend to zero in a weak sense while its points stay far from zero. The same happens to sequences of functionals. In many spaces, two such sequences can still meet: the values $f_n(x_n)$ need not go to zero. In $\ell^2$, the unit vectors $e_n$ and the functionals "take the $n$-th coordinate" are both weakly null, yet each value is $1$. Spaces with the Dunford–Pettis property forbid this.

A function is superadditive when the whole is worth at least the sum of its parts. The square function on $[0, \infty)$ is an example: $(3 + 4)^2 = 49$ is more than $3^2 + 4^2 = 25$. The number of pairs that can be formed from a group of people is another: two groups merged have all the old pairs and some new ones.

The trefoil is the simplest real knot: tie an overhand knot in a string and glue the ends together. Any picture of it has at least three crossings, and the standard picture has three lobes like a clover leaf. It comes in two mirror forms, left-handed and right-handed, and these cannot be moved into each other.

In projective space, any set cut out locally by holomorphic equations is cut out globally by polynomial equations. So every compact complex submanifold of projective space, however it is presented, is an algebraic variety. This is a striking rigidity: holomorphic functions are much more flexible than polynomials, but the compactness of projective space forces algebraicity.

A wavelet is a small wave: it wiggles up and down and then dies out. Stretching it gives slow wiggles for coarse features, and squeezing it gives fast wiggles for fine detail. Sliding these copies along the line, one can match the features of a signal wherever they occur. The simplest wavelet is the Haar wavelet, which is $+1$ on $[0, \tfrac12)$, $-1$ on $[\tfrac12, 1)$ and $0$ elsewhere.

In a linear integral equation the unknown is multiplied by a kernel and then integrated. In a Urysohn equation the kernel may depend on the unknown in any way, for example through $\sin(x\, \varphi(t))$. The equation still says that the unknown at a point is a weighted total of its values elsewhere. For instance, $\varphi(x) = x$ solves $\varphi(x) = x + \int_0^1 \sin\bigl(x(\varphi(t) - t)\bigr)\, dt$, since the sine is then $0$.

However you colour the positive integers with finitely many colours, one colour contains arithmetic progressions of every length. The theorem says more: for a given length and number of colours, a fixed finite stretch $1, \ldots, N$ already suffices.

Unlike sine and cosine, there is no minus sign here. Each of $\sinh$ and $\cosh$ is the derivative of the other.

The clique number of a finite graph is the size of its largest clique. It is a lower bound for the chromatic number. The node clique number gives the precise definition.

Each product $hk$ can be written in several ways, because you can move an element of $H \cap K$ from one factor to the other. Each element of $HK$ has exactly $|H \cap K|$ such ways. So you divide the number of pairs, $|H| \, |K|$, by $|H \cap K|$.

Think of $\mu$ as a distribution of electric charge along the real line. The Stieltjes transform at a point $z$ of the plane adds up the contributions $1/(t - z)$ of all charges, a kind of complex potential. When $z$ comes close to the real line at a point $x$, the imaginary part feels mostly the charge near $x$, so it reveals the density of $\mu$ there. For one unit charge at $0$, the transform is $-1/z$.

To decide whether a system of differential equations has many solutions, Cartan counts how many conditions each step of building a solution imposes. The test compares this count with the actual number of conditions defining the possible tangent planes of solutions. If the two agree, the system is in good shape: no hidden conditions will appear later.

Think of a ball in a bowl that is shaken by an outside force. Without shaking, the ball rolls back to the bottom. With shaking, the ball stays close to the bottom, and the distance depends on how hard the shaking is. For the system $\dot x = -x + u$, the state satisfies $\abs{x(t)} \le e^{-t} \abs{x(0)} + \sup \abs{u}$. So the effect of the start dies out, and what remains is bounded by the input.

Many problems ask for the solutions of an equation that depends on a parameter, such as the shape of a beam under a load. Usually the solution changes smoothly with the parameter, but at special values several solutions can appear. The Lyapunov–Schmidt method splits the unknown into a few "dangerous" directions and all the others. The other directions can be solved for, so only the few dangerous directions are left. The problem then becomes a question about a small number of equations, which can often be solved by hand.

A Lie group is a group whose elements form a smooth space, like the rotations of the plane or of space. Smoothness lets one use calculus on the group. Hilbert asked whether smoothness is automatic: if a group is only known to look locally like ordinary space, with continuous operations, must it already be smooth? The answer turned out to be yes, so for such groups continuity forces smoothness.

Cylindrical coordinates describe a point of space by the polar coordinates of its projection to the $xy$-plane and by its height $z$ above that plane. They simplify problems with symmetry about an axis, such as integrals over cylinders and cones.