A product metric is a metric on the Cartesian product of finitely many metric spaces, made from the metrics of the factors. Common choices take the maximum, the sum, or the Euclidean combination of the distances in the factors. All these choices give the product topology.

Try to form "the set of all sets that do not contain themselves". Ask whether this set contains itself: both answers lead to a contradiction. So some collections are too large to be sets.

The Hessian matrix of a real-valued function of $n$ variables at a point is the $n \times n$ matrix of its second partial derivatives at that point. It describes the curvature of the function, and it is the main tool of the second derivative test for local extrema.

If you measure in centimeters instead of meters, every value is multiplied by $100$. The variance is then multiplied by $100^2 = 10000$, because it is measured in squared units.

In the integers, Euclid's lemma says that a prime that divides a product divides a factor. In a unique factorisation domain, the same thing is true for every irreducible element. In $\mathbb{Z}[\sqrt{-5}]$ this fails, and so factorisation is not unique there.

A series converges conditionally when it converges but the series of absolute values diverges. The sum of such a series depends on the order of its terms. The node conditional convergence gives the precise definition.

A point outside the closed ball is at some positive distance beyond its edge. A small ball about that point stays outside, so the outside is open.

If a series of nonnegative terms is smaller term by term than a series with a finite sum, it has a finite sum too. If a smaller series already has an infinite sum, the larger one has an infinite sum as well.

The number of Sylow $p$-subgroups is one more than a multiple of $p$. This simple fact, together with a divisibility condition, leaves only a few possible values. Often the only possible value is $1$, and then the Sylow subgroup is normal.

The indefinite integral of a function is the family of all its antiderivatives on an interval. It is written as one antiderivative plus an arbitrary constant. The node indefinite integral gives the precise definition.

On the unit circle, $dz$ is $i z \, d\theta$, so $dz / z$ is just $i \, d\theta$. Going once around adds up $i \, d\theta$ over a full turn of $2\pi$, which gives $2 \pi i$.

In a principal ideal domain, every irreducible element behaves like a prime number. That is the key to unique factorisation. So every Euclidean domain, such as the integers, has unique factorisation.

The second derivative of a function is the derivative of its derivative. It measures how the slope changes, and so how the graph bends. The node second derivative gives the precise definition.

To show that two collections have the same size, you can pair each element of one with exactly one element of the other. For example, the subsets of $\{1, 2, 3\}$ pair with the words of length 3 in the letters 0 and 1. A subset gives the word with 1 in the positions of its elements.

On a disc, each holomorphic function is the derivative of another holomorphic function. You can build it by integrating along straight lines from the center, as in the real case $F(x) = \int_c^x f(t) \, dt$.

A solution of a polynomial congruence modulo $p$ often gives solutions modulo $p^2$, $p^3$, and so on. The condition $f'(r) \not\equiv 0 \pmod p$ says that $r$ is a simple root modulo $p$. Under this condition, each step of the lifting has exactly one answer.

Replace the graph by its tangent line, and find where the line crosses the axis. Repeat from there. Near a simple root, each step roughly doubles the number of correct digits.

A linear equation with constant coefficients is a linear differential equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = b(x)$ with real constants $a_0, \dots, a_{n-1}$. The roots of its characteristic polynomial give all solutions of the homogeneous equation. See the definition.

A metric space is a set with a distance function between its points. The distance is zero only from a point to itself, it is symmetric, and it satisfies the triangle inequality. Metric spaces are the setting for limits, continuity and completeness beyond the real line.

Start with a $1$ at the top. Each new row starts and ends with $1$, and each inside number is the sum of the two numbers above it. Row 4 is $1, 4, 6, 4, 1$, and these are the coefficients of $(x + y)^4$.