The theorem says that the order of two differentiations does not matter, when $f$ is regular enough. For $f(x, y) = x^3 y^2$, differentiating first in $x$ and then in $y$ gives $6 x^2 y$, and the other order gives $6 x^2 y$ too.

At each step, we compute the slope at the current point and walk a short distance $h$ along it. The result is a broken line that stays close to the true solution when $h$ is small. For $y' = y$, $y(0) = 1$, the method gives $y_k = (1 + h)^k$, and with $h = 1/k$ this tends to $e$ at time $1$.

The integers modulo $n$ form the ring $\mathbb{Z}/n\mathbb{Z}$ of the $n$ residue classes modulo $n$, with the addition and the multiplication of representatives. The definition of the integers modulo n gives the precise form.

The p-norms give different ways to measure the length of a vector from its coordinates. For $p = 1$, you add the absolute values. For $p = 2$, you get the usual length. For $p = \infty$, you take the largest coordinate.

The rate of growth is the square of the size, so the growth feeds itself very fast. The solution reaches infinity in finite time, unlike exponential growth.

A block diagonal matrix is a square block matrix whose diagonal blocks are square and whose other blocks are all zero.

For each prime $p$, some number $g$ has powers that run through all of $1, 2, \dots, p - 1$ modulo $p$. For $p = 11$, the number $2$ works: its powers are $2, 4, 8, 5, 10, 9, 7, 3, 6, 1$.

The set difference of two sets is the set of the elements of the first set that are not elements of the second set.

If we know the lengths of all vectors, then we also know all inner products. In the real case, we only need the lengths of the two diagonals $x + y$ and $x - y$. So lengths determine angles.

A decimal expansion writes a nonnegative real number as an integer part and an infinite sequence of digits after the decimal point. It represents the number as a sum of multiples of powers of $10$. For example, $1/4 = 0.25000\dots$ and $1/3 = 0.333\dots$.

A recursive rule tells you how to get the next term from the terms you already have. Start with $a_0 = 1$ and the rule $a_{n+1} = 2 a_n$, and you get $1, 2, 4, 8, \dots$.

Exchanging rows and columns does not change the determinant. So we can expand, reduce or test a determinant by columns as well as by rows. Every rule for rows has a twin rule for columns.

The slope field of a first-order equation $y' = f(x, y)$ is a picture with a short segment of slope $f(x, y)$ at many points $(x, y)$ of the plane. The graphs of the solutions are tangent to these segments. See the definition.

A linear subspace of a vector space is a subset that contains the zero vector and is closed under addition and scalar multiplication. It is a vector space with the same operations.

The coordinate space $F^n$ is the space of lists of $n$ numbers. We add two lists position by position, and we scale a list by scaling each entry. Every finite-dimensional vector space looks like one of these spaces after we choose a basis.

Over a field, you can divide any polynomial by any nonzero polynomial and get a remainder of smaller degree. This is the same situation as division with remainder for integers. So all the theory of the Euclidean algorithm works for polynomials over a field.

A closed set in a topological space is a set whose complement is open. In a metric space, the closed sets are the sets that contain the limits of all their convergent sequences.

When you add three numbers, you can add the first two first or the last two first. For example, $(2 + 3) + 4 = 5 + 4 = 9$ and $2 + (3 + 4) = 2 + 7 = 9$.

The rational canonical form is a standard representative of each similarity class, over any field. It uses companion matrices instead of Jordan blocks. It needs no roots of polynomials, only divisibility of polynomials.

The transitive closure of a relation is the smallest transitive relation that contains it.