# Miscellaneous Thoughts

Last Modified 2022-03-20

If a first-order ODE can be written in the form

for some functions * separable*. Such an ODE admits a one-parameter family of solutions

where

If a first-order ODE can be written in the form

for some functions * linear*. Such an ODE admits a one-parameter family of solutions

where * homogeneous*, and its corresponding family of solutions simplifies to

If a first-order ODE can be written in the form

where * exact*. Such an ODE admits a one-parameter family of solutions obtained by solving the equation

for

The method of * variation of parameters* is used to construct a particular solution to an inhomogeneous linear ODE

when we already know a fundamental set of complementary solutions

In particular, variation of parameters tells us that

is a solution to the inhomogeneous ODE, where the functions

**Definition: Relation, Binary Relation, $a \mathrel{R} b$**

Let * relation* between

*, is a subset of the Cartesian product*

**binary relation****Definition: Reflexive, Symmetric, Antisymmetric, Transitive**

Let

- A binary relation
$R \subseteq A \times A$ isif**reflexive**$a \mathrel{R} a$ for all$a \in A$ . - A binary relation
$R \subseteq A \times A$ isif**symmetric**$a \mathrel{R} b$ implies$b \mathrel{R} a$ for all$a, b \in A$ . - A binary relation
$R \subseteq A \times A$ isif**antisymmetric**$a \mathrel{R} b$ and$b \mathrel{R} a$ implies$a = b$ for all$a, b \in A$ . - A binary relation
$R \subseteq A \times A$ isif**transitive**$a \mathrel{R} b$ and$b \mathrel{R} c$ implies$a \mathrel{R} c$ for all$a, b, c \in A$ .

**Definition: Equivalence Relation**

An * equivalence relation* on a set

**Definition: Converse, Inverse**

Let * converse* or

*of a relation*

**inverse**In other words,

**Definition: Function, Map, Domain, Codomain, $f: A \to B$, $f(a)$, Maps**

Let * function* from

*, is a relation*

**map***of*

**domain***of*

**codomain**

**maps****Definition: Injective, Injection, $f: A \injto B$**

Let * injective* if

*, and we write*

**injection**In other words, a function is injective if it maps distinct elements of its domain to distinct elements of its codomain.

**Definition: Surjective, Surjection, $f: A \surjto B$**

Let * surjective* if, for every

*, and we write*

**surjection****Definition: Bijective, Bijection, $f: A \bijto B$**

Let * bijective* if it is both injective and surjective. We call a bijective function a

*, and we write*

**bijection****Only Bijections have Inverses**

**Theorem:** Let

*Proof:* Suppose

Conversely, suppose

**Definition: Composition, $g \circ f$**

Let * composition* of two functions

**Composition Preserves Injectivity and Surjectivity**

**Theorem:** Let

- If
$f$ and$g$ are both injective, then$g \circ f$ is injective. - If
$f$ and$g$ are both surjective, then$g \circ f$ is surjective.

*Proof:*

- Let
$a_1, a_2 \in A$ be given. If$a_1 \ne a_2$ , then by the injectivity of$f$ , we have$f(a_1) \ne f(a_2)$ . It follows by the injectivity of$g$ that$g(f(a_1)) \ne g(f(a_2))$ , or equivalently,$(g \circ f)(a_1) \ne (g \circ f)(a_2)$ . - Let
$c \in C$ be given. By the surjectivity of$g$ , there exists$b \in B$ such that$g(b) = c$ . Now, by the surjectivity of$f$ , there exists$a \in A$ such that$f(a) = b$ . Hence, we have$(g \circ f)(a) = g(f(a)) = g(b) = c$ . □

**Definition: Hausdorff Space**

A * Hausdorff space* is a topological space

**Definition: Basis**

Let * basis* of

In other words, a basis is a collection of open sets that allows every open set to be written as a union of basis elements.

**Definition: Second-Countable**

A topological space is * second-countable* if it admits a countable basis.

**Definition: Manifold, Dimension**

A * manifold* of

**dimension***, is a second-countable locally Euclidean Hausdorff space.*

$\boldsymbol{n}$ -manifoldIf manifolds were not required to be Hausdorff, then the line with two origins would be a one-dimensional manifold. Similarly, if manifolds were not required to be second-countable, then the long line would be a one-dimensional manifold.

**Definition: Factorial**

Let * factorial* of

We define

**Definition: Gauss’s Pi Function**

* Gauss’s pi function* is the function

Note that

converges absolutely.

For

By putting these results together, we conclude that Gauss’s pi function interpolates the factorial function.

**Bohr–Mollerup Theorem**

**Theorem:** The gamma function is the unique function

$f(1) = 1$ .$f(x + 1) = xf(x)$ for all$x > 0$ .$\log f$ is convex.