lacunary - Definition Index

An A/B test is a two-sample statistical inference problem. When we're running an A/B test, we have two groups of users, group A and group B.

We change some functionality for group A and leave it stable for group B. We want to see if there is a statistically significant difference between group A and group B for some metric.

TODO: finish this. Formalize the two-sample setup in terms of the two random variables $X$ and $Y$, and describe the CUPED variance-reduction technique the page is named for.

Abelian groups are groups whose operation is commutative. For $a,b\in G,a\ast b=b\ast a$.

The subgroup of $S_n$ consisting of all even permutations of $n$ letters is the alternating group $A_n$ on $n$ letters. If $n\ge 2$, then this set forms a subgroup of $S_n$ of order $n!/2$.

We say that a complex function $f$ is analytic in an open set $S$ if it has a complex derivative at every point of $S$. In other words, when studying complex functions, we consider differentiability on open sets rather than on specific points.

A complex function $f$ is said to be analytic at a point $z$ if it is analytic in some neighborhood of $z$. However, even here, the point being analytic depends on the neighborhood around the point being complex differentiable.

The Archimedean property is that given two positive numbers $x$ and $y,$ there is an integer $n$ such that

A set $A$ is said to be at most countable if $A$ is finite or countable.

Given the open or closed ball with center $\mathbf{x}$ and radius $r$ is defined as the set of points $\mathbf{y}$ such that or $|\mathbf{x}-\mathbf{y}|\le r,$ respectively.

A collection $V_{\alpha }$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x\in X$ and every open set $G\subset X$ such that $x\in G,$ we have $x\in V_{\alpha }\subset G$ for some $\alpha .$ In other words, every open set in $X$ is the union of a subcollection of $V_{\alpha }.$

A Bernoulli Process must possess the following properties:

1. The experiment consists of repeated trials.
2. Each trial results in a boolean outcome, which can be considered true or false, success or failure, etc.
3. The probability of success, $p$, remains constant across trials.
4. Repeated trials are independent.

In a @dynamical-system with a smoothly varying @parameter $r,$ a sudden change in qualitative or topological behavior of the system, especially the creation, annihilation, or change of stability in its fixed points, is called a bifurcation.

The parameter values at which bifurcations occur are called bifurcation points.

Binary Classification is the task of putting things into one of two categories (each called a class.)

For $n,k\in \mathbb{N},n\ge k\ge 0,$ the binomial coefficient is the coefficient of the $x^k$ term in the polynomial expansion of the binomial power $(1+x{)}^n.$ It is denoted as $(\genfrac{}{}{0ex}{}{n}{k})$ and can be expressed through @factorial notation as

$$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!}.$$

A binomial distribution aggregates the outcomes of a Bernoulli process across multiple trials to tell you how likely it is to achieve a certain number of successes. The number $X$ of successes in $n$ Bernoulli trials is called a binomial random variable, and its probability distribution is what we call the binomial distribution.

If we let $n$ be the number of trials, $x$ be the number of successes, and $p$ be the probability of success on a given trial, and $q=1-p$ be the probability of failure on a given trial, then the probability distribution of the random variable $X$ is

$$b(x,n,p)=(\genfrac{}{}{0ex}{}{n}{x})p^x{q}^{n-x},x=0,1,2,\dots ,n.$$

To break this down some, there are $(\genfrac{}{}{0ex}{}{n}{x})$ ways to have $x$ successes out of $n$ trials (different orderings). For each, there are $x$ independent events that occur with a probability of $p$ and $n-x$ independent events that occur with a probability of $q=1-p.$

The mean and variance of the binomial distribution $b(x,n,p)$ are

$$\mu =np,\text{and}{\sigma }^2=npq.$$

We can use a summation over the formula above to find the probability of there being between $a$ and $b$ successes:

$$\sum _{x=a}^b(\genfrac{}{}{0ex}{}{n}{x})p^x{q}^{n-x}.$$

A vector with a fixed endpoint is called a bound vector.

The points on a manifold whose neighborhoods are homeomorphic to a neighborhood in a half $k$-ball form the boundary of the manifold. Formally,

$$\partial M=\{p\in M|\text{there exists a chart }(U,p)\text{ with }\phi (p)\in {\mathbb{R}}^{k-1}\times 0\subset {\mathbb{H}}^k\}.$$

That is, $p$ is a boundary point, if, in local coordinates, it maps to the edge of the half-space model.

$E$ is bounded if there is a real number $M$ and a point $q\in X$ such that for all $p\in E.$

The sequence $\{p_n\}$ is said to be bounded if its range (sequence) is bounded.

Let $E_0$ be the interval $[0,1].$ Remove the segment $(\frac{1}{3},\frac{2}{3}),$ and let

$$E_1=[0,\frac{1}{3}]\cup [\frac{2}{3},1].$$

Similarly, remove the middle thirds of these intervals, and let

$$E_2=[0,\frac{1}{9}]\cup [\frac{2}{9},\frac{3}{9}]\cup [\frac{6}{9},\frac{7}{9}]\cup [\frac{8}{9},1].$$

We can continue this forever, and we get a nested sequence $\{E_n\}$ of compact sets $E_n$ where:

(a) $E_{n+1}\subset E_n.$

(b) $E_n$ is the union of $2^n$ intervals, each of length $1/3^n.$

Finally, the set

$$P=\bigcap _{n=1}^{\mathrm{\infty }}{E}_n$$

is called the Cantor set.

The capacity $C$ of a discrete channel is given by

$$C=\underset{t\to \mathrm{\infty }}{lim}\frac{\log N(t)}{t},$$

where $N(t)$ is the number of allowed @signals of duration $t.$

Given two sets, $A$ and $B$, if there is a bijection (a one-to-one mapping of $A$ onto $B$) between $A$ and $B$, we say $A$ and $B$ have the same cardinal number, or that $A$ and $B$ are equivalent. We denote this as $A\sim B$.

Let's say we have an integral

$$\begin{array}{cc}{\int }_A^{B}f(x)dx& \text{(a)}\end{array}$$

whose integrand becomes infinite at a point $a$ in the interval of integration.

Then, (a) means, by definition

$${\int }_A^{B}f(x)dx=\underset{ϵ\to 0}{lim}{\int }_A^{a-ϵ}f(x)dx+\underset{\eta \to 0}{lim}{\int }_{a+\eta }^{B}f(x)dx.$$

However, it may be the case that neither limit exists when both $ϵ$ and $\eta$ approach $0$ independently, but that the limit

$$\underset{ϵ\to 0}{lim}[{\int }_A^{a-ϵ}f(x)dx+{\int }_{a+ϵ}^{B}f(x)dx]$$

does exist.

This limit is called the Cauchy principal value of the integral and is written as

$$\text{pr. v. }{\int }_A^{B}f(x)dx.$$

A sequence $\{p_n\}$ in a metric space $X$ is said to be a Cauchy sequence if for every there is an integer $N$ such that is $n\ge N$ and $m\ge N.$

The center of a group $G$ is all the elements that commute with all elements of $G$:

$$Z(G)=\{z\in G|zg=gz\text{ for all }g\in G\}.$$

The characteristic equation tells us how to find the @eigenvalues and is given by:

$${\lambda }^2-\tau \lambda +\mathrm{\Delta }=0.$$

The circle in which the Taylor series converges to the function is called the circle of convergence for the Taylor series.

A class is a collection of individuals or individual objects. A class can be defined either by extension (specifying members) or by intension (specifying conditions).

Classification is the activity of assigning objects to some pre-existing classes or categories.

A set $E$ is closed if every limit point of $E$ is a point of $E.$

If $X$ is a metric space, $E\subset X,$ and $E^{\prime }$ denotes the set of all limit points of $E$ in $X,$ then the closure of $E$ is the set $\bar{E}=E\cup E^{\prime }.$

Real numbers $a_1,\dots ,a_n$ are commensurable if there exists a common measure such that each $a_i$ is an integer multiple of it: $$a_i=k_{i}m,k_i\in \mathbb{Z},i=1,\dots ,n.$$ Equivalently, all pairwise ratios are rational: $a_i/a_j\in \mathbb{Q}$ for all $i,j$ (with $a_j\ne 0$).

A ring in which multiplication is commutative is called a commutative ring.

The commutator subgroup of $G$ is the group $C$ generated by all elements of the set

$$\{ab{a}^{-1}{b}^{-1}|a,b\in G\}.$$

A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. More explicitly, the requirement is that if $\{G_{\alpha }\}$ is an open cover of $K,$ then there are finitely many indicies ${\alpha }_1,\dots ,{\alpha }_n$ such that $$K\subset G_{{\alpha }_1}\cup \cdots \cup G_{{\alpha }_n}.$$

The complement of $E$ (denoted by $E^c$) is the set of all points $p\in X$ such that $p\notin E.$

A metric space in which every cauchy sequence converges is said to be complete.

The derivative of a complex function $f$ with respect to $z$ at $z_0$ is defined as

provided the limit exists.

In the above definition of complex derivative, if the limit exists, $f$ is said to be differentiable at the point $z_0$.

A complex-valued function $f$ of a complex variable $z$ is a rule that assigns a complex number $f(z)$ to each complex number $z$ in a set $S$. We call such a function a complex function.

A complex number is one which can be written in the form $a+bi$, where $a$ and $b$ are real numbers. The set of all complex numbers is denoted $\mathbb{C}$.

Let $\mathbf{f}$ be a function that maps an open set $E\subset R^n$ into $R^m.$ Let $\{{\mathbf{e}}_1,\dots ,{\mathbf{e}}_n\}$ and $\{{\mathbf{u}}_1,\dots ,{\mathbf{u}}_n\}$ be the standard bases of $R^n$ and $R^m.$ The components of $\mathbf{f}$ are the real functions $f_1,\dots ,f_m$ defined by

$$\mathbf{f}(\mathbf{x})=\sum _{i=1}^m{f}_i(\mathbf{x}){\mathbf{u}}_i(\mathbf{x}\in E),$$

or, equivalently, by $f_i(\mathbf{x})=\mathbf{f}(\mathbf{x})\cdot {\mathbf{u}}_i,1\le i\le m.$

Suppose $X,Y,Z$ are metric spaces, $E\subset X,$ $f:E\to Y,$ $g:f(E)\to Z,$ $h:E\to Z$ with

$$h(x)=g(f(x))(x\in E).$$

The function $h$ is called the composition or the composite of $f$ and $g.$ The notation

$$h=g\circ f$$

is frequently used.

A function $\phi$ is said to be concave if $-\phi$ is a convex function.

A point $p$ in a metric space $X$ is said to be a condensation point of a set $E\subset X$ if every neighborhood of $p$ contains uncountably many points of $E.$

Let $X$ and $Y$ be random variables, not necessarily @independent. The probability that $Y$ takes the value $y$ when $X$ takes the value $x$ (for any specific $x\in \mathcal{X},y\in \mathcal{Y}$) is called the conditional probability $p(y|x)$ and is given by:

$$p(y|x)=\frac{p(x,y)}{p(x)},p(x)=\sum _{y}p(x,y).$$

(Note that $p(x,y)$ is the @joint-probability of $X=x,Y=y.$)

We define the conditional entropy of $Y$ given $X,$ $H(Y|X),$ as the average of the entropy of $Y$ for each value $x$ of $X$ weighted according to the probability of getting that particular $x.$ That is,

$$H(Y|X)=-\sum _{x\in \mathcal{X},y\in \mathcal{Y}}p(x,y)\log p(y|x).$$

From this, we can give the formula for conditional probability. The probability of $A$ given $B$ is

$$P(A|B)=\frac{P(A\text{and}B)}{P(B)}.$$

That is to say, the probability of $A$ occurring given $B$ has occurred is the portion of times $B$ occurs that $A$ also occurs.

If $P(A)=P(A|B)$, then $A$ and $B$ are independent events, and $P(A)\cdot P(B)=P(A\text{and}B).$

A confidence interval tells us how likely it is that a population parameter falls within a specified range, based on a statistic calculated from a sample of data. This interval provides a range of values that, with a certain level of confidence (usually expressed as a percentage like $95\mathrm{\%}$ or $99\mathrm{\%}$), is believed to encompass the true parameter value. The width of the interval gives an idea of the precision of our estimate, with narrower intervals representing more precise estimates.

To say that the range of values has a certain level of confidence means that if we were to repeat the experiment or sampling process many times (theoretically an infinite number of times), the true population parameter would fall within that interval in the stated percentage of all trials. For example, a $95\mathrm{\%}$ confidence level means that $95$ out of $100$ such confidence intervals would contain the true population parameter (which is unknown).

If you were to repeat the sampling process an infinite number of times, each time calculating a new $95\mathrm{\%}$ confidence interval using the same method, about $95\mathrm{\%}$ of these intervals would contain the true population parameter. Each interval is calculated from a different sample and might be different in range, but the method of calculation ensures that $95\mathrm{\%}$ of these intervals will capture the true parameter value.

If $X$ is a metric space, a set $E\subset X$ is said to be connected if $E$ is not a union of two nonempty separated sets.

A set $S$ is said to be connected if every pair of points in $S$ can be joined by a finite number of line segments joined end to end that lie entirely within $S$.

Systems for which a conserved quantity exist are called conservative systems.

Given a system $\dot{\mathbf{x}}=\mathbf{f}\mathbf{(}\mathbf{x}\mathbf{)},$ a conserved quantity is a @real-valued continuous function $E(\mathbf{x})$ that is constant on trajectories, i.e. $dE/dt=0.$ We also require that $E(\mathbf{x})$ is nonconstant on every open set, to avoid trivial examples such as $E(\mathbf{x})=0.$

Suppose $X$ and $Y$ are metric spaces, $E\subset X,p\in E,$ and $f:E\to Y.$ Then $f$ is said to be continuous at $p$ if for every there exists a such that

for all points $x\in E$ for which

If $f$ is continuous at every point of $E,$ then $f$ is said to be continuous on $E$.

A random variable is called a continuous random variable if it takes values on a continuous scale.

If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space.

A differentiable function $\mathbf{f}$ of an open set $E\subset R^n$ into $R^m$ is said to be continuously differentiable in $E$ if ${\mathbf{f}}^{\prime }$ is a continuous function of $E$ into $L(R^n,R^m).$ More explicitly, it is required that to every $\mathbf{x}\in E$ and to every corresponds a such that

if $\mathbf{y}\in E$ and

If this is the case, we also say that $\mathbf{f}$ is a ${\mathcal{C}}^{\prime }$-mapping or that $\mathbf{f}\in {\mathcal{C}}^{\prime }(E).$

Given a smooth curve $C$ in a domain $D$ and $f(z)=u(x,y)+v(x,y)i$ a continuous function defined along $C$, we say that

$${\int }_{C}f(z)dz={\int }_C(u+vi)(dx+dyi)={\int }_C(udx-vdy)+i{\int }_C(vdx+udy).$$

We can parameterize $C$ as

$$C:z=z(t)=x(t)+y(t)i,\alpha \le t\le \beta$$

Then, $z^{\prime }(t)=x^{\prime }(t)+i{y}^{\prime }(t),$

and by the chain rule,

$$x^{\prime }(t)=\frac{dx}{dt}dt,y^{\prime }(t)=\frac{dy}{dt}dt.$$

Therefore,

$$z^{\prime }(t)=\frac{dz}{dt}=\frac{dx}{dt}+i\frac{dy}{dt}$$

or equivalently

$$dz=z^{\prime }(t)dt.$$

Then we have

$${\int }_{C}f(z)dz={\int }_{\alpha }^{\beta }f[z(t)]z^{\prime }(t)dt.$$

A sequence $\{p_n\}$ in a metric space $X$ is said to converge if there is a point $p\in X$ with the following property: For every there is an integer $N$ such that $n\ge N$ implies that

If a sequence converges, it is said to be a convergent sequence.

A convex combination is a linear combination of points where all @coefficients are non-negative and sum to 1.

Let $E\subseteq {\mathbb{R}}^k$ be a convex set. A function $\phi :E\to \mathbb{R}$ is convex if

$$\phi (\lambda \mathbf{x}+(1-\lambda )\mathbf{y})\le \lambda \phi (\mathbf{x})+(1-\lambda )\phi (\mathbf{y})$$

whenever $\mathbf{x},\mathbf{y}\in E$ and

In geometric terms, this means a @chord joining any two points on the graph of $\phi$ lies on or above the graph between them.

A set $E\subset {\mathbb{R}}^k$ is said to be convex if

$$\lambda \mathbf{x}+(1-\lambda )\mathbf{y}\in E$$

whenever $\mathbf{x},\mathbf{y}\in E,$ and

In geometric terms, this means a set is convex if we can connect any two points in the set with a line segment whose points are all within the set.

Sums of independent random variables are called convolutions.

If $X$ and $Y$ are continuous, the probability density function of $X+Y$ is given by

$$f_{X+Y}(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X(t)f_Y(x-t)dt.$$

If they are discrete, the probability mass function of $X+Y$ is given by

$$f_{X+Y}(n)=\sum _k{f}_X(k)f_Y(n-k).$$

The correlation coefficient of $X$ and $Y$, ${\rho }_{x,y},$ is given by

$${\rho }_{x,y}=\frac{\mathrm{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}.$$

Let $H$ be a subgroup of $G$. Given $a\in G$, the subset $aH=\{ah|h\in H\}$ of $G$ is the left coset of $H$ containing $a$, while the subset $Ha=\{ha|h\in H\}$ is the right coset of $H$ containing $a$.

A set $A$ is said to be countable if there exists a bijection between $A$ and the set of all positive integers , that is, if

Covariance is a measure of the joint variability of two random variables.

$$\mathrm{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])].$$

If large $X$ values go with large $Y$ values and small $X$ goes with small $Y$, covariance will be positive. The covariance will be negative if large $X$ values go with small $Y$ values and vice-versa. An alternative, equivalent form of covariance is:

$$\mathrm{Cov}(X,Y)=E[XY]-E[X]E[Y].$$

The cross product $\mathbf{a}\times \mathbf{b}$ (read "a cross b") of two vectors $\mathbf{a}$ and $\mathbf{b}$ is the vector $\mathbf{v}$ denoted by

$$\mathbf{v}=\mathbf{a}\times \mathbf{b}.$$

I. If $\mathbf{a}=\mathbf{0}$ or $\mathbf{b}=\mathbf{0},$ then we define $\mathbf{v}=\mathbf{a}\times \mathbf{b}=\mathbf{0}.$

II. If both vectors are nonzero vectors, then vector $\mathbf{v}$ has the length

$$\begin{array}{cc}|\mathbf{v}|=|\mathbf{a}\times \mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin \theta ,& \text{(1)}\end{array}$$

where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}.$ Furthermore, $\mathbf{a}$ and $\mathbf{b}$ form the sides of a parallelogram on a plane in space. The area of this parallelogram is precisely given by (1), such that the length $|\mathbf{v}|$ of the vector $\mathbf{v}$ is equal to the area of the parallelogram.

III. If $\mathbf{a}$ and $\mathbf{b}$ lie in the same straight line, i.e. $\mathbf{a}$ and $\mathbf{b}$ have the same or opposite directions, then $\theta$ is $0^{\circ }$ or ${180}^{\circ }$ so that $\sin \theta =0.$ In that case $|\mathbf{v}|=0,$ so that $\mathbf{v}=\mathbf{a}\times \mathbf{b}=\mathbf{0}.$

IV. If cases I and III do not occur, then $\mathbf{v}$ is a nonzero vector. The direction of $\mathbf{v}=\mathbf{a}\times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ such that $\mathbf{a},\mathbf{b},\mathbf{v},$ precisely in this order, form a right-handed triple

Given two discrete probability distributions, $p$ and $q,$ that share a support $\mathcal{X},$ cross-entropy measures the average number of bits needed to represent an event drawn from $\mathcal{X}$ when the coding scheme is optimized for an estimated distribution $q$ rather than the true distribution $p.$ Its formula is very similar to that of entropy, except surprisal is calculated using $q$ while expectation uses $p$:

$$H(p,q)=-\sum _{x\in \mathcal{X}}p(x)\log q(x).$$

We can write it in expectation form as

$$H(p,q)={\mathbb{E}}_p[-\log q],$$

where ${\mathbb{E}}_p$ is expectation under $p.$

Sometimes we want to know the probability that a random variable will be less than or equal to some real number $x.$ If we let $F(x)=P(X\le x)$ for all real $x$, we define $F(x)$ to be the cumulative distribution function of the random variable $X.$ More formally, the cumulative distribution function $F(x)$ of a discrete random variable $X$ with a probability distribution $f(x)$ is:

For a vector field $\mathbf{F}=(F_1,F_2,F_3)$ defined in three-dimensional space ${\mathbb{R}}^3$, with each component function $F_i$ depending on the variables $x$, $y$, and $z$, the curl of $\mathbf{F}$ is defined as

$$\mathrm{curl}\mathbf{F}=\mathrm{\nabla }\times \mathbf{F}=(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z})\mathbf{i}+(\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x})\mathbf{j}+(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y})\mathbf{k}$$

A permutation $\sigma \in S_n$ is a cycle if it has at most one orbit containing more than one element. The length of a cycle is the number of elements in its largest orbit.

A cyclic group is a group where there exists some $g\in G$ such that every element in $G$ can be generated from the group operation applied to $g$.

That is, $G=\{g^n|n\in \mathbb{Z}\}$ when we think of the operation as multiplication, or $G=\{ng|n\in \mathbb{Z}\}$ when we think of the operation as addition.

A cycle in $S_n$ may be written as $(a_1,a_2,\cdots ,a_k)$ - this is called cyclic notation. It represents the permutation that sends $a_1\to a_2\to \cdots \to a_k\to a_1$.

Any subgroup of a cyclic group is also cyclic - a cyclic subgroup.

In Euclidean space ${\mathbb{R}}^n$ with coordinates $(x_1,\dots ,x_n)$ and standard basis $({\mathbf{e}}_1,\dots ,{\mathbf{e}}_n),$ del is a vector operator whose $x_1,\dots ,x_n$ components are the partial derivative operators $\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_n};$ that is

$$\mathrm{\nabla }=\sum _{i=1}^n{\mathbf{e}}_i\frac{\partial }{\partial x_i}=(\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_n}).$$

$E$ is dense in $X$ if every point of $X$ is a limit point of $E,$ or a point of $E$ (or both.)

A derivation (or proof) in $\mathcal{F}$ is a finite sequence of formulas

$${\phi }_1,{\phi }_2,\dots ,{\phi }_n$$

such that for each $i\le n$,

1. Either ${\phi }_i\in A$, or
2. ${\phi }_i$ follows from earlier formulas in the sequence by an inference rule in $R$.

While the equation defined in (a) only applies to a specific point $z_0$, we can drop the subscript to get

which defines the complex derivative of $f$ at the point $z$. This equation defines the complex function $f^{\prime }$, known as the derivative function.

Let $E$ be a nonempty subset of a metric space $X,$ and let $S$ be the set of all real numbers of the form $d(p,q),$ with $p,q\in E.$ The supremum of $S$ is called the diameter of $E.$

A difference equation has the general form

$${\mathbf{x}}_{\mathbf{n}\mathbf{+}\mathbf{1}}=\mathbf{f}\mathbf{(}{\mathbf{x}}_{\mathbf{n}}\mathbf{,}\mathbf{n}\mathbf{)},$$

where $n$ is a time-like variable, $\mathbf{f}$ and ${\mathbf{x}}_{\mathbf{n}}$ may be vectors, $n=0,1,2,\cdots ={\mathbb{N}}_0.$

Suppose $E$ is an open set in $R^n,$ $\mathbf{f}:E\to R^m,$ and $\mathbf{x}\in E.$ If there exists a linear transformation $A:R^n\to R^m$ such that

$$\begin{array}{cc}\underset{\mathbf{h}\to \mathbf{0}}{lim}\frac{|\mathbf{f}(\mathbf{x}+\mathbf{h})-\mathbf{f}(\mathbf{x})-A\mathbf{h}|}{|\mathbf{h}|}=0,& \text{(14)}\end{array}$$

then we say that $\mathbf{f}$ is differentiable at $\mathbf{x}$ and we write

$${\mathbf{f}}^{\prime }(\mathbf{x})=A.$$

If $\mathbf{f}$ is differentiable at every $\mathbf{x}\in E,$ we say that $\mathbf{f}$ is differentiable in $E.$

Note that in (14), $\mathbf{h}\in R^n.$ If $|\mathbf{h}|$ is small enough, then $\mathbf{x}+\mathbf{h}\in E,$ because $E$ is open. Therefore, $\mathbf{f}(\mathbf{x}+\mathbf{h})$ is defined, $\mathbf{f}(\mathbf{x}+\mathbf{h})\in R^m,$ and since $A\in L(R^n,R^m),$ $A\mathbf{h}\in R^m.$ Therefore,

$$\mathbf{f}(\mathbf{x}+\mathbf{h})-\mathbf{f}(\mathbf{x})-A\mathbf{h}\in R^m.$$

The norm in the numerator of (14) is that of $R^m,$ while the norm in the denominator is the $R^n$-norm.

We can also rewrite (14) as

$$\begin{array}{cc}\mathbf{f}(\mathbf{x}+\mathbf{h})-\mathbf{f}(\mathbf{x})={\mathbf{f}}^{\prime }(\mathbf{x})\mathbf{h}+\mathbf{r}(\mathbf{h}),& \text{(17)}\end{array}$$

where the remainder $\mathbf{r}(\mathbf{h})$ satisfies

$$\underset{\mathbf{h}\to \mathbf{0}}{lim}\frac{|\mathbf{r}(\mathbf{h})|}{|\mathbf{h}|}=0.$$

This means that for a fixed $\mathbf{x}$ and a small $\mathbf{h},$ the left side of (17) is approximately equal to ${\mathbf{f}}^{\prime }(\mathbf{x})\mathbf{h},$ that is, to the value of a linear transformation applied to $\mathbf{h}.$

The direction of a vector can be specified by the angle between it and some fixed reference, such as the $x$-axis.