lacunary - Definition Index

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

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

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

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

Given $\vec{x} \in \mathbb{R}^k, r > 0,$ the open or closed ball with center $\vec{x}$ and radius $r$ is defined as the set of points $\vec{y}$ such that $|\vec{x} - \vec{y}| < r$ or $|\vec{x} - \vec{y}| \leq r,$ 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 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,

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

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 $d(p, q) < M$ 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

$$ P = \bigcap_{n=1}^\infty E_n $$

is called the Cantor set.

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 $A \sim B$.

A sequence $\{p_n\}$ in a metric space $X$ is said to be a Cauchy sequence if for every $\epsilon > 0$ there is an integer $N$ such that $d(p_n, p_m) < \epsilon$ is $n \geq N$ and $m \geq N.$

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

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

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 $E'$ denotes the set of all limit points of $E$ in $X,$ then the closure of $E$ is the set $\closure{E} = E \cup E'.$

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

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

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.

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

$$ \vec{f}(\vec{x}) = \sum_{i=1}^m f_i(\vec{x})\vec{u}_i \quad (\vec{x} \in E), $$

or, equivalently, by $f_i(\vec{x}) = \vec{f}(\vec{x}) \cdot \vec{u}_i, 1 \leq i \leq 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

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

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

$$ h = g \circ f $$

is frequently used.

A 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 $E.$

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 $S$.

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 $\epsilon > 0$ there exists a $\delta > 0$ such that

$$ d_Y(f(x), f(p)) < \epsilon $$

for all points $x \in E$ for which $d_X(x, p) < \delta.$

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

A differentiable function $\vec{f}$ of an open set $E \subset R^n$ into $R^m$ is said to be continuously differentiable in $E$ if $\vec{f}'$ is a continuous function of $E$ into $L(R^n, R^m).$ More explicitly, it is required that to every $\vec{x} \in E$ and to every $\epsilon > 0$ corresponds a $\delta > 0$ such that

$$ ||\vec{f}'(\vec{y}) - \vec{f}'(\vec{x})|| < \epsilon $$

if $\vec{y} \in E$ and $|\vec{x} - \vec{y}| < \delta.$

If this is the case, we also say that $\vec{f}$ is a $\mathscr{C}'$-mapping or that $\vec{f} \in \mathscr{C}'(E).$

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 $\epsilon > 0,$ there is an integer $N$ such that $n \geq N$ implies that $d(p_n, p) < \epsilon.$

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

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

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

whenever $\vec{x}, \vec{y} \in E,$ and $0 < \lambda < 1.$

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.

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 $\mathbb{Z}_{>0}$, that is, if $A \sim \mathbb{Z}_{>0}.$

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

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

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

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

$$ \tag{1} |\vec{v}| = |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin{\theta}, $$

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}.$ Furthermore, $\vec{a}$ and $\vec{b}$ 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 $|\vec{v}|$ of the vector $\vec{v}$ is equal to the area of the parallelogram.

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

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

For a vector field $\vec{F} = (F_1, F_2, F_3)$ 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 $\vec{F}$ is defined as

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

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.

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

A permutation is a bijection $\phi : A \to A$, that is, a bijection from a set onto itself.

Let $\sigma$ be a permutation of $A$. The equivalence classes in $A$ determined by the equivalence relation "~" are the orbits of $\sigma$.

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

In Euclidean space $\mathbb{R}^n$ with coordinates $(x_1, \dots, x_n)$ and standard basis $(\vec{e}_1, \dots, \vec{e}_n),$ 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

$$ \nabla = \sum_{i = 1}^n \vec{e}_i \frac{\partial}{\partial x_i} = \left ( \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n} \right ). $$

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

The derivative of a function $f$ at a point $a$ is defined as: $$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ provided this limit exists.

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 $E.$

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

$$ \lim_{\vec{h} \to \vec{0}} \frac{|\vec{f}(\vec{x} + \vec{h}) - \vec{f}(\vec{x}) - A\vec{h}|}{|\vec{h}|} = 0, \tag{14} $$

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

$$ \vec{f}'(\vec{x}) = A. $$

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

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

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

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

$$ \vec{f}(\vec{x} + \vec{h}) - \vec{f}(\vec{x}) = \vec{f}'(\vec{x})\vec{h} + \vec{r}(\vec{h}), \tag{17} $$

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

$$ \lim_{\vec{h} \to \vec{0}} \frac{|\vec{r}(\vec{h})|}{|\vec{h}|} = 0. $$

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

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

Using the setup from the definition of component, let us fix an $\vec{x} \in E$ (with $E \subset R^n$,) and let $\vec{u} \in R^n$ be a unit vector. Then,

$$ \lim_{t \to 0} \frac{f(\vec{x} + t \vec{u}) - f(\vec{x})}{t} = (\nabla f)(\vec{x}) \cdot \vec{u} $$

is called the directional derivative of $f$ at $\vec{x}$ in the direction of the unit vector $u$ and is denoted as $D_{\vec{u}}f(\vec{x}).$

The discrete metric is defined as:

$$ d(x, y) = \begin{cases} 0 & \text{if } x = y \\ 1 & \text{if } x \neq y \end{cases} $$

Two or more than two cycles are disjoint if no element appears in more than one cycle.

If a sequence $\{p_n\}$ does not converge, it is said to diverge.

For a vector field $\vec{F} = (F_1, F_2, F_3)$ defined in three-dimensional space $\mathbb{R}^3$, the divergence is defined as

$$ \div \vec{F} = \nabla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}. $$

Let $R$ be a ring with unity. If every nonzero element of $R$ is a unit (has a multiplicative inverse), then $R$ is called a division ring.

A domain is an open connected set. Note that this is not the same as a domain of definition of a function.

If $f$ is a function from the set $A$ to the set $B,$ the set $A$ is called the domain of $f.$

The objects that make up a set are called its elements or its members.

A permutation of a finite set is even if it is the product of an even number of transpositions.

A commutative division ring is called a field.

A set $A$ is said to be finite if $A \sim \mathbb{N}_n$ for some $n.$

A vector whose endpoints are not fixed at particular points is called a free vector.

Consider two sets, $A$ and $B,$ whose elements may be any objects whatsoever, and suppose that with each element $x$ of $A$ there is associated, in some manner, any element of $B,$ which we denote by $f(x).$ Then $f$ is said to be a function from $A$ to $B.$

Given a scalar function $f : \mathbb{R}^n \to \mathbb{R}$, the gradient of $f$, denoted as $\nabla f$, is defined as the vector of its partial derivatives. Specifically, for a function $f(x_1, x_2, \cdots, x_n)$, the gradient is given by

$$ \nabla f = \begin{bmatrix} \frac{\partial f}{\partial x_1} \\ \frac{\partial f}{\partial x_2} \\\ \vdots \\\ \frac{\partial f}{\partial x_n} \end{bmatrix} $$

There is also a concept of a greatest lower bound, which is a lower bound that is greater than or equal to every other lower bound. The greatest lower bound is also known as the infimum.

A group, is a set $G$ together with a binary operation $*$ such that:

Closure: The set is closed under the binary operation. For all $a, b \in G, a * b \in G$.

Associativity: The binary operation is associative on the set. For all $a, b, c \in G, (a * b) * c = a * (b * c)$.

Identity: The set contains an identity element, denoted $e$. For all $a \in G, a * e = a$.

Inverses: All elements in the set have inverse elements in the set, denoted using $a^{-1}$. For all $a \in G$ there exists $a^{-1} \in G$ such that $a * a^{-1} = e$.

This set/operation combination $G$ is commonly denoted as the pair $(G, *)$.

A half-open interval $(a,b]$ or $[a, b)$ is the set of all real numbers such that $a < x \leq b$ or $a \leq x < b,$ respectively.

Suppose $f ~ : ~ \mathbb{R}^n \to \mathbb{R}$ is a function taking as input a vector $\vec{x} \in \mathbb{R}^n$ and outputting a scalar $f(\vec{x}) \in \mathbb{R}.$ If all second-order partial derivatives of $f$ exist, then the Hessian matrix $\vec{H}$ of $f$ is a square $n \times n$ @matrix, usually defined and arranged as

$$ \mathbf H_f= \begin{bmatrix} \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\[2.2ex] \dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\[2.2ex] \vdots & \vdots & \ddots & \vdots \\[2.2ex] \dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2} \end{bmatrix}. $$

That is, the @entry of the $i$th row and the $j$th column is

$$ (\vec{H}_f)_{i,j} = \frac{\partial^2 f}{\partial x_i \partial x_j}. $$

For a function $f ~ : ~ \mathbb{R}^3 \to \mathbb{R},$ this is

$$ \vec{H} = \begin{bmatrix} \dfrac{\partial^2 f}{\partial x^2} & \dfrac{\partial^2 f}{\partial x \partial y} & \dfrac{\partial^2 f}{\partial x \partial z}\\ \dfrac{\partial^2 f}{\partial y \partial x} & \dfrac{\partial^2 f}{\partial y^2} & \dfrac{\partial^2 f}{\partial y \partial z}\\ \dfrac{\partial^2 f}{\partial z \partial x} & \dfrac{\partial^2 f}{\partial z \partial y} & \dfrac{\partial^2 f}{\partial z^2}\\ \end{bmatrix}. $$

A homeomorphism is a @bijective and continuous function between @topological-spaces that has a continuous @inverse-function.

A homomorphism is a map $\phi : G \to G'$ between groups (not necessarily a bijection), $\langle G, * \rangle$ and $\langle G', *' \rangle,$ that satisfies the homomorphism property:

$$ \phi(a * b) = \phi(a) *' \phi(b), ~ \forall a, b \in G. $$

Let $\vec{v}$ be the velocity vector of of the motion of particles in a fluid. If $\div{\vec{v}} = 0,$ then the fluid has constant density and is said to be incompressible.

The number of left cosets of a subgroup $H$ in a group $G$ is the index $(G:H)$ of $H$ in $G$.

A set $A$ is said to be infinite it is not finite.

If $\vec{x}$ and $\vec{y}$ are vectors in $\mathbb{R}^n,$ then their inner product is defined as

$$ \vec{X} \cdot \vec{y} = \sum_{i = 1}^n x_i y_i. $$

A point $p$ is an interior point of $E$ if there is a neighborhood $N$ of $p$ such that $N \subset E.$

A interval $[a, b]$ is the set of all real numbers $x$ such that $a \leq x \leq b.$

If $f: A \to B$ and $E \subset(B),$ then $f^{-1}(E)$ denotes the set of all $x \in A$ such that $f(x) \in E.$ We call $f^{-1}(E)$ the inverse image of $E$ under $f.$

If the curl of a vector field is $\vec{0},$ i.e. if $\curl{\vec{v}} = 0,$ the field is said to be irrotational.

If $p \in E$ and $p$ is not a limit point of $E,$ then $p$ is called an isolated point of E.

Let $\vec{f} ~ : ~ \mathbb{R}^n \to \mathbb{R}^m$ be a function such that each of its first-order partial derivatives exists on $\mathbb{R}^n.$ This function takes a point $\vec{x} = (x_1, \dots, x_n) \in \mathbb{R}^n$ as input and produces the vector $\vec{f}(\vec{x}) = (f_1(\vec{x}), \dots, f_m(\vec{x})) \in \mathbb{R}^m$ as output. Then the Jacobian matrix of $\vec{f},$ denoted $\vec{J}_{\vec{f}},$ is the $m \times n$ @matrix whose $(i,j)$ @entry is $\frac{\partial f_i}{\partial x_j};$ explicitly

$$ \begin{bmatrix} \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n} \end{bmatrix} = \begin{bmatrix} \nabla^{\mathsf{T}} f_1 \\ \vdots \\ \nabla^{\mathsf{T}} f_m \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix}, $$

where $\nabla^{\mathsf{T}} f_i$ is the @transpose (@row-vector) of the gradient of the $i$-th component.

Given $\vec{a}, \vec{b} \in \mathbb{R}^k,$ if $\vec{a}_i < \vec{b}_i$ for all $i = 1, 2, \dots, k,$ then the set of all points $\vec{x}$ who satisfy $\vec{a}_i \leq \vec{x}_i \leq \vec{b}_i,$ $i = 1, 2, \dots, k,$ is called a k-cell. So, a 1-cell is an interval, a 2-cell is a rectangle, and so on.

The kernel of a homomorphism $\phi$ is the set of elements that $\phi$ sends to $e'$, and it is denoted by $\ker{(\phi)}$. It is a normal subgroup of $G$.

The Laplace operator is a second-order differential operator in the $n$-dimensional Euclidean space, defined as the divergence $(\nabla \cdot)$ of the gradient $(\nabla f$). Thus, if $f$ is a twice differentiable real-valued function, then the Laplacian of $f$ is the real-valued function defined by

$$ \Delta f = \nabla^2 f = \nabla \cdot \nabla f. $$

The Laplacian of $f$ is the sum of all the unmixed second partial derivatives in the Cartesian coordinates $x_i$:

$$ \nabla^2 f = \sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2}. $$

In two dimensions, using Cartesian coordinates, the Laplace operator is given by

$$ \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} $$

and in three dimensions by

$$ \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} $$

The number $L$ is said to be the least upper bound of the set $A$ if it's an upper bound of $A$ and if $L \leq m$ for all upper bounds $m$ of $A$.

The least upper bound is also known as the supremum.

Let $S$ be a surface represented by $f(x, y, z) = c,$ where $c$ is constant and $f$ is differentiable. Such a surface is called a level surface of $f,$ and for different $c,$ we get different level surfaces.

Let $X$ and $Y$ be metric spaces; suppose $E \subset X,$ $f : E \to Y,$ and $p$ is a limit point of $E.$ We write $f(x) \to q$ as $x \to p,$ or

$$ \lim_{x \to p} f(x) = q $$

if there is a point $q \in Y$ with the following property: For every $\epsilon > 0,$ there exists a $\delta > 0$ such that

$$ d_Y(f(x), q) < \epsilon $$

for all points $x \in E$ for which

$$ 0 < d_X(x,p) < \delta. $$

If a sequence $\{p_n\}$ converges to $p,$ we say that $p$ is the limit of $\{p_n\},$ denoted as:

$$ \lim_{n \to \infty} p_n = p. $$

A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q \neq p$ such that $q \in E.$

If $f$ is defined on a smooth curve C given by $x = x(t), y = y(t), a \le t \le b$, and $f$ if defined on $C$, then the line integral of $f$ along $C$ is defined as:

$$ \int_C f(x,y) ds = \lim_{n \to \inf} \sum_{i=1}^n f(x_i, y_i) \Delta s_i $$

A line integral of a vector function $\vec{F}(\vec{r})$ over a curve $C: \vec{r}(t)$ is defined by

$$ \int_{C} \vec{F}(\vec{r}) \cdot d \vec{r} = \int_{a}^{b} \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) dt \tag{a} $$

where $\vec{r}(t)$ is the parametric representation of $C.$

Writing (a) in terms of components, with $d \vec{r} = [dx, dy, dz]$ and $' = d/dt,$ we get

$$ \int_{C} \int_{C} \vec{F}(\vec{r}) \cdot d \vec{r} = \int_{C} (F_1 dx + F_2 dy + F_3 dz) = \int_{a}^{b} (F_1 x' + F_2 y' + F_3 z') dt). $$

Let $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n} \in \mathbb{R}^n$ and $c_1, c_2, \cdots, c_3 \in \mathbb{R}.$ Then, the vector

$$ \vec{v} = c_1 \vec{v_1} + c_2 \vec{v_2} + \cdots + c_n \vec{v_n} $$

is called a linear combination of $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}.$

Let $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n} \in \mathbb{R}^n.$ The set of all linear combinations of $\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}$ is called their span, denoted $\Span{\left(\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}\right)}.$

That is:

$$ \Span{\left(\vec{v_1}, \vec{v_2}, \cdots, \vec{v_n}\right)} = \left\{ \vec{v} \in \mathbb{R}^n : \vec{v} = c_1 \vec{v_1} + c_2 \vec{v_2} + \cdots + c_n \vec{v_n} ~ \text{for some scalars} ~ c_1, c_2, \cdots, c_n \right\} $$

The number $m$ is said to be a lower bound of a nonempty set $A$ if $x \geq m$ for all $x \in A.$

Let $\vec{x} = (x_1, x_2, \cdots, x_n) \in \mathbb{R}^n$. The magnitude, or length, $\vec{x}$ is denoted as $|| \vec{x} ||$ and is defined as:

$$ ||\vec{x}|| = \sqrt{ {x_1}^2 + {x_2}^2 + \cdots + {x_n}^2 } = \sqrt{\vec{x} \cdot \vec{x}} = \sqrt{\sum_{i=1}^n x_i^2} $$

A manifold is a @topological-space that resembles @euclidean-space near each point. That is, an $n$-dimensional manifold is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of $n$-dimensional Euclidean space.

The membership criterion for a set $X$ is a statement of the form $x \in X \iff P(x),$ where $P(x)$ is a proposition that is true for precisely for those objects $x$ that are elements of $X,$ and no others.

A set $X,$ whose elements we'll call points, together with a distance function $d: X \times X \to \mathbb{R}$ is called a metric space and the distance function $d$ is called a metric, if the following conditions, called the metric axioms, hold for $p, q, r \in X:$

• If $p \neq q, d(p,q) > 0.$ (distance is always positive between two distinct points.)

• $d(p,p) = 0.$ (distance is always zero between a point and itself.)

• $d(p,q) = d(q,p).$ (the distance from $p$ to $q$ is the same as the distance from $q$ to $p$.)

• $d(p,q) \leq d(p,r) + d(r,p)$ (triangle inequality.)

We can denote a metric space on set $X$ with metric $d$ as the tuple $(X, d).$

For some element $a$ in a ring with unity $R$ where $1 \neq 0$, if $a^{-1} \in R$ such that $aa^{-1} = a^{-1}a = 1$, $a^{-1}$ is said to be the multiplicative inverse of $a.$

A neighborhood, or r-neighborhood of $p$ is a set $N_r(p)$ consisting of all $q$ such that $d(p, q) < r$ for some $r > 0.$ This subset of $X$ is all the points within a circle of radius $r$ - the open ball of radius $r$ centered at $p.$

A subgroup $H$ of a group $G$ is normal if its left and right cosets coincide, that is, if $gH = Hg$, i.e. $gHg^{-1} = H$, for all $g \in G$.

The normal derivative is the directional derivative in the direction of the @normal-vector.

A permutation of a finite set is odd if it is the product of an odd number of transpositions.

A set $E$ is open if every point of $E$ is an interior point of $E.$

An open cover of a set $E$ in a metric space $X$ is a collection $\{G_\alpha\}$ of open subsets of $X$ such that $E \subset \bigcup_\alpha G_\alpha.$

Suppose $E \subset Y \subset X,$ and $X$ is a metric space. We say that $E$ is open relative to $Y$ if to each $p \in E$ there is associated an $r > 0$ such that $q \in E, q \in Y$ whenever $d(p, q) < r.$

The order of a finite group is the number of its elements. The order of group $G$ is denoted as $\ord{(G)}$ or $\|G\|$. The order of an element $a$ (also called period length or period) is the number of elements in the subgroup generated by $a$, and is denoted by $\ord{(a)}$ or $\|a\|$.

Two vectors $\vec{x}$ and $\vec{y}$ are said to be parallel if $\vec{x}$ is a scalar multiple of $\vec{y}$, i.e., if there exists some scalar $c$ where $\vec{x} = c \vec{y}.$

Using the setup from the definition of component, for $\vec{x} \in E, 1 \leq i \leq m, 1 \leq j \leq n,$ we define

$$ (D_j f_i)(\vec{x}) = \lim_{t \to 0} \frac{f_i(\vec{x} + t \vec{e}_j) - f_i(\vec{x})}{t}, $$

provided the limit exists. Writing $f_i(x_1, \dots, x_n)$ in place of $f_i(\vec{x}),$ we see that $D_j f_i$ is the derivative of $f_i$ with respect to $x_j,$ keeping the other variables fixed. The notation

$$ \frac{\partial f_i}{\partial x_j} $$

is therefore often used in place of $D_j f_i,$ and $D_j f_i$ is called a partial derivative.

We say that line integral 4.7 is independent of path in a domain $D$ if for any two points $A$ and $B$ in $D$, the value of the line integral is the same for all piecewise smooth curves in $D$ from $A$ to $B$.

$E$ is perfect if $E$ is closed and if every point of $E$ is a limit point of $E.$

Two vectors $\vec{u}$ and $\vec{v}$ are said to be perpendicular or orthogonal if the angle between them is $\pi/2$ radians, or, equivalently, if the inner product $\vec{u} \cdot \vec{v} = 0.$

A potential function is a scalar function whose gradient is a vector field. They allow representing certain vector fields in a simpler, more fundamental form.

In other words, given a vector field $\vec{F},$ a potential function $\phi$ is a scalar function such that

$$ \vec{F} = - \grad{\phi} \quad \text{or} \quad \vec{F} = \grad{\phi}. $$

The set of all values of a function $f$ is called the range of $f.$

The set of all points $p_n$ of a sequence $\{p_n\} (n = 1, 2, 3, \dots)$ is the range of $\{p_n\}.$

The real numbers $\mathbb{R}$ are a set of objects along with two binary operations $+ : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and $\cdot : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ that satisfy the 9 field axioms along with the Order axiom and the Completeness axiom. That is, the reals are an ordered, complete field.

A real sequence of numbers if a function $f$ from $\mathbb{N}$ to $\mathbb{R}$.

A ring is a set together with two binary operations $+$ and $\cdot$, which we will call addition and multiplication, such that the following axioms are satisfied:

1. $\langle R, + \rangle$ is an abelian group.

2. Multiplication is associative.

3. For all $a,b,c \in R$, the left distributive law and the right distribute law hold, i.e.

$$ a \cdot (b + c) = (a \cdot b) + (a \cdot c), \quad (a + b) \cdot c = (a \cdot c) + (b \cdot c). $$

A ring homomorphism $\phi : R \to R'$ must satisfy the following two properties:

1. $\phi{(a+b)} = \phi{(a)} + \phi{(b)}.$

2. $\phi{(ab)} = \phi{(a)}\phi{(b)}.$

A ring that has a multiplicative identity element is called a ring with unity.

A scalar is an element of a field used to define a vector space.

Let $P$ be any point in a domain of definition. Then we define a scalar function $f,$ whose values are scalars, that is,

$$ f = f(p) $$

that depends on $P.$

A scalar function depends only on the point $P,$ not on the coordinate system chosen to represent it.

A segment $(a, b)$ is the set of all real numbers $x$ such that $a < x < b.$

A metric space is called separable if it contains a countable dense subset.

Two subsets $A$ and $B$ of a metric space $X$ are said to be separated if both $A \cap \closure{B}$ and $\closure{A} \cap B$ are empty, i.e., if no point of $A$ lies in the closure of $B$ and no point of $B$ lies in the closure of $A.$

A sequence is a function $f$ defined on the set $J$ of all positive integers.

A set is a collection of objects, considered as a whole.

A group $G$ is simple if it has no proper nontrivial normal subgroups, that is, if $|G| > 1$ and the only normal subgroups of $G$ are $\{e\}$ and $G$ itself.

A function with a continuous first derivative is said to be smooth.

For a parametrically defined function where $x = f(t), y = g(t)$, both $f'$ and $g'$ must be continuous.

A subgroup $H$ of a group $G$ is a subset of $G$ group together with the same operation as $G$ that still forms a group. The identity element of $G$ must also be the identity element of $H$.

Given a sequence $\{p_n\},$ consider a sequence $\{n_k\}$ of positive integers, such that $n_1 < n_2 < n_3 < \cdots.$ Then the sequence $\{p_{n_i}\}$ is called a subsequence of $\{p_n\}.$

If a subsequence $\{p_{n_i}\}$ of $\{p_n\}$ converges, its limit (sequence) is called a subsequential limit of $\{p_n\}.$

If $X$ and $Y$ are sets such that every element of $X$ is also an element of $Y,$ then we say $X$ is a subset of $Y,$ denoted as $X \subset Y.$ Formally,

$$X \subset Y \iff (\forall x)(x \in X \implies x \in Y)$$

If $X$ and $Y$ are sets such that every element of $Y$ is also an element of $X,$ then we say $X$ is a superset of $Y,$ denoted as $X \supset Y.$ This is the same as $Y \subset X.$

Given a piecewise smooth surface $S,$ we can parameterize it as

$$ \vec{r}(u,v) = \langle x(u,v), y(u,v), z(u,v) \rangle = x(u,v) \vec{i} + y(u,v) \vec{j} + z(u,v) \vec{k}, $$

with $u_0 \leq u < u_1$ and $v_0 \leq v \leq v_1$ (i.e. $u$ and $v$ vary over a region $R$ in the $uv$-plane).

Now, $S$ has a @normal-vector and unit @normal-vector, respectively, as

$$ \vec{N} = \vec{r}_u \times \vec{r}_v, \quad \vec{n} = \frac{1}{|\vec{N}|} \vec{N} $$

at every point, except perhaps for some edges or cusps, such as for cubes and cones. For a given vector function $\vec{F}$ we can now define the surface integral over $S$ by

$$ \iint_S \vec{F} \cdot \vec{n} dA = \iint_R \vec{F}(\vec{r}(u,v)) \cdot \vec{N}(u,v) du dv. $$

Given a tangent plane to a surface $S$ at $P,$ the normal to this plane (the straight line through $P$ perpendicular to the tangent plane) is called the surface normal to $S$ at $P.$

A vector in the direction of the surface normal of a surface $S$ at point $P$ is called a surface normal vector of $S$ at $P.$

If $S$ is a level surface of a function $f,$ and $P$ is a point of $S,$ then the set of all tangent vectors of all curves passing through $P$ will generally form a plane, called the tangent plane of $S$ at $P.$

If $M$ is locally given by a smooth parametrization

$$ \varphi : U \subset \mathbb{R}^k \to M \subset \mathbb{R}^n, $$

and $p = \varphi(u_0)$, then

$$ T_p M = \operatorname{span}\left\{ \frac{\partial \varphi}{\partial u_1}(u_0), \frac{\partial \varphi}{\partial u_2}(u_0), \dots, \frac{\partial \varphi}{\partial u_k}(u_0) \right\}. $$

So it’s the collection of all possible velocity vectors of curves on the manifold passing through $p,$ and is a $k$-dimensional linear subspace of $\mathbb{R}^n.$

For a space curve given parametrically by $\vec{r}(t),$ the tangent vector (or specifically, the unit tangent vector) at the point $\vec{r}(t)$ is the unit vector defined by:

$$ \vec{T}(t) = \frac{\vec{r}'(t)}{||\vec{r}'(t)||} $$

If $f$ is a sequence denoted as $\{x_n\},$ the values of $f,$ that is, the elements $x_n,$ are called the terms of the sequence.

The derivative defined above in differentiable is called the total derivative of $\vec{f}$ at $\vec{f},$ or the differential of $\vec{f}$ at $\vec{x}.$

A cycle of length 2 is a transposition.

A set $A$ is said to be uncountable if it is neither finite nor countable.

If $a$ has a multiplicative inverse in $R,$ $a$ is said to be a unit in $R$.

If a vector has a length of 1, i.e. if $||x|| = 1$, we say the vector is a unit vector.

The element $1$ is also called unity.

The number $m$ is said to be an upper bound of a nonempty set $A$ if $x \leq m$ for all $x \in A$.

If $f$ is a function from the set $A$ to the set $B,$ the elements $f(x) \in B$ are called the values of $f.$

A vector is an element in a vector space.

We perform vector addition by adding two vectors $\vec{x} = (x_1, x_2, \cdots, x_n)$ and $\vec{y} = (y_1, y_2, \cdots, y_n)$ according to the following rule:

$$ \vec{x} + \vec{y} = (x_1 + y_1, x_2 + y_2, \cdots, x_n + y_n) $$

that is, by making a new vector where the coordinates are the sums of the respective coordinates in the vectors being summed.

Two vectors are equal if they have the same coordinates (or equivalently, the same length and direction).

We say that a vector function defines a vector field in a domain of definition.

Let $P$ be any point in a domain of definition. Then we define a vector function $\vec{v}$ whose values are vectors, that is,

$$ \vec{v} = \vec{v}(P) = [v_1(P), v_2(P), v_3(P)] $$

that depends on points $P$ in space. A vector function depends only on the point $P,$ not on the coordinate system chosen to represent its components.

We can multiply vectors by a scalar. Given the scalar $c$:

$$ c \vec{x} = (c x_1, c x_2, \cdots, c x_n) $$

A vector space over a field $F$ is a set $V$ together with two operations: - Vector addition: $V \times V \to V$ - Scalar multiplication: $F \times V \to V$

satisfying the following axioms:

1. $(V, +)$ is an abelian group

2. Scalar multiplication is associative: $a(bv) = (ab)v$

3. Distributive laws hold

4. Identity: $1v = v$ for all $v \in V$

Similarly, vector subtraction can be performed as:

$$ \vec{x} - \vec{y} = (x_1 - y_1, x_2 - y_2, \cdots, x_n - y_n) $$

A vector space over a field $F$ is a set $V$ together with two operations: - Vector addition: $V \times V \to V$ - Scalar multiplication: $F \times V \to V$

satisfying the following axioms: 1. $(V, +)$ is an abelian group 2. Scalar multiplication is associative: $a(bv) = (ab)v$ 3. Distributive laws hold 4. Identity: $1v = v$ for all $v \in V$

If $T$ is a closed, bounded, three-dimensional region of space, and $f(x,y,z)$ is defined and continuous in a domain containing $T,$ then the volume integral of $f(x,y,z)$ over the region $T$ is denoted by

$$ \iiint_T f(x,y,z) dx dy dz = \iiint_T f(x,y,z) dV. $$

Such integrals can be evaluated via three successive integrations.

The zero vector $(0, 0, \cdots, 0)$ is denoted as $\vec{0}$ and has no direction.