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# Function (mathematics)

In mathematics, a function[1] was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f(x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f.[2] The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).

A function is uniquely represented by its graph which is the set of all pairs (x, f(x)). When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere. For example, graphs of functions are commonly used in newspapers for representing the evolution of price indexes and stock market indexes

Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said that they are "the central objects of investigation" in most fields of mathematics.[3]

Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output
The red curve is the graph of a function, because any vertical line has exactly one crossing point with the curve.
A function that associates any of the four colored shapes to its color.

## Definition

Diagram of a function, with domain X={1, 2, 3} and codomain Y={A, B, C, D}, which is defined by the set of ordered pairs {(1,D), (2,C), (3,C)}. The image/range is the set {C,D}.

This diagram, representing the set of pairs {(1,D), (2,B), (2,C)}, does not define a function. One reason is that 2 is the first element in more than one ordered pair, (2, B) and (2, C), of this set. Two other reasons, also sufficient by themselves, is that neither 3 nor 4 are first elements (input) of any ordered pair therein.

Intuitively, a function is a process that associates to each element of a set X a single element of a set Y.

Formally, a function f from a set X to a set Y is defined by a set G of ordered pairs (x, y) such that xX, yY, and every element of X is the first component of exactly one ordered pair in G.[4][note 1] In other words, for every x in X, there is exactly one element y such that the ordered pair (x, y) belongs to the set of pairs defining the function f. The set G is called the graph of the function. Formally speaking, it may be identified with the function, but this hides the usual interpretation of a function as a process. Therefore, in common usage, the function is generally distinguished from its graph. Functions are also called maps or mappings, though some authors make some distinction between "maps" and "functions" (see #Map vs function).

In the definition of function, X and Y are respectively called the domain and the codomain of the function f. If (x, y) belongs to the set defining f, then y is the image of x under f, or the value of f applied to the argument x. Especially in the context of numbers, one says also that y is the value of f for the value x of its variable, or, still shorter, y is the value of f of x, denoted as y = f(x).

Two functions f and g are equal if their domain and codomain sets are the same and their output values agree on the whole domain. Formally, f = g if f(x) = g(x) for all xX, where f:XY and g:XY.[5][6][note 2]

The domain and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one knows only that the domain is contained in a larger set. Typically, this occurs in mathematical analysis, where "a function from X to Y " often refers to a function that may have a proper subset of X as domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable, and this phrase does not mean that the domain of the function is the whole set of the real numbers, but only that the domain is a set of real numbers that contains a non-empty open interval; such a function is then called a partial function. For example, if f is a function that has the real numbers as domain and codomain, then a function mapping the value x to the value ${\displaystyle g(x)={\tfrac {1}{f(x)}}}$  is a function g from the reals to the reals, whose domain is the set of the reals x, such that f(x) ≠ 0.

The range of a function is the set of the images of all elements in the domain. However, range is sometimes used as a synonym of codomain, generally in old textbooks.

### Relational approach

Any subset of the Cartesian product of a domain ${\displaystyle X}$  and a codomain ${\displaystyle Y}$  is said to define a binary relation ${\displaystyle R\subseteq (X\times Y)}$  between these two sets. It is immediate that an arbitrary relation may contain pairs that violate the necessary conditions for a function, given above.

A univalent relation is a relation such that

${\displaystyle (x,y)\in R\;\land \;(x,z)\in R\quad \Rightarrow \quad y=z.}$

Univalent relations may be identified to functions whose domain is a subset of X.

A left-total relation is a relation such that

${\displaystyle \forall x\in X\;\exists y\in Y\colon (x,y)\in R.}$

Formally, functions may be identified to relations that are both univalent and left total. Violating the left-totality is similar to giving a convenient encompassing set instead of the true domain, as explained above.

Various properties of functions and function composition may be reformulated in the language of relations. For example, a function is injective if the converse relation ${\displaystyle R^{\text{T}}\subseteq (Y\times X)}$  is univalent, where the converse relation is defined as ${\displaystyle R^{\text{T}}=\{(y,x):(x,y)\in R\}.}$ [7]

### As an element of a Cartesian product over a domain

A function is sometimes identified with an element of Cartesian products over the domain of the function. Namely, given sets ${\displaystyle X,Y}$ , a function ${\displaystyle f:X\to Y}$  from a set X to a set Y is an element of the Cartesian products of copies of ${\displaystyle Y}$ 's

${\displaystyle \prod _{X}Y}$

over the index set ${\displaystyle X}$ : for each ${\displaystyle x\in X}$ , the x-th coordinate of the element is the value ${\displaystyle f(x)}$ . It reflects the intuition that for each ${\displaystyle x\in X}$ , the function picks some element ${\displaystyle y\in Y}$ , namely, ${\displaystyle f(x)}$ . (This point of view is used for example in the discussion of a choice function.)

Note: infinite Cartesian products are often simply "defined" as sets of functions.[8]

## Notation

There are various standard ways for denoting functions. The most commonly used notation is functional notation, which defines the function using an equation that gives the names of the function and the argument explicitly. This gives rise to a subtle point, often glossed over in elementary treatments of functions: functions are distinct from their values. Thus, a function f should be distinguished from its value f(x0) at the value x0 in its domain. To some extent, even working mathematicians will conflate the two in informal settings for convenience, and to avoid the use of pedantic language. However, strictly speaking, it is an abuse of notation to write "let ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$  be the function f(x) = x2 ", since f(x) and x2 should both be understood as the value of f at x, rather than the function itself. Instead, it is correct, though pedantic, to write "let ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$  be the function defined by the equation f(x) = x2, valid for all real values of x ".

This distinction in language and notation becomes important in cases where functions themselves serve as inputs for other functions. (A function taking another function as an input is termed a functional.) Other approaches to denoting functions, detailed below, avoid this problem but are less commonly used.

### Functional notation

As first used by Leonhard Euler in 1734,[9] functions are denoted by a symbol consisting generally of a single letter in italic font, most often the lower-case letters f, g, h. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). By convention, in this case, a roman type is used, such as "sin" for the sine function, in contrast to italic font for single-letter symbols.

The notation (read: "y equals f of x")

${\displaystyle y=f(x)}$

means that the pair (x, y) belongs to the set of pairs defining the function f. If X is the domain of f, the set of pairs defining the function is thus, using set-builder notation,

${\displaystyle \{(x,f(x)):x\in X\}.}$

Often, a definition of the function is given by what f does to the explicit argument x. For example, a function f can be defined by the equation

${\displaystyle f(x)=\sin(x^{2}+1)}$

for all real numbers x. In this example, f can be thought of as the composite of several simpler functions: squaring, adding 1, and taking the sine. However, only the sine function has a common explicit symbol (sin), while the combination of squaring and then adding 1 is described by the polynomial expression ${\displaystyle x^{2}+1}$ . In order to explicitly reference functions such as squaring or adding 1 without introducing new function names (e.g., by defining function g and h by ${\displaystyle g(x)=x^{2}}$  and ${\displaystyle h(x)=x+1}$ ), one of the methods below (arrow notation or dot notation) could be used.

Sometimes the parentheses of functional notation are omitted when the symbol denoting the function consists of several characters and no ambiguity may arise. For example, ${\displaystyle \sin x}$  can be written instead of ${\displaystyle \sin(x).}$

### Arrow notation

For explicitly expressing domain X and the codomain Y of a function f, the arrow notation is often used (read: "the function f from X to Y" or "the function f mapping elements of X to elements of Y"):

${\displaystyle f\colon X\to Y}$

or

${\displaystyle X~{\stackrel {f}{\to }}~Y.}$

This is often used in relation with the arrow notation for elements (read: "f maps x to f(x)"), often stacked immediately below the arrow notation giving the function symbol, domain, and codomain:

${\displaystyle x\mapsto f(x).}$

For example, if a multiplication is defined on a set X, then the square function ${\displaystyle \operatorname {sqr} }$  on X is unambiguously defined by (read: "the function ${\displaystyle \operatorname {sqr} }$  from X to X that maps x to xx")

{\displaystyle {\begin{aligned}\operatorname {sqr} \colon X&\to X\\x&\mapsto x\cdot x,\end{aligned}}}

the latter line being more commonly written

${\displaystyle x\mapsto x^{2}.}$

Often, the expression giving the function symbol, domain and codomain is omitted. Thus, the arrow notation is useful for avoiding introducing a symbol for a function that is defined, as it is often the case, by a formula expressing the value of the function in terms of its argument. As a common application of the arrow notation, suppose ${\displaystyle f\colon X\times X\to Y;\;(x,t)\mapsto f(x,t)}$  is a two-argument function, and we want to refer to a partially applied function ${\displaystyle X\to Y}$  produced by fixing the second argument to the value t0 without introducing a new function name. The map in question could be denoted ${\displaystyle x\mapsto f(x,t_{0})}$  using the arrow notation for elements. Note that the expression ${\displaystyle x\mapsto f(x,t_{0})}$  (read: "the map taking x to ${\displaystyle f(x,t_{0})}$ ") represents this new function with just one argument, whereas the expression ${\displaystyle f(x_{0},t_{0})}$  refers to the value of the function f at the point ${\displaystyle (x_{0},t_{0})}$ .

### Index notation

Index notation is often used instead of functional notation. That is, instead of writing f(x), one writes ${\displaystyle f_{x}.}$

This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence, and, in this case the element ${\displaystyle f_{n}}$  is called the nth element of sequence.

The index notation is also often used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map ${\displaystyle x\mapsto f(x,t)}$  (see above) would be denoted ${\displaystyle f_{t}}$  using index notation, if we define the collection of maps ${\displaystyle f_{t}}$  by the formula ${\displaystyle f_{t}(x)=f(x,t)}$  for all ${\displaystyle x,t\in X}$ .

### Dot notation

In the notation ${\displaystyle x\mapsto f(x),}$  the symbol x does not represent any value, it is simply a placeholder meaning that, if x is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, x may be replaced by any symbol, often an interpunct "". This may be useful for distinguishing the function f(⋅) from its value f(x) at x.

For example, ${\displaystyle a(\cdot )^{2}}$  may stand for the function ${\displaystyle x\mapsto ax^{2}}$ , and ${\displaystyle \textstyle \int _{a}^{\,(\cdot )}f(u)\,du}$  may stand for a function defined by an integral with variable upper bound: ${\displaystyle \textstyle x\mapsto \int _{a}^{x}f(u)\,du}$ .

### Specialized notations

There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis, linear forms and the vectors they act upon are denoted using a dual pair to show the underlying duality. This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.

## Map vs function

A function is often also called a map or a mapping. But some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure; e.g., a group homomorphism between groups can simply be called a map between those groups for the sake of succinctness. See also: maps of manifolds. Some authors[10] reserve the word mapping to the case where the codomain Y belongs explicitly to the definition of the function. In this sense, the graph of the mapping recovers the function as the set of pairs.

Because the term "map" is synonymous with "morphism" in category theory, the term "map" can in particular emphasize the aspect that a function is a morphism in the category of sets: in the informal definition of function ${\displaystyle f:X\to Y}$ , it is a subset of ${\displaystyle X\times Y}$  consisting of all the pairs ${\displaystyle (x,f(x))}$  for ${\displaystyle x\in X}$ . In this sense, the function doesn't capture the information of which set ${\displaystyle Y}$  is used as the co-domain. Only the range ${\displaystyle f(X)}$  is determined by the function.

Some authors, such as Serge Lang,[11] use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of the fields R or C) and the term mapping for more general functions.

In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.

A partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, with the same meaning. All these usages can be applied to "maps" as general functions or as functions with special properties.

## Specifying a function

Given a function ${\displaystyle f}$ , by definition, to each element ${\displaystyle x}$  of the domain of the function ${\displaystyle f}$ , there is a unique element associated to it, the value ${\displaystyle f(x)}$  of ${\displaystyle f}$  at ${\displaystyle x}$ . There are several ways to specify or describe how ${\displaystyle x}$  is related to ${\displaystyle f(x)}$ , both explicitly and implicitly (or axiomatically). Often, the specification or description is referred to as the definition of the function ${\displaystyle f}$ .

• On a finite set, a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. E.g., if ${\displaystyle A=\{1,2,3\}}$ , then one can define a function ${\displaystyle f:A\to \mathbb {R} }$  by ${\displaystyle f(1)=2,f(2)=3,f(3)=4.}$
• Depending on a function, if possible, one might use a (closed) formula to define the function. For example, in the above example, ${\displaystyle f}$  can be defined by the formula ${\displaystyle f(n)=n+1}$ , for ${\displaystyle n\in \{1,2,3\}}$ .
• One often combines arithmetic or algebraic operations to define a new function; for example, a function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ , defined by ${\displaystyle f(x)={\sqrt {1+x^{2}}}}$ , is defined by a combination of the polynomial ${\displaystyle 1+x^{2}}$  and the square root function.
• The following is an example of an implicit definition of a function: using calculus, one can show that, for each real number ${\displaystyle x}$ , ${\displaystyle y^{2}=x^{2}+1}$  has two distinct real roots, a positive and a negative one. One can then define ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$  by saying that ${\displaystyle f(x)}$  is the positive root of the equation ${\displaystyle y^{2}=x^{2}+1}$  (cf. #Multi-valued functions). This is the same function as the previous one. The natural logarithm is often defined by an appeal to the implicit function theorem.
• A power series can be used to define a function on the domain in which the series converges: for example, the exponential function is given by ${\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}}$ .
• In general, given a function ${\displaystyle f}$ , there might not be a (closed) formula or condition that can be used to define it. An elementary function[note 3] is a function that can be defined by a finite number of applications of arithmetic operations and composition involving only constants, arithmetic operations, exponential functions, logarithms, and roots of polynomials.
• A choice function in the axiom of choice is an example of a function that exists only by postulation.

A recurrence relation is sometimes used to define a function. That this can be done relies formally on the following theorem.

Recursion theorem[12] — Let ${\displaystyle \varphi :X\to X}$  be a function from a nonempty set ${\displaystyle X}$  to itself and ${\displaystyle x_{0}\in X}$  an element. Then there exists a unique function ${\displaystyle f:\mathbb {N} \to X}$  such that (1) ${\displaystyle f(0)=x_{0}}$  and (2) ${\displaystyle f(n+1)=\varphi (f(n))}$  for each integer ${\displaystyle n\geq 0}$ .

The theorem is a consequence of the axiom of induction of the natural numbers in the Peano axioms (there is also a transfinite version of the above theorem; see transfinite recursion.) Another inductive way of specifying a function is through analytic continuation. For example, for a complex variable ${\displaystyle s}$ , the series

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}}$

can be shown to converge if the real part of ${\displaystyle s}$  is strictly greater than 1. Hence, it defines a function on the domain ${\displaystyle \{s\in \mathbb {C} |\operatorname {Re} (s)>1\}}$ . This function is analytic and has a unique analytic extension defined on ${\displaystyle \mathbb {C} -\{1\}}$ , called the Riemann zeta function. Note the value ${\displaystyle \zeta (s)}$  is given by the above series if ${\displaystyle \operatorname {Re} (s)>1}$  but is not so in general.

Differential equations are sometimes used to define a function. For example, the natural logarithm is the solution of the initial value problem ${\displaystyle {\frac {d(\ln |x|)}{dx}}={\frac {1}{x}}}$  such that ${\displaystyle \ln(1)=0}$ .

## Representing a function

A graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.

### Graphs and plots

The function mapping each year to its US motor vehicle death count, shown as a line chart

The same function, shown as a bar chart

Given a function ${\displaystyle f\colon X\to Y,}$  its graph is, formally, the set

${\displaystyle G=\{(x,f(x)):x\in X\}.}$

In the frequent case where X and Y are subsets of the real numbers (or may be identified with such subsets, e.g. intervals), an element ${\displaystyle (x,y)\in G}$  may be identified with a point having coordinates x, y in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a plot that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the graph of the function. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the square function

${\displaystyle x\mapsto x^{2},}$

consisting of all points with coordinates ${\displaystyle (x,x^{2})}$  for ${\displaystyle x\in \mathbb {R} ,}$  yields, when depicted in Cartesian coordinates, the well known parabola. If the same quadratic function ${\displaystyle x\mapsto x^{2},}$  with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates ${\displaystyle (r,\theta )=(x,x^{2}),}$  the plot obtained is Fermat's spiral.

### Tables

A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function ${\displaystyle f\colon \{1,\ldots ,5\}^{2}\to \mathbb {R} }$  defined as ${\displaystyle f(x,y)=xy}$  can be represented by the familiar multiplication table

y
x
1 2 3 4 5
1 1 2 3 4 5
2 2 4 6 8 10
3 3 6 9 12 15
4 4 8 12 16 20
5 5 10 15 20 25

On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:

x sin x
1.289 0.960557
1.290 0.960835
1.291 0.961112
1.292 0.961387
1.293 0.961662

Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.

### Bar chart

Bar charts are often used for representing functions whose domain is a finite set, the natural numbers, or the integers. In this case, an element x of the domain is represented by an interval of the x-axis, and the corresponding value of the function, f(x), is represented by a rectangle whose base is the interval corresponding to x and whose height is f(x) (possibly negative, in which case the bar extends below the x-axis).

## General properties

This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.

### Canonical functions

Some functions are uniquely defined by their domain and codomain, and are sometimes called canonical:

• For every set X, there is a unique function, called the empty function from the empty set to X. This function is not interesting by itself, but useful for simplifying statements, similarly as the empty sum (equal to 0) and the empty product equal to 1.
• For every set X and every singleton set {s}, there is a unique function, called the canonical surjection, from X to {s}, which maps to s every element of X. This is a surjection (see below), except if X is the empty set.
• Given a function ${\displaystyle f\colon X\to Y,}$  the canonical surjection of f onto its image ${\displaystyle f(X)=\{f(x)\mid x\in X\}}$  is the function from X to f(X) that maps x to f(x)
• For every subset X of a set Y, the canonical injection of X into Y is the injective (see below) function that maps every element of X to itself.
• The identity function of X, often denoted by ${\displaystyle \operatorname {id} _{X}}$  is the canonical injection of X into itself.

### Function composition

Given two functions ${\displaystyle f\colon X\to Y}$  and ${\displaystyle g\colon Y\to Z}$  such that the domain of g is the codomain of f, their composition is the function ${\displaystyle g\circ f\colon X\rightarrow Z}$  defined by

${\displaystyle (g\circ f)(x)=g(f(x)).}$

That is, the value of ${\displaystyle g\circ f}$  is obtained by first applying f to x to obtain y =f(x) and then applying g to the result y to obtain g(y) = g(f(x)). In the notation the function that is applied first is always written on the right.

The composition ${\displaystyle g\circ f}$  is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both ${\displaystyle g\circ f}$  and ${\displaystyle f\circ g}$  satisfy these conditions, the composition is not necessarily commutative, that is, the functions ${\displaystyle g\circ f}$  and ${\displaystyle f\circ g}$  need not be equal, but may deliver different values for the same argument. For example, let f(x) = x2 and g(x) = x + 1, then ${\displaystyle g(f(x))=x^{2}+1}$  and ${\displaystyle f(g(x))=(x+1)^{2}}$  agree just for ${\displaystyle x=0.}$

The function composition is associative in the sense that, if one of ${\displaystyle (h\circ g)\circ f}$  and ${\displaystyle h\circ (g\circ f)}$  is defined, then the other is also defined, and they are equal. Thus, one writes

${\displaystyle h\circ g\circ f=(h\circ g)\circ f=h\circ (g\circ f).}$

The identity functions ${\displaystyle \operatorname {id} _{X}}$  and ${\displaystyle \operatorname {id} _{Y}}$  are respectively a right identity and a left identity for functions from X to Y. That is, if f is a function with domain X, and codomain Y, one has ${\displaystyle f\circ \operatorname {id} _{X}=\operatorname {id} _{Y}\circ f=f.}$

### Image and preimage

Let ${\displaystyle f\colon X\to Y.}$  The image by f of an element x of the domain X is f(x). If A is any subset of X, then the image of A by f, denoted f(A) is the subset of the codomain Y consisting of all images of elements of A, that is,

${\displaystyle f(A)=\{f(x)\mid x\in A\}.}$

The image of f is the image of the whole domain, that is f(X). It is also called the range of f, although the term may also refer to the codomain.[13]

On the other hand, the inverse image, or preimage by f of a subset B of the codomain Y is the subset of the domain X consisting of all elements of X whose images belong to B. It is denoted by ${\displaystyle f^{-1}(B).}$  That is

${\displaystyle f^{-1}(B)=\{x\in X\mid f(x)\in B\}.}$

For example, the preimage of {4, 9} under the square function is the set {−3,−2,2,3}.

By definition of a function, the image of an element x of the domain is always a single element of the codomain. However, the preimage of a single element y, denoted ${\displaystyle f^{-1}(x),}$  may be empty or contain any number of elements. For example, if f is the function from the integers to themselves that map every integer to 0, then f−1(0) = Z.

If ${\displaystyle f\colon X\to Y}$  is a function, A and B are subsets of X, and C and D are subsets of Y, then one has the following properties:

• ${\displaystyle A\subseteq B\Longrightarrow f(A)\subseteq f(B)}$
• ${\displaystyle C\subseteq D\Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)}$
• ${\displaystyle A\subseteq f^{-1}(f(A))}$
• ${\displaystyle C\supseteq f(f^{-1}(C))}$
• ${\displaystyle f(f^{-1}(f(A)))=f(A)}$
• ${\displaystyle f^{-1}(f(f^{-1}(C)))=f^{-1}(C)}$

The preimage by f of an element y of the codomain is sometimes called, in some contexts, the fiber of y under f.

If a function f has an inverse (see below), this inverse is denoted ${\displaystyle f^{-1}.}$  In this case ${\displaystyle f^{-1}(C)}$  may denote either the image by ${\displaystyle f^{-1}}$  or the preimage by f of C. This is not a problem, as these sets are equal. The notation ${\displaystyle f(A)}$  and ${\displaystyle f^{-1}(C)}$  may be ambiguous in the case of sets that contain some subsets as elements, such as ${\displaystyle \{x,\{x\}\}.}$  In this case, some care may be needed, for example, by using square brackets ${\displaystyle f[A],f^{-1}[C]}$  for images and preimages of subsets, and ordinary parentheses for images and preimages of elements.

### Injective, surjective and bijective functions

Let ${\displaystyle f\colon X\to Y}$  be a function.

The function f is injective (or one-to-one, or is an injection) if f(a) ≠ f(b) for any two different elements a and b of X. Equivalently, f is injective if, for any ${\displaystyle y\in Y,}$  the preimage ${\displaystyle f^{-1}(y)}$  contains at most one element. An empty function is always injective. If X is not the empty set, and if, as usual, the axiom of choice is assumed, then f is injective if and only if there exists a function ${\displaystyle g\colon y\to X}$  such that ${\displaystyle g\circ f=\operatorname {id} _{X},}$  that is, if f has a left inverse. The axiom of choice is needed, because, if f is injective, one defines g by ${\displaystyle g(y)=x}$  if ${\displaystyle y=f(x),}$  and by ${\displaystyle g(y)=x_{0}}$ , if ${\displaystyle y\not \in f(X),}$  where ${\displaystyle x_{0}}$  is an arbitrarily chosen element of X.

The function f is surjective (or onto, or is a surjection) if the range equals the codomain, that is, if f(X) = Y. In other words, the preimage ${\displaystyle f^{-1}(y)}$  of every ${\displaystyle y\in Y}$  is nonempty. If, as usual, the axiom of choice is assumed, then f is surjective if and only if there exists a function ${\displaystyle g\colon y\to X}$  such that ${\displaystyle f\circ g=\operatorname {id} _{Y},}$  that is, if f has a right inverse. The axiom of choice is needed, because, if f is injective, one defines g by ${\displaystyle g(y)=x,}$  where ${\displaystyle x}$  is an arbitrarily chosen element of ${\displaystyle f^{-1}(y).}$

The function f is bijective (or is bijection or a one-to-one correspondence) if it is both injective and surjective. That is f is bijective if, for any ${\displaystyle y\in Y,}$  the preimage ${\displaystyle f^{-1}(y)}$  contains exactly one element. The function f is bijective if and only if it admits an inverse function, that is a function ${\displaystyle g\colon y\to X}$  such that ${\displaystyle g\circ f=\operatorname {id} _{X},}$  and ${\displaystyle f\circ g=\operatorname {id} _{Y}.}$  (Contrarily to the case of injections and surjections, this does not require the axiom of choice.)

Every function ${\displaystyle f\colon X\to Y}$  may be factorized as the composition is of a surjection followed by an injection, where s is the canonical surjection of X onto f(X), and i is the canonical injection of f(X) into Y. This is the canonical factorization of f.

"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement "f maps X onto Y" differs from "f maps X into B" in that the former implies that f is surjective), while the latter makes no assertion about the nature of f the mapping. In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage to be more symmetrical.

### Restriction and extension

If ${\displaystyle f\colon X\to Y}$  is a function, and S is a subset of X, then the restriction of f to S, denoted f|S, is the function from S to Y that is defined by

${\displaystyle f_{|S}(x)=f(x)\quad {\text{for all }}x\in S.}$

This often used for define partial inverse functions: if there is a subset S of a function f such that f|S is injective, then the canonical surjection of f|S on its image f|S(S) = f(S) is a bijection, which has an inverse function from f(S) to S. This is in this way that inverse trigonometric functions are defined. The cosine function, for example, is injective, when restricted to the interval (–0, π); the image of this restriction is the interval (–1, 1); this defines thus an inverse function from (–1, 1) to (–0, π), which is called arccosine and denoted arccos.

Function restriction may also be used for "gluing" functions together: let ${\displaystyle \textstyle X=\bigcup _{i\in I}U_{i}}$  be the decomposition of X as a union of subsets. Suppose that a function ${\displaystyle f_{i}\colon U_{i}\to Y}$  is defined on each ${\displaystyle U_{i},}$  such that, for each pair of indices, the restrictions of ${\displaystyle f_{i}}$  and ${\displaystyle f_{j}}$  to ${\displaystyle U_{i}\cap U_{j}}$  are equal. Then, this defines a unique function ${\displaystyle f\colon X\to Y}$  such that ${\displaystyle f_{|U_{i}}=f_{i}}$  for every i. This is generally in this way that functions on manifolds are defined.

An extension of a function f is a function g such that f is a restriction of g. A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.

Here is another classical example of a function extension that is encountered when studying homographies of the real line. An homography is a function ${\displaystyle h(x)={\frac {ax+b}{cx+d}}}$  such that adbc ≠ 0. Its domain is the set of all real numbers different from ${\displaystyle -d/c,}$  and its image is the set of all real numbers different from ${\displaystyle a/c.}$  If one extends the real line to the projectively extended real line by adding to the real numbers, one may extend h for being a bijection of the extended real line to itself, by setting ${\displaystyle h(\infty )=a/c}$  and ${\displaystyle h(-d/c)=\infty .}$

## Multivariate function

A binary operation is a typical example of a bivariate, function which assigns to each pair ${\displaystyle (x,y)}$  the result ${\displaystyle x\circ y}$ .

A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time and its speed.

More formally, a function of n variables is a function whose domain is a set of n-tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. More generally, every mathematical operation is defined as a multivariate function.

The Cartesian product ${\displaystyle X_{1}\times \cdots \times X_{n}}$  of n sets ${\displaystyle X_{1},\ldots ,X_{n}}$  is the set of all n-tuples ${\displaystyle (x_{1},\ldots ,x_{n})}$  such that ${\displaystyle x_{i}\in X_{i}}$  for every i with ${\displaystyle 1\leq i\leq n}$ . Therefore, a function of n variables is a function

${\displaystyle f\colon U\to Y,}$

where the domain U has the form

${\displaystyle U\subseteq X_{1}\times \cdots \times X_{n}.}$

When using function notation, one usually omits the parentheses surrounding tuples, writing ${\displaystyle f(x_{1},x_{2})}$  instead of ${\displaystyle f((x_{1},x_{2})).}$

In the case where all the ${\displaystyle X_{i}}$  are equal to the set ${\displaystyle \mathbb {R} }$  of real numbers, one has a function of several real variables. If the ${\displaystyle X_{i}}$  are equal to the set ${\displaystyle \mathbb {C} }$  of complex numbers, one has a function of several complex variables.

It is common to also consider functions whose codomain is a product of sets. For example, Euclidean division maps every pair (a, b) of integers with b ≠ 0 to a pair of integers called the quotient and the remainder:

{\displaystyle {\begin{aligned}{\text{Euclidean division}}\colon \quad \mathbb {Z} \times (\mathbb {Z} \setminus \{0\})&\to \mathbb {Z} \times \mathbb {Z} \\(a,b)&\mapsto (\operatorname {quotient} (a,b),\operatorname {remainder} (a,b)).\end{aligned}}}

The codomain may also be a vector space. In this case, one talks of a vector-valued function. If the domain is contained in a Euclidean space, or more generally a manifold, a vector-valued function is often called a vector field.

## In calculus

The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus (see History of the function concept). At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to functions of several variables and to function of a complex variable. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.

Functions are now used throughout all areas of mathematics. In introductory calculus, when the word function is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.

### Real function

Graph of a linear function

Graph of a polynomial function, here a quadratic function.

Graph of two trigonometric functions: sine and cosine.

A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval. In this section, these functions are simply called functions.

The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous, differentiable, and even analytic. This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.

Functions enjoy pointwise operations, that is, if f and g are functions, their sum, difference and product are functions defined by

{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(f-g)(x)&=f(x)-g(x)\\(f\cdot g)(x)&=f(x)\cdot g(x)\\\end{aligned}}.}

The domains of the resulting functions are the intersection of the domains of f and g. The quotient of two functions is defined similarly by

${\displaystyle {\frac {f}{g}}(x)={\frac {f(x)}{g(x)}},}$

but the domain of the resulting function is obtained by removing the zeros of g from the intersection of the domains of f and g.

The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function ${\displaystyle x\mapsto {\frac {1}{x}},}$  whose graph is a hyperbola, and whose domain is the whole real line except for 0.

The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that is differentiable in any open interval in which the original function is continuous. For example, the function ${\displaystyle x\mapsto {\frac {1}{x}}}$  is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for x = 1, is a differentiable function called the natural logarithm.

A real function f is monotonic in an interval if the sign of ${\displaystyle {\frac {f(x)-f(y)}{x-y}}}$  does not depend of the choice of x and y in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function f is monotonic in an interval I, it has an inverse function, which is a real function with domain f(I) and image I. This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function.

Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation

${\displaystyle y''+y=0}$

such that

${\displaystyle \sin 0=0,\quad \cos 0=1,\quad {\frac {\partial \sin x}{\partial x}}(0)=1,\quad {\frac {\partial \cos x}{\partial x}}(0)=0.}$

### Function of several real or complex variables

Functions of several real (or complex) variables are functions which domain consists of tuples of real (or complex) numbers. This is, the domain is a subset of ${\displaystyle \mathbb {R} ^{n}}$  or ${\displaystyle \mathbb {C} ^{n}}$ . For example, the division operation is a function that inputs a pair of real or complex numbers, one to be the dividend and the other the divisor, and returns their quotient ${\displaystyle (a,b)\mapsto a/b}$ .

In the case of functions of several variables, the functional notation can be used in two ways. The symbol ${\displaystyle f(x)}$  could denote a function of several variables, if ${\displaystyle x}$  is assumed to be denoting tuples of real or complex numbers ${\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})}$ . Alternatively, the components of the tuple can be made explicit in the notation when this is written as ${\displaystyle f(x_{1},x_{2},\ldots ,x_{n})}$ . The repeated parentheses ${\displaystyle f((x_{1},x_{2},\ldots ,x_{n}))}$  that would result from syntactically replacing ${\displaystyle x}$  by ${\displaystyle (x_{1},x_{2},\ldots ,x_{n})}$  in the symbol ${\displaystyle f(x)}$  are generally avoided in mathematics.

### Vector-valued function

When the elements of the co-domain of a function are vectors the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. The function that associates to each point of a fluid its velocity vector is a vector-valued function.

Some vector-valued function are defined on a subset of ${\displaystyle \mathbb {R} ^{n}}$  or other spaces that share geometric or topological properties similar to ${\displaystyle \mathbb {R} ^{n}}$ , like manifolds. These vector-valued functions are given the name vector fields.

## Function space

In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact set) form a function space that is at the basis of the theory of distributions.

Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.

## Multi-valued functions

Together, the two square roots of all nonnegative real numbers form a single smooth curve.

Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point ${\displaystyle x_{0},}$  there are several possible starting values for the function.

For example, in defining the square root as the inverse function of the square function, for any positive real number ${\displaystyle x_{0},}$  there are two choices for the value of the square root, one of which is positive and denoted ${\displaystyle {\sqrt {x_{0}}},}$  and another which is negative and denoted ${\displaystyle -{\sqrt {x_{0}}}.}$  These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive x, one value for 0 and no value for negative x.

In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps y to a root x of ${\displaystyle x^{3}-3x-y=0}$  (see the figure on the right). For y = 0 one may choose either ${\displaystyle 0,{\sqrt {3}},{\text{ or }}-{\sqrt {3}}}$  for x. By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval [–2, 2] and the image is [–1, 1]; for the second one, the domain is [–2, ∞) and the image is [1, ∞); for the last one, the domain is (–∞, 2] and the image is (–∞, –1]. As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single multi-valued function of y that has three values for –2 < y < 2, and only one value for y ≤ –2 and y ≥ –2.

Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets i for the square root of –1; while, when extending through complex numbers with negative imaginary parts, one gets i. There are generally two ways of solving the problem. One may define a function that is not continuous along some curve, called a branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.

## In foundations of mathematics and set theory

The definition of a function that is given in this article requires the concept of set, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.

For example, the singleton set may be considered as a function ${\displaystyle x\mapsto \{x\}.}$  Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definition for these weakly specified functions.[14]

These generalized functions may be critical in the development of a formalization of foundations of mathematics. For example, the Von Neumann–Bernays–Gödel set theory, is an extension of the set theory in which the collection of all sets is a class. This theory includes the replacement axiom, which may be interpreted as "if X is a set, and F is a function, then F[X] is a set".

## In computer science

In computer science, functions are callable units of code. For example, the code

   define square_function(x):
return x*x

can be called by the expression square_function(x), where x can be replaced, for example, by numeric literals as in square_function(2) or identifiers of other valid input.

The calling of the unit of code consists in the expression square_function(x) being replaced by the execution of the code inside the function's definition. In the example, it would be replaced by executing the evaluation of the expression x*x. This expression in turn, could be indicating to compute the multiplication of x by itself, in the case of x representing a numeric value. The evaluation would then compute the square of x. The result of this evaluation is called the return value of the function.

As in the example, some functions in computer science behave like mathematical functions. The computer science function of the example would normally behave for numerical inputs like the mathematical function ${\displaystyle f(x)=x^{2}}$ . On the other hand, not all functions in computer science behave like mathematical functions. Functions in computer science could not have a return value. The return value could not be unique for a given input as they are required to be for mathematical functions. This is due to the execution of the unit of code depending also on the state of the computing machine at the time in which the call is executed.

A function may[clarification needed] be described by an algorithm, and any kind of algorithm may[clarification needed] be used. Sometimes, the definition of a function may involve elements or properties that can be defined, but not computed. For example, if one considers the set ${\displaystyle {\mathcal {P}}}$  of the programs in a given programming language that take an integer as input. The terminating function is the function that returns 1 if a program of ${\displaystyle {\mathcal {P}}}$  runs forever when executed on a given integer input, and returns 0 otherwise. It is a basic theorem of computability theory that there does not exist an algorithm for computing this function. More generally, computability theory is the study of computable functions, that is, the functions that can be computed by an algorithm.

The above ways of defining functions define them "pointwise", that is, each value is defined independently of the other values. This is not necessarily the case.

## Notes

1. ^ The sets X, Y are parts of data defining a function; i.e., a function is a set of ordered pairs ${\displaystyle (x,y)}$  with ${\displaystyle x\in X,y\in Y}$ , together with the sets X, Y, such that for each ${\displaystyle x\in X}$ , there is a unique ${\displaystyle y\in Y}$  with ${\displaystyle (x,y)}$  in the set.
2. ^ This follows from the axiom of extensionality, which says two sets are the same if and only if they have the same members. Some authors drop codomain from a definition of a function, and in that definition, the notion of equality has to be handled with care; see, for example, https://math.stackexchange.com/questions/1403122/when-do-two-functions-become-equal
3. ^ Here "elementary" has not exactly its common sense: although most functions that are encountered in elementary courses of mathematics are elementary in this sense, some elementary functions are not elementary for the common sense, for example, those that involve roots of polynomials of high degree.
1. ^ The words map, mapping, transformation, correspondence, and operator are often used synonymously. Halmos 1970, p. 30.
2. ^ MacLane, Saunders; Birkhoff, Garrett (1967). Algebra (First ed.). New York: Macmillan. pp. 1–13.
3. ^ Spivak 2008, p. 39.
4. ^ Hamilton, A. G. (1982). Numbers, sets, and axioms: the apparatus of mathematics. Cambridge University Press. p. 83. ISBN 978-0-521-24509-8.
5. ^ Apostol 1981, p. 35.
6. ^ Kaplan 1972, p. 25.
7. ^ Gunther Schmidt( 2011) Relational Mathematics, Encyclopedia of Mathematics and its Applications, vol. 132, sect 5.1 Functions, pp. 49–60, Cambridge University Press ISBN 978-0-521-76268-7 CUP blurb for Relational Mathematics
8. ^ Halmos, Naive Set Theory, 1968, sect.9 ("Families")
9. ^ Ron Larson, Bruce H. Edwards (2010), Calculus of a Single Variable, Cengage Learning, p. 19, ISBN 978-0-538-73552-0
10. ^ T. M. Apostol (1981). Mathematical Analysis. Addison-Wesley. p. 35.
11. ^ Lang, Serge (1971), Linear Algebra (2nd ed.), Addison-Wesley, p. 83
12. ^ Jacobson, § 0.4. pg 16.
13. ^ Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, p. 15. ISO 80000-2 (ISO/IEC 2009-12-01)
14. ^ Gödel 1940, p. 16; Jech 2003, p. 11; Cunningham 2016, p. 57


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# Geometrical interpretation of dot product (trying to prove)

1. Aug 14, 2009

### Дьявол

Hi! I am trying to find out where:

$$cos\theta=\frac{a \cdot b}{|a||b|}$$

came from.

Here is mine geometrical interpretation of scalar projection:

Now, (pr means projection)

$$pr_{\overrightarrow{A}} \overrightarrow{B} = p\overrightarrow{B_0}$$

and

$$cos\theta=\frac{|pr_{\overrightarrow{A}} \overrightarrow{B}|}{|A|}$$

$$cos\theta=\frac{|pB_0|}{|A|}=\frac{|p|}{|A|}$$

How do

$$cos\theta=\frac{a_xb_x+a_yb_y+a_zb_z}{|a||b|}$$

for 3 dimensional coordinate system, wher |a| and |b| are modulo of "a" and "b" ?

Regards.

2. Aug 15, 2009

### GPPaille

Your use of the projection is recursive, since the projection of a vector use the fact that cos(theta) = dot(A,B)/|A||B|. Wikipedia offers a simple proof, look for the dot product. And the proof is valid in every dimension, not only in 2D.

3. Aug 15, 2009

### Дьявол

Ohh, thank you. The proof is very simple. And why my use of the projection is recursive? The scalar projection equals |A| cos(θ).

4. Aug 15, 2009

### GPPaille

But the dot product is hidden in |p|, and the way we compute the projection using the dot product uses the fact that we know that it's equal to cos(theta)/|A||B|. Explicitly, it looks like this:

1- Hypothesis: A dot B = cos(theta)/|A||B|
2- Use it to define projection
3- Use projection arguments to prove 1.

5. Aug 15, 2009

### arildno

An inner product space is a particular type of normed spaces, thus it makes little meaning to say that the inner product is "hidden" within the concept of length.

6. Aug 15, 2009

### GPPaille

Here, p is a scalar, that comes magically from the projection. And if it don't use the dot product to compute p, then I don't see the link between the dot product and cos(theta).

7. Aug 15, 2009

### Дьявол

Thanks for the replies. Now I understand the dot product. But what about the cross product?

$$|\overrightarrow{c}|=|\overrightarrow{a}||\overrightarrow{b}|sin(\overrightarrow{a},\overrightarrow{b})$$

$$\overrightarrow{c} \cdot \overrightarrow{a} = \overrightarrow{0}$$

and

$$\overrightarrow{c} \cdot \overrightarrow{b} = \overrightarrow{0}$$

How is possible that the area of the parallelogram is the magnitude of a x b or c ??

Regards.

8. Aug 15, 2009

### HallsofIvy

The length of a cross product, $a\times b$ is, as you say, $|a||b|sin(\theta)$ where $\theta$ is the angle between vectors a and b. In addition, the area of a parallelogram is "height times base" where "height" is measured perpendicular to the "base". If you take, for example, the vector b as base, then you can drop a perpendicular from the tip of a to b to get a right triangle- the length of the perpendicular is "height" and, since the right triangle has angle $\theta$ and hypotenuse of length |a|, that perpendicular is the "opposite leg" so $h/|a|= sin(\theta)$ and $h= |a|sin(\theta)$.

9. Aug 16, 2009

### Дьявол

Thanks for the reply.
Yes, I know that the area of parallelogram is A=aha, and $sin\theta=\frac{h_a}{b}$, so that $h_a=bsin\theta$ and $A=absin\theta$. What I do not understand is why the vector "c" got magnitude of the area of the parallelogram between a and b, i.e the magnitude of the vector is $A=absin\theta$. Why the magnitude isn't a*b or some other value?

Regards.

10. Aug 16, 2009

### GPPaille

Using Lagrange's identity identity, you get

$$|a \times b|^2 = (a \cdot a)(b \cdot b) - (a \cdot b)(a \cdot b)$$

Which is simply equal to

$$|a \times b|^2 = |a|^2|b|^2 - |a|^2|b|^2cos^2(\theta)$$
$$|a \times b|^2 = |a|^2|b|^2 (1 - cos^2(\theta))$$
$$|a \times b|^2 = |a|^2|b|^2 sin^2(\theta)$$

11. Aug 16, 2009

### Дьявол

Thanks for the reply. I know that, but why the magnitude of the a x b, is $|a||b|sin\theta$? If we suppose that c is vector so that it is normal to a and b (i.e c . a =0 and c.b=0) and a x b = c, why the magnitude of c (i.e |c| ) is $|c|=|a||b|sin\theta$??

Regards.

12. Aug 16, 2009

### GPPaille

Then I really don't see what you want to know. I proved by an algebraic property that $$|c|=|a||b|sin\theta$$ and I did not suppose anything.

13. Aug 16, 2009

### Дьявол

Yes, its actually that's what I do not understand. Look at the picture. The magnitude of |c| equals the area of the parallelogram.

I already proved that $$A=|a||b|sin\theta$$ where A is the area of the parallelogram. But what I do not understand how A and c are linked together?

Regards.

P.S Here is thehttp://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Cross_product_parallelogram.svg/180px-Cross_product_parallelogram.svg.png" [Broken]

Last edited by a moderator: May 4, 2017
14. Aug 16, 2009

### GPPaille

ok! So we need to do it in three steps:

1- Show that the area of a parallelogram in 2D is the determinant of a 2x2 matrix containing the two side vectors.
2- Show that the area of a parallelogram in 3D is the squared sum of his projection on the three axis planes.
3- Make the final link.

1- I won't do it here, because it's long, but you can look here: http://mathforum.org/library/drmath/view/71204.html . Geometric/Algebraic proof, the boring part.

2- We now project the parallelogram on the 3 axis planes (YZ-plane, ZX-plane, XY-plane) by eliminating the components corresponding to the projection axis. E.g. if $A=(a_x,a_y,a_z), B=(b_x,b_y,b_z)$, then $A_x=(0,a_y,a_z), B_x=(0,b_y,b_z)$. The area of a projected parallelogram is simply the determinant of the 2x2 matrix containing the remaining components. You need to follow the even permutation rule to put the components in the matrix. E.g. For X, put Y first, then Z. For Y, put Z then X. For Z, put X then Y. For what follows, P is the area of the 3D parallelogram, and $P_x,P_y,P_z$ are the areas of the projected parallelograms.

Here's the most interesting part. In 3 dimensions, there's a duality between a line and a plane (This duality can be generalized in every dimension but let's concentrate on this particular case, for more, look at Exterior Algebra). And we can compute the area of a parallelogram with a Pythagorean-like formula:

$$P^2=P^2_x+P^2_y+P^2_z=(P_x,P_y,P_z)^2$$

Without going into endless algebraic computation, you can convince yourself that it's correct by looking at the trivial cases where the parallelogram is parallel to a axis plane. Then consider a rectangular shape inclined in only one direction. Once you pictured this second example, the formula should be clear.

3- If we look carefully, $P=(P_x,P_y,P_z)=A \times B$, so each component of the cross product contains the area of the parallelogram projected

I don't think I can go more geometric than this.

15. Aug 18, 2009

### Дьявол

Sorry for being late, but I got problems on college.

1. Yes, I completely understand it.

2. Ohhh, I know what you mean.

$$\overrightarrow{a} \times \overrightarrow{b} = (\begin{vmatrix} a_y & a_z \\ b_y & b_z \end{vmatrix}, \begin{vmatrix} a_z & a_x \\ b_z & b_x \end{vmatrix}, \begin{vmatrix} a_x & a_y \\ b_x & b_y \end{vmatrix} )$$

I know how to prove it by solving:

$$\overrightarrow{a} \times \overrightarrow{b}=(a_x \overrightarrow{i} +a_y \overrightarrow{j} + a_z \overrightarrow{k}) \times (b_x \overrightarrow{i} +b_y \overrightarrow{j} + b_z \overrightarrow{k})$$

Thank you very much.

Anyway, do you have some picture for the projection of the parallelogram on the planes?

Can somebody explain why:

$$\overrightarrow{i} \times \overrightarrow{k} = - \overrightarrow{j}$$

and not

$$\overrightarrow{i} \times \overrightarrow{k} = \overrightarrow{j}$$

Is this taken by convention or something?

Regards.

16. Aug 18, 2009

### GPPaille

Sorry, I don't have any picture, but it's simple to draw it on paper. Just take the YZ-plane projection and draw the two vectors $(a_y,a_z), (b_y,b_z)$. It's like when you look at it from the X axis.

$\overrightarrow{i} \times \overrightarrow{k} = - \overrightarrow{j}$ because it's the algebraic area, which can be negative. This has to do with the orientation of the parallelogram which has two sides. It's not a convention, since you can compute it with $\overrightarrow{i} \times \overrightarrow{k} = (1,0,0) \times (0,0,1) = (0,-1,0)$. This come from the definition of the cross product.

17. Aug 19, 2009

### Дьявол

Thanks, I know that is simple to draw it on paper, but I can't flag the answer $P=\sqrt{P_{x}^{2} + P_{y}^{2}+P_{z}^{2}}$ as obvious.

Here is the picture:

I chose random vectors a = (2,1,-1) and b = (-1,2,1).

The green one is |a x b|, the other ones are the parallelograms in XY, XZ and YZ planes.

I made |i|=|j|=|k|=1 cm (you can see on the diagram small dots on the x,y,z axis)

Now for the second question.

Also I understand the cross product respect to the right-handed coordinate system:

i = (1,0,0) and k=(0,0,1)

So c. i = (a,b,c).(1,0,0)=0

which a=0

c.k=(a,b,c).(0,0,1)=0

which gave us c=0

Now we got the normal vector c=(0,b,0)

Since we know that |c|=1 we can conclude that |b|=1, which $b=\pm 1$

Now we need to get in mind the (iii) rule, and that is the right-handed coordinate system.

Here I am stuck.

Regards.

Last edited by a moderator: May 4, 2017
18. Aug 19, 2009

### GPPaille

1-
Take an easier example, like the rectangular inclined parallelogram define by the two vectors $\vec{A}=(a,0,0), \vec{B}=(0,b,c)$. Only two projections has non-null area. Defining $h=\sqrt{b^2 + c^2}$, we know that $P = ah$ since it's rectangular. But

$$P = ah = a\sqrt{b^2 + c^2} = \sqrt{(ab)^2 + (ac)^2} = \sqrt{P_z^2 + P_y^2}$$

That's is really not a proof by I think it's enough to be convinced of the geometry behind this formula.

2-
You don't have to use you fingers to know that it's equal to -1, just do some matrix determinants.

$$(1,0,0)\times(0,0,1) = \left( \begin{vmatrix}0&0\\0&1\end{vmatrix}, \begin{vmatrix}0&1\\1&0\end{vmatrix}, \begin{vmatrix}1&0\\0&0\end{vmatrix} \right) = (0,-1,0)$$


text:
# Taking notes and blogging with style

## TL;DR

I explore a set-up to share notes among all my devices (namely my Laptop, tablet and smart-phone) and make them available on the cloud, plus how to even work with them from Emacs. I even show how I use the same system for blogging. If you are a OSX and Emacs use and also happen to have a device from Apple (e.g. Macbook Pro or iPhone) read below.

I have always been obsessed with personal productivity, and even though I sometimes thing that the whole thing is pointless, I keep trying to improve my "workflow" to make me more productive. Around 6 months ago I found a way to combine some applications together to take notes everywhere. Thus, I would like to share with the world.

I have separated the configuration by levels of "geekness". Level 1 is for more normal people while upper levels are more for über geeks (those which use Emacs / Org Mode). I will also show how I how use the same note taking system to even write the blog you are reading right now.

I know many of the current Org-users that read this might think I am not using the complete power of org-mode, and that org-mode offer other alternatives, but this way it is simple enough for me and everyone is free to adapt it as they want (isn't org-mode flexibility all about?).

## Intro

You can safely skip this part if you want to get the set-up running. Here I just want or rant a little bit about my experience in the note taking field.

As I mentioned before, I have always been obsessed with personal productivity. In particular to-do list and note taking. Back in the days, I used simple text files for storing my notes, then I switched to personal wiki solutions like the once popular Tomboy . Then I switched to integrated solutions like org-mode . I think org-mode is pure awesomeness but I just do not like as the notes are stored and accessed. The main issues I have found when using org-mode are the following:

• In org-mode the usual set-up implies all the notes are stored in one or some files in a tree fashion.
• Search in Org-Mode is not as flexible and quick as I would like it to be.
• Accessing my org files from other devices different of my computer was a PITA. and MobileOrg did not cover my needs (and was also difficult to Set-up).
• Furthermore, Integrating org-mode although possible is sometimes too difficult or cumbersome.

However, org-mode has a lot of things that make it rock:

• Can use the awesome Emacs to take notes.
• Simple format for storing information that can be used with or without syntax highlight.
• The ability to export to many formats including Latex, PDF and HTML.
• Syntax for highlighting when exporting.
• Executing of source code via Org Babel
• And more, much more.

So I stared looking initially for ways to be able to access my notes from everywhere. Once I got that figured out I started looking for ways to use org-mode to takes notes while at the same time being able to have read write access from wherever I wanted. And below you will find my solution.

## Set-Up

Before starting to describe I want to clarify what hardware I possess. The setup might be replicated with a different set of devices or operating system combination. I will try to add the alternatives.

• Laptop with Mac OSX: Macbook Pro
• Cellphone: iPhone (>=4)

(Yeah, yeah… you can call me Apple fan boy but I'm not!)

## LvL 1: Notes in all devices

In order to achieve this we only need a couple of applications. Notational Velocity and SimpleNote.

### Notational Velocity & SimpleNote

Notational Velocity is an open source light weight text editor with really nice functionalities. It features a simple interface with a search bar and a list of notes.

Notational Velocity

Notational Velocity allows the user to select in which directory to store the notes. So if we select a Dropbox folder we can browse it basically from everywhere.

To access the notes in other devices, I use SimpleNote (if you don't know this application you should totally check it out). Notational Velocity can sync your notes with SimpleNote. The Web version of SimpleNote is free but you have to pay for this app in order to use it in iOS device. It is cheap and totally make easier to share your computer notes with you iOS device(s).

Here is actually how SimpleNote looks on the iPad:

So by using Dropbox, Notational Velocity and SimpleNote you can:

• Take plain text notes using a free cool app.
• Access it everywhere using Dropbox (if you need to, but this is more of a emergency).
• Take notes online using the Simple note Web interface or any of the clients it has. Notational velocity will sync back to your hard drive and Dropbox folder.
• Sync and take notes using any or your devices iPhone or Emacs. I guess this configuration will also work or Android based devices, but I have no experience with Android based devices. However by this point, the main idea should be clear and should not be hard to replicate.

So you can stop reading here if you are not a user of Emacs (or if this level of Geeknes is enough for you).

## LvL 2: Using Emacs to take notes

OK if you are reading this that means that you are an active user of Emacs and that you want to see how you can integrate the previous Workflow with it. Well it is actually quite easy.

There is a special mode in Emacs called Deft with basically provides Emacs a similar interface for note taking as Notational Velocity does.

Deft Mode

Deft configuration is simple (add this to .emacs ):

(when (require 'deft nil 'noerror)
(setq
deft-extension "txt"
deft-directory "~/Dropbox/Notational Data/"
deft-text-mode 'org-mode
deft-use-filename-as-title t
)
(global-set-key (kbd "<C-f9>") 'deft))

This should be self-explanatory, but the important things are the :deft-directory option, which I select the same directory on my Dropbox folde. The :deft-text-mode option allows me to chose a default mode for my note files.

## LvL 3: Writing a blog from everywhere

So far we have the ability of writing and reading or text or org based mode from everywhere. This can be used to write draft of posts for your blog from basically everywhere. In my personal case I use Jekyll to generate this blog. Jekyll natively support markup formats like Markdown or Textile, but you can use raw HTML as well. This last feature allows me to write my posts using org-mode format and export them to HTML. This can be easily automated by defining a project.

(setq org-publish-project-alist
'(

("mf" :components ("mf-org"
"mf-img"))

("mf-org"
:base-directory "/Users/miguel/Dropbox/Notational Data"
:recursive t
:base-extension "blog.org.txt"
:publishing-directory "/Users/miguel/Dropbox/blog-stuff/mfcabrera.com/_posts"
:site-root "http://mfcabrera.com"
:publishing-function org-html-publish-to-html
:section-numbers nil
:table-of-contents nil
:auto-index nil
:auto-preamble nil
:body-only t
:auto-postamble nil)

("mf-img"
:base-directory "/Users/miguel/Dropbox/blog-stuff/source/"
:recursive t
:exclude "^publish"
:base-extension "jpg\\|gif\\|png|jpeg"
:publishing-directory "/Users/miguel/Dropbox/blog-stuff/mfcabrera.com/files/images"

:publishing-function org-publish-attachment)

))

There are three project mf-org and mf-img and mf. With mf-org I define base directory (where I have all my notes). :base-extension tells org-mode to only take into account files ending in .blog.org.txt (So I don't publish all my notes). With :body-only I tell org to export only the body of the HTML. mf-img takes images from a directory and copies to /files/images/ directory using the function :org-publish-attachment.

This allow me for example to start writing a blog-post using my iPad (or even my cellphone) and then when I get access to a computer just format it a bit and generate it. However, this apparently has not made my blog more active. However, I will try to actually do it more often now that I finally finished my M.Sc. degrees :).

Comments? ideas? improvements? Comment or write me an e-mail.


text:
Mind Matters Natural and Artificial Intelligence News and Analysis

# Bitcoin: What’s Good, What’s Bad and What’s the Future?

Bitcoins are currently bouncing in the range of $30,000 to$40,000 each

Bitcoin has been having quite the wild ride over the last several months, and especially the last several weeks. During 2020, the price of a single bitcoin dipped below $5,000, then soared past$20,000 in December. It is currently bouncing in the range of $30,000 to$40,000.
So what is the attraction of the digital currency Bitcoin? Why is the value of a bitcoin so high and what are people doing with it?

To begin with, keep in mind that a “single bitcoin” is a bit of a misnomer. While a single bitcoin is indeed expensive, coins can be split. The smallest unit in Bitcoin is known as a satoshi, which is 0.00000001 bitcoin. While the quoted price is always given for full coins, one doesn’t have to have the cash for a full coin in order to use the system.

So why did bitcoins soar in value?

There are many reasons but ultimately, the bitcoin’s price depends on how likely the market thinks it is to become an established currency. To understand this, let’s look a little bit about the mechanics of Bitcoin and the monetary system.

First of all, Bitcoin was designed to be anti-inflationary. There are roughly 18 million total bitcoins in circulation. The maximum number of bitcoins allowed by the Bitcoin system is roughly 21 million. So, imagine if Bitcoin replaced the US dollar.

Depending on how Bitcoin would be integrated into the monetary system, it might represent basic currency (known as M1), currency plus deposits ( M2 ), or any directly spendable asset (MZM).
M1 currently represents $6.7 trillion. If Bitcoin were to replace M1, it would be replacing$6.7 trillion worth of money. That means that each coin would be worth over $300,000. Likewise, replacing M2 would make coins worth over$900,000, and replacing MZM would make coins worth over a million dollars each.

So the question becomes, is Bitcoin likely to become the next major currency? Its current usage right now makes me question it. A tiny number of merchants take Bitcoin, and even then it is usually more as a novelty and an advertising strategy than a serious move towards cryptocurrency. There are about 350,000 Bitcoin transactions each day. However, most Bitcoin transactions are speculation and trading, not actual economic transactions.

In comparison, Visa processes the same number of transactions every six seconds, and all of those transactions are economic in nature.

In fact, Bitcoin’s seeming main purpose at present is underworld transactions. Bitcoin is being used as the payment system of choice for hackers holding computers hostage. On the more extreme end, Bitcoin is being used as the payment processing system for assassinations. In a more morally grey area, Bitcoin is being used to subvert governmental policies and controls on commerce, or even being used by governments to subvert what they consider to be oppressive trade policies or sanctions of other nations. Likewise, many who are opposed to the US lockdowns are considering it in the US for the same reasons.

In any case, the extended lockdowns have probably sparked increased usage of Bitcoin for business owners opposed to the lockdown, which may have contributed to its recent rise. Additionally, the waves of anti-government sentiment have probably caused additional increases in Bitcoin values, as well as an increasing distrust of the way that the US government is handling the economy.

Given all these considerations, I think the chance of Bitcoin replacing a significant fraction of legitimate economic activity is unlikely. In addition to the reasons I’ve outlined before, I think that the non-inflationary nature of Bitcoin is just as much of a problem as a solution. Because most of the Bitcoin that will ever exist is already on the network, this means that it will generate a huge disparity between early and late adopters, essentially driving an economic wedge in the economy that serves no economic purpose. With a gold standard, if the price of money gets too high, people can invest in mining. With a fiat standard, if the purchasing power of money gets too high, the government can inflate the currency. With Bitcoin, there is no available offsetting process. An increase in the need for money simply establishes and fortifies the power of those who simply already hold cash. It doesn’t go to those who build, invent, and maintain society, but merely to the cash hoarders.

Because of this, I don’t think Bitcoin will ever make the transition to a primary currency.

However, I do think that, as government regulations of commerce become more odious to the populations at large, a variety of crypto alternatives will be developed to meet those needs.

You may also enjoy: One of Bitcoin’s wilder rides:

Hackers seize popular Twitter accounts for giant Bitcoin scam Scammers have gained access to a number of high-profile Twitter accounts, including those of Warren Buffett, Bill Gates, Kanye West, and Elon Musk Jonathan Bartlett:


text:
jcYoon's Physics
Newsgroup and Email Discussion

Author Message
jcyoon

Joined: 08 Aug 2006
Posts: 213

 Posted: Tue Oct 24, 2006 12:14 pm    Post subject: Parity Violation at E158 Fri 2004-10-01 8:51 AM sjbth@slac.stanford.edu Dear Professor Brodsky, Recently, I have noticed that you have cited E158 paper “Observation of Parity Nonconservation in Moller Scattering” in your preprint. However, there is one thing I could not figure out in E158 paper. As long as I know, the helicity of a massive particle is not definite, since we can always find a boosted frame of reference in which the helicity is the opposite. Therefore, E158 measurment of helicity of electron or muons is only that of laboratory frame and it implies Lorentz violation. I would like to appreciate if you can explain this issue. I sincerely apologize for your time and effort if you find it unworthy of your attention. I am looking forward to hearing from you. Thanks, J.C. Yoon
Stanley J. Brodsky

Joined: 28 Sep 2006
Posts: 6

 Posted: Tue Oct 24, 2006 12:17 pm    Post subject: RE: Parity Violation at E158 Fri 2004-10-01 3:22 PM Dear J.C., Recently, I have noticed that you have cited E158 paper "Observation of Parity Nonconservation in Moller Scattering" in your preprint. However, there is one thing I could not figure out in E158 paper. As long as I know, the helicity of a massive particle is not definite, since we can always find a boosted frame of reference in which the helicity is the opposite. JC: Therefore, E158 measurement of helicity of electron or muons is only that of laboratory frame and it implies Lorentz violation. I would like to appreciate if you can explain this issue. I sincerely apologize for your time and effort if you find it unworthy of your attention. I am looking forward to hearing from you. Sjb: The helicity is the spin projection of the particle in the direction of momentum the particle. It can be defined covariantly so it has the same value in any Lorentz frame. The covariant definition is given in the Bjorken and Drell book and slide 8 from: http://urhic.phys.jyu.fi/JSS04/Heppelmann/3DiracSpin.ppt. Note that both $s^\mu$ and $p^\mu$ transform as four-vectors. I am sending this correspondence to my colleague Dae Sung Hwang in case he can add more clarification. Hope this response is helpful. Best regards, Stan
jcyoon

Joined: 08 Aug 2006
Posts: 213

 Posted: Tue Oct 24, 2006 12:19 pm    Post subject: RE: Parity Violation at E158 Sat 2004-10-02 12:08 AM Dear Professor Brodsky, Thanks you for your reply and time. It looks like the point is whether $s^\mu$ transforms as four-vectors. Let us consider a particle with spin and momentum. If we boost a frame faster than this particle, will it change its spin? If we think of spin as rotation, then the answer is no and we may refer to peskin & shroeder p47. The property of spin is well understood as an angular momentum, if it is true that spin transforms as four-vector, it means that our understanding the spin as the analogy of angular momentum is incorrect, since transform does not conserve the angular momentum. Therefore, it is doubtful that $s^\mu$ transforms as four-vectors. I would like to appreciate your response and I will be looking forward to hearing from you. Thanks, J.C. Yoon
Dae Sung Hwang

Joined: 24 Oct 2006
Posts: 1

 Posted: Tue Oct 24, 2006 12:26 pm    Post subject: RE: Parity Violation at E158 Sat 2004-10-02 12:34 AM Dear J.C. Yoon, [quote="J.C. Yoon"] Therefore, it is doubtful that $s^\mu$ transforms as four-vectors.[/quote] Please let me write below what I think: When we write $s^\mu$, it means that $s^\mu$ is a four-vector which transforms as a four-vector. As one sees in page 282 of Bjorken and Drell Relativistic Quantum Mechanics, $s^\mu$ is $(0 , {\hat s})$ in the rest frame of the particle. Then, $s^\mu$ in another frame is obtained by Lorentz transforming $(0 , {\hat s})$ to this another frame from the rest frame of the particle. I hope that what I wrote above is useful. If it is not clear, please let me know. Sincerely, Dae Sung Hwang
Stanley J. Brodsky

Joined: 28 Sep 2006
Posts: 6

 Posted: Tue Oct 24, 2006 12:28 pm    Post subject: RE: Parity Violation at E158 Sat 2004-10-02 1:13 AM Dear J.C., I agree with Dae Sung. Best regards, Stan
jcyoon

Joined: 08 Aug 2006
Posts: 213

 Posted: Tue Oct 24, 2006 12:29 pm    Post subject: RE: Parity Violation at E158 Sat 2004-10-02 3:40 AM Dear Professor Brodsky, I made a mistake in understanding the notation of $s^{mu}$ in the Bjorken and Drell. Thank Dae Sung for correcting my mistake and I agree with him. However, my point is still valid. Let us look at Eq.(7.94) page 141 in Bjorken and Drell. We can see the right-handedness change into the left-handedness as we boost a frame of reference so that it changes the sign of $\beta_{i} = \vec p_i \over E_i$. Therefore, it is not Lorentz-invariant but covariant. Let us say we have physical observations of left-handed and right-handed electrons in the lab frame. Since it is massive, both could be observed as that of right-handed electron from a certain frame of refence. Therefore, our analysis is based on a specific frame of reference does not hold in other frame and this explanation violates Lorentz invariance, since it is not a frame-independent intrinsic property. Thanks, J.C. Yoon
Stanley J. Brodsky

Joined: 28 Sep 2006
Posts: 6

 Posted: Tue Oct 24, 2006 12:44 pm    Post subject: RE: Parity Violation at E158 Sat 2004-10-02 6:05 AM Dear JC: [quote="J.C. Yoon"] I made a mistake in understanding the notation of $s^{mu}$ in the Bjorken and Drell. Thank Dae Sung for correcting my mistake and I agree with him. However, my point is still valid. Let us look at Eq.(7.94) page 141 in Bjorken and Drell. We can see the right-handedness change into the left-handedness as we boost a frame of reference so that it changes the sign of $\beta_{i} = \vec p_i \over E_i$. Therefore, it is not Lorentz-invariant but covariant. Let us say we have physical observations of left-handed and right-handed electrons in the lab frame. Since it is massive, both could be observed as that of right-handed electron from a certain frame of refence. Therefore, our analysis is based on a specific frame of reference does not hold in other frame and this explanation violates Lorentz invariance, since it is not a frame-independent intrinsic property.[/quote] Sjb: The helicity projectors $$\Sigma_\pm = {1\over 2}{1 \pm s^\mu \gamma^\mu \gamma_5}$$ are Lorentz scalars formed from the dot product of $s^\mu$ with $\gamma^\mu \gamma_5$ which also transforms as a four-vector. Best regards, Stan
jcyoon

Joined: 08 Aug 2006
Posts: 213

 Posted: Tue Oct 24, 2006 12:45 pm    Post subject: RE: Parity Violation at E158 Sat 2004-10-02 7:22 AM Dear Professor Brodsky, Thanks for your patience with my questions. I agree that the helicity projector is Lorentz scalar, since $s^\mu \gamma^\mu \gamma_5$ is a scalar as a production of vector, vector and scalar. But, the definition of helicity is not exactly up to this projector, but determined from whether spin orientation is parallel or antiparallel to momentum, which makes left- and right-handedness frame-denpendent and it is still consistent since it keeps helicity projector Lorentz scalar due to $\gamma^\mu$. Thanks, J.C. Yoon
Stanley J. Brodsky

Joined: 28 Sep 2006
Posts: 6

 Posted: Tue Oct 24, 2006 12:46 pm    Post subject: RE: Parity Violation at E158 Sat 2004-10-02 7:35 AM Dear JC, You may be confusing chirality and helicity. You need to use the helicity projectors for massive particles. Best regards, Stan
jcyoon

Joined: 08 Aug 2006
Posts: 213

 Posted: Tue Oct 24, 2006 12:49 pm    Post subject: RE: Parity Violation at E158 Sun 2004-10-03 1:38 AM Dear Professor Brodsky, I may be confusing with chirality and helicity. I would like to apologize for this confusion and let me clear these things up. Chirality operator: ${1\over 2}{1 \pm \gamma_5}$ Helicity operator: {1\over 2}{1 \pm p^{\hat} \Sigma} Spin projection operator $$\Sigma_\pm = {1\over 2}{1 \pm s^\mu \gamma^\mu \gamma_5}$$ I agree we need helicity operator with a massive particle and, in fact, my argument is mainly concerned with the difference between chirality and helicity questioning whether the approximation of helicity to chirality in the massless limit is theoretically impeccable. Thanks, J.C. Yoon
Stanley J. Brodsky

Joined: 28 Sep 2006
Posts: 6

 Posted: Tue Oct 24, 2006 12:50 pm    Post subject: RE: Parity Violation at E158 Sun 2004-10-03 5:00 AM Dear J.C., I am beginning to see your point. I was confusing the helicity operator with the spin operator which is a Lorentz scalar. As you note, the helicity operator is defined (e.g., to classify Dirac spinors) in a frame like the CM. You would like to have a covariant definition. I had thought that people always use the spin operator to do this. Is there a problem with this? Maybe Dae Sung can clarify. Here is a paper which might be useful. http://arxiv.org/PS_cache/hep-ph/pdf/0008/0008055.pdf Best regards, Stan
jcyoon

Joined: 08 Aug 2006
Posts: 213

 Posted: Tue Oct 24, 2006 12:51 pm    Post subject: RE: Parity Violation at E158 Thu 2004-10-07 3:54 AM Dear Professor Brodsky, I am sorry for the late reply. I guess that you would agree that the reason the spin operator was brought up is due to the shortcomings of the helicity operator. For a massive particle, its definition, for example, is ambiguous in the rest frame and depends on the frame of reference. Unfortunately, I believe people used rather the helicity than the covariant spin operators in their measurement and theory. Therefore, we have a Lorentz-violating theoretical explanation of left-right asymmetry since this could be observed as right-right asymmetry in a boosted frame. And it is difficult to be justified as an approximation since theory does not provide the exact prediction so that we could estimate how accurate this approximation is. Thanks, J.C. Yoon
Stanley J. Brodsky

Joined: 28 Sep 2006
Posts: 6

 Posted: Tue Oct 24, 2006 12:53 pm    Post subject: RE: Parity Violation at E158 Thu 2004-10-07 7:41 AM Dear J. C., [quote="J.C. Yoon"] I guess that you would agree that the reason the spin operator was brought up is due to the shortcomings of the helicity operator. For a massive particle, its definition, for example, is ambiguous in the rest frame and depends on the frame of reference. Unfortunately, I believe people used rather the helicity than the covariant spin operators in their measurement and theory. Therefore, we have a Lorentz-violating theoretical explanation of left-right asymmetry since this could be observed as right-right asymmetry in a boosted frame. And it is difficult to be justified as an approximation since theory does not provide the exact prediction so that we could estimate how accurate this approximation is.[/quote] Sjb: It seems like a good idea for you to repeat the calculation for E158 using the spin projection method and quantify any difference. Best regards, Stan
jcyoon

Joined: 08 Aug 2006
Posts: 213

 Posted: Tue Oct 24, 2006 12:54 pm    Post subject: RE: Parity Violation at E158 Wed 2005-06-01 5:09 AM Dear Professor Brodsky, Recently, I have posted a preprint that is related to our previous discussion. "Lorentz violation of the standard model"( http://arxiv.org/abs/hep-ph/0502142 ). It is quite frustrating to argue against what most people think, but I could not find any fallacy of scientific method in my argument. Though I know it sounds outrageous and it may waste your time, I would like to ask you a favor to read this preprint and let me know your opinion. If you could point out where my argument fails so that I could stop this nonsense, I will deeply appreciate it with all respect. In advance, I would like to apologize any time you would waste if it is just a nonsense. Thanks, J.C. Yoon
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text:
# Conditionals: \renewcommand{\index}

I'm working on a book project which is divided in several parts and chapters. At the end of some chapters (let's say at the end of chapter 2 and 3) I want an index which lists all the keyword mentioned since the last index.

The output should look like this

\documentclass{book}

\usepackage{blindtext}

\usepackage{imakeidx}
\makeindex[name=1]
\makeindex[name=2]

\begin{document}

\blindtext[2]

\index[1]{Aaa}\index[1]{Bbb}\index[1]{Caa} % keywords for INDEX 1

\blindtext[2]

\index[1]{Aaa}\index[1]{Ddd}\index[1]{Eee} % keywords for INDEX 1

\printindex[1] % INDEX 1

\index[2]{Aaa}\index[2]{Ddd}\index[2]{Fff} % keywords for INDEX 2

\printindex[2] % INDEX 2

\end{document}

The problem is that the book is already written and I don't want to change all \index{} commands to \index[1]{} or \index[2]{} manually. The idea is to redefine the \index{} command depending on the chapter number.

I'm looking for something like:

\if \thechapter < 3
\renewcommand{\index}{\index[1]}
\else
\renewcommand{\index}{\index[2]}
\fi

The code above doesn't work and I don't know why.

My file looks now like this

\documentclass{book}

\usepackage{blindtext}

\usepackage{imakeidx}
\makeindex[name=1]
\makeindex[name=2]

% here is something missing like:
%
% \if \thechapter < 3
%   \renewcommand{\index}{\index[1]}
% \else
%   \renewcommand{\index}{\index[2]}
% \fi

\renewcommand{\index}

\begin{document}

\blindtext[2]

\index{Aaa}\index{Bbb}\index{Caa} % keywords for INDEX 1

\blindtext[2]

\index{Aaa}\index{Ddd}\index{Eee} % keywords for INDEX 1

\printindex[1] % INDEX 1

\index{Aaa}\index{Ddd}\index{Fff} % keywords for INDEX 2

\printindex[2] % INDEX 2

\end{document}
• Your approach is syntactically and logically wrong: \if \thechapter can't work and even if it would work, at the time of writing the command it does not what you want – user31729 Aug 4 '16 at 7:23

This is a code similar to the other answer to a question by the OP I provided here

First grab the original definition of \index, taking care of the optional argument introduced by imakeidx and then branching into the original version if \index[...]{foo} is used or checking for the chapter number with

\ifnum\value{chapter} < 3
\else
\fi

Of course, this does work for this particular purpose only.

\documentclass{book}

\usepackage{blindtext}
\usepackage{letltxmacro}
\usepackage{imakeidx}
\makeindex[name=1]
\makeindex[name=2]

\makeatletter

\LetLtxMacro\latex@@index\index
\renewcommand{\index}[2][]{%
\def\@firstarg{#1}%
\ifx\@firstarg\empty
\ifnum\value{chapter} < 3
\latex@@index[1]{#2}%
\else
\latex@@index[2]{#2}%
\fi
\else
\latex@@index[#1]{#2}%
\fi
}
\makeatother

\begin{document}

\blindtext[2]

\index{Aaa}\index{Bbb}\index{Caa} % keywords for INDEX 1

\blindtext[2]

\index{Aaa}\index{Ddd}\index{Eee} % keywords for INDEX 1

\printindex[1] % INDEX 1

\index{Aaa}\index{Ddd}\index{Fff} % keywords for INDEX 2

\printindex[2] % INDEX 2

\end{document}

You can simply change the default value for the optional argument into the current chapter number.

\documentclass{book}

\usepackage[noautomatic]{imakeidx}

\indexsetup{level=\section*,toclevel=section,noclearpage}

\makeindex[name=1,intoc]
\makeindex[name=2,intoc]
\makeindex[name=3,intoc]

\usepackage{blindtext}

\makeatletter
% The default value for the optional argument is the jobname.
% Change it into the current chapter number.
\renewcommand{\index}{%
\@bsphack
\@ifnextchar[{\@index}{\@index[\arabic{chapter}]}%
}
\makeatother

\begin{document}

\tableofcontents

\blindtext[2]

\index{Aaa}\index{Bbb}\index{Caa} % keywords for INDEX 1

\printindex[2] % INDEX 1

\blindtext[2]

\index{Aaa}\index{Ddd}\index{Eee} % keywords for INDEX 2

\printindex[2] % INDEX 2

\index{Aaa}\index{Ddd}\index{Fff} % keywords for INDEX 3

\printindex[3] % INDEX 3

\end{document}

I also added the necessary bits in order for the local indices to be printed correctly. The noautomatic option is mandatory, because otherwise the index files will be closed upon processing the first \printindex command.

For a sequence of indices not tied to chapters, a different strategy can be used: we define a counter that's stepped at each printing of an index and use that instead of the chapter counter like in the previous solution.

\documentclass{book}

\usepackage[noautomatic]{imakeidx}

\indexsetup{
othercode={\stepcounter{printindex}},
}

\makeindex[name=1,intoc]
\makeindex[name=2,intoc]

\usepackage{blindtext}

\makeatletter
% The default value for the optional argument is the jobname.
% Change it into the current printindex value, that's stepped
% at each \printindex command.
\newcounter{printindex}
\setcounter{printindex}{1}
\renewcommand{\index}{%
\@bsphack
\@ifnextchar[{\@index}{\@index[\arabic{printindex}]}%
}
\makeatother

\begin{document}

\tableofcontents

\blindtext[2]

\index{Aaa}\index{Bbb}\index{Caa} % keywords for INDEX 1

\blindtext[2]

\index{Aaa}\index{Ddd}\index{Eee} % keywords for INDEX 2

\printindex[1] % INDEX 1


text:
# Gorenstein dimension of modules over homomorphisms

Research paper by Lars Winther Christensen, Srikanth Iyengar

Indexed on: 18 Nov '05Published on: 18 Nov '05Published in: Mathematics - Commutative Algebra

#### Abstract

Given a homomorphism of commutative noetherian rings R --> S and an S-module N, it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup{m | Tor^R_m(E,N) \noteq 0} where E is the injective hull of the residue field of R. This result is analogous to a theorem of Andr\'e on flat dimension.


text:
40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
 Coating Ring-down Measurement Lab elog Not logged in
Message ID: 319     Entry time: Fri Feb 24 14:44:22 2017
 Author: Gabriele Type: General Category: General Subject: Disk model

It's possible to build an analytical model of the resonant frequencies of a simple thin disk. For example, see J. Sound and Vibrations 188, 685 (1995), section 2

The solutions are given in term of Bessel functions:

$W_{mn}(r, \theta)=\left[ J_m(\lambda_{mn} r/a) + C_{mn} I_m(\lambda_{mn} r/a)\right] \cos m \theta$

where J is a Bessel function of the first kind, and I a modified Bessel function of the first kind, a is the disk radius.

The coefficient Cmn and the eigenvalue can be found as solution of the following two equations

Then the eigenfrequencies are given by

$f_{mn} = \frac{\lambda_{mn}^2}{2\pi a^2}\sqrt{\frac{D}{\rho h}}$

where rho is the material density, h the disk thickness and D the flexural rigidity

$D =\frac{E h^3}{12(1-\nu^2)}$

where E is the material Young's modulus and nu the Poisson's ratio.

From all those results we can conclude that the frequency scaling with respect to disk radius and thickness are very simple:

• the frequencies scale linearly with the thickness
• the frequencies scale inversely proportionally to the square of the radius

Also, the frequencies scales as sqrt(E/rho)

$f_{nm} \propto \sqrt{\frac{E}{\rho}} \frac{h}{a^2}$

The dependency on the Poisson's ratio is more complex since nu is involved in the eigenfrequency equation shown above.

Unfortunately the thin disk model does not exactly match the COMSOL results: deviations of few tens of Hz are present, probaly due to the thin disk approximation. The COMSOL model is more accurate to match the experimental frequencies.

However, I checked that the eigenfrequencies predicted by COMSOL also scales as predicted with thickness and radius.

Using the measurements on the six samplex we got from Mark Optics, after annealing, I was able to tune the COMSOL model to fit all measured frequencies within 6 Hz. I chose to change the disk thickness (since diameter and Young'r modulus are degenerate) and the Poisson's ratio.

Here is an example of the difference between the measured and modeled frequencies:

The table below summarizes the best fit for each of the disks

SN Thickness [mm] Poisson's ratio
S1600519 1.0194 0.1669
S1600520 1.0186 0.1663
S1600521 1.0217 0.1669
S1600522 1.0223 0.1635
S1600523 1.0227 0.1655
S1600524 1.0209 0.1654

Since the material is the same, I would expect the Poisson's ratio to be constant. So for future modeling I'm using the average of the values above: 0.166

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