curve

Flux, or diffusion, for gaseous molecules can be related to the function: where N is the total number of gaseous particles, k is Boltzmann's constant, T is the relative temperature in kelvins, and σab is the mean free path between the molecules a and b. Chemical molar flux of a component A in an isothermal, isobaric system is also defined in ...

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Flux, Flux - Flux definition and theorems, Flux - Thermal systems, Flux - Chemical diffusion, Flux - Flux definition and theorems, Flux - Maxwell's equations, Flux - Poynting vector

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A deck transformation or automorphism of a cover p : C → X is a homeomorphism f : C → C such that p o f = p. The set of all deck transformations of p forms a group under composition, the deck transformation group Aut(p). Every deck transformation permutes the elements of each fiber. This defines a group action of the deck transformation group on each fiber. Note that by the unique lifting property, if f is not the id ...

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Covering map, Covering map - Examples, Covering map - Elementary properties, Covering map - Universal covers, Covering map - Deck transformation group regular covers, Covering map - Monodromy action, Covering map - Group structure redux

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A cover q : D → X is a universal cover iff D is simply connected. The name comes from the following important property: if p : C → X is any cover of X with C connected, then there exists a covering map f : D → C such that p o f = q. This can be phrased as "The universal cover of < ...

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Covering map, Covering map - Examples, Covering map - Elementary properties, Covering map - Universal covers, Covering map - Deck transformation group regular covers, Covering map - Monodromy action, Covering map - Group structure redux

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In category theory, there are covariant functors and contravariant functors. The dual space of a vector space is a standard example of a contravariant functor. Some constructions of multilinear algebra are of 'mixed' variance, which prevents them from being functors. The distinction between homology theory and cohomology theory in topology is that homology is a covariant functor, while cohomology is a contravariant functor (it was suggested in a book, Hilton & Wylie, that contrahomology was therefore a better term for cohomolog ...

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Covariance and contravariance, Covariance and contravariance - Informal usage, Covariance and contravariance - Example: covariant basis vectors in Euclidean R³, Covariance and contravariance - What 'contravariant' means, Covariance and contravariance - Usage in tensor analysis, Covariance and contravariance - Algebra and geometry

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Most mathematicians define a binary relation (and hence a function) as an ordered triple (X, Y, G), where X and Y are the domain and codomain sets, and G is the graph of f. However, some mathematicians define a relation as being simply the set of pairs G, without explicitly giving the domain and co-domain. There are advantages and disadvantages to each definition, but either of them is satisfactory for most uses of functions in mathematics. The explicit domain and ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Choose a point O and a line L not passing through O. Point O and line L define the plane in which the cissoid will be drawn. Draw a line M passing through O and perpendicular to line L. Let point P be the intersection of lines L and M. Bisect line OP at point A. Draw a circle centered at A and with radius APSee also:

Cissoid of Diocles, Cissoid of Diocles - Construction, Cissoid of Diocles - Delian problem, Cissoid of Diocles - Roulette, Cissoid of Diocles - The cissoid of Diocles as a pedal curve, Cissoid of Diocles - Inversion

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This curve is also a roulette. Take two congruent parabolas, set them vertex-to-vertex, and roll one along the other; the vertex of the rolling parabola will trace the cissoid. Figure 1. A pair of parabolas — shown in black — face each other symmetrically: one on top and one on the bottom. Then the top parabola is rolled without slipping along the bottom one, and its successive positions are shown in green and blue. Then the path traced by the vertex of the top parabola is it rolls is a roulette — shown in reddish color — which ...

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Cissoid of Diocles, Cissoid of Diocles - Construction, Cissoid of Diocles - Delian problem, Cissoid of Diocles - Roulette, Cissoid of Diocles - The cissoid of Diocles as a pedal curve, Cissoid of Diocles - Inversion

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The notion of function is generalizes to the notion of morphism in the context of category theory. A category is a collection of objects and morphisms, each morphism is an ordered triple (X, Y, f), where f is a rule connecting domain X and codomain Y, and X and Y are objects in the collection. Ordinary functions are sometimes referred to as morphisms in a concrete category. ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups. Instead of by a pair (A, b), Euclidean group elements can also be represented as square matrices of size n + 1, as explained for the affine group. In the terms of the Erlangen programme, Euclidean geometry is therefore a specialisation of affine geometry. All affine theorems apply; the extra fa ...

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Euclidean group, Euclidean group - Subgroup structure matrix and vector representation, Euclidean group - Subgroups, Euclidean group - Relation to the affine group, Euclidean group - Rigid body motions, Euclidean group - Overview of isometries in up to three dimensions, Euclidean group - Commuting isometries, Euclidean group - Conjugacy classes

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Function mathematics - Injective surjective bijective. Three important properties that a function may have are: injective (or one-to-one, or an injection) if it associates different arguments to different values; i.e., if f(a) = f(b) implies a = b, for any arguments a and b; surjective (or onto, or a surjection) if its range is equal to its codomain; in other words, if for every y in the ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The condition for a binary relation f from X to Y to be a function can be split into two conditions: f is total, or entire: for each x in X, there exists some y in Y such that x is related to y. f is single-valued: for each x in X, there is at most one y in Y such that x is related to y. In some contexts, a relation that satisfies condition (1), but not necessarily (2) ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows: (f + g)(x) = f(x) + g(x) (f × g)(x) = f(x) × < ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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THEOREM: The pedal curve of a parabola with respect to its vertex is a cissoid of Diocles. Proof: Any parabola can be rotated and translated so that it will end up being described by the equation whose slope at point (x, y) is given by the derivative Then the set of points which form the line tangent to the parabola at point (b, a b2) is and the set of points which form the line which is perpendicular to L< ...

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Cissoid of Diocles, Cissoid of Diocles - Construction, Cissoid of Diocles - Delian problem, Cissoid of Diocles - Roulette, Cissoid of Diocles - The cissoid of Diocles as a pedal curve, Cissoid of Diocles - Inversion

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In common physics usage, the adjective covariant may sometimes be used informally as a synonym for invariant (or equivariant, in mathematicians' terms). For example, the Schrödinger equation does not keep its written form under the coordinate transformations of special relativity; thus one might say that the Schrödinger equation is not covariant. By contrast, the Klein-Gordon equation and the Dirac equation take the same form in any coordinate frame of special relativity: thus, one might say that these equations are covari ...

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Covariance and contravariance, Covariance and contravariance - Informal usage, Covariance and contravariance - Example: covariant basis vectors in Euclidean R³, Covariance and contravariance - What 'contravariant' means, Covariance and contravariance - Usage in tensor analysis, Covariance and contravariance - Algebra and geometry

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Main article: continuity (topology) The above definitions of continuous functions can be generalized to functions from one topological spaces to another in a natural way; a function f : X → Y, where X and Y are topological spaces, is continuous iff for every open set V ⊆ Y, f −1(V) is open in X. ...

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Continuous function, Continuous function - Real-valued continuous functions, Continuous function - Epsilon-delta definition, Continuous function - Heine definition of continuity, Continuous function - Examples, Continuous function - Facts about continuous functions, Continuous function - Continuous functions between metric spaces, Continuous function - Continuous functions between topological spaces, Continuous function - Continuous functions between partially ordered sets

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The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a composite function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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One often extends the concept (and notation) of image of an argument to sets of arguments. Namely, if A is any subset of the domain X, the image of A under f is the subset of Y defined f(A) = {f(x) | x is in A} So, for example, the image of {-3,2,3} under the squaring function sqr is sqr({-3,2, 3}) = {4, 9}. This extension is consistent as long as no subset of the domain is also an element of the domain. A ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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Informally, a restriction of a function f is the result of trimming its graph to a smaller domain. More precisely, if f is a function from a X to Y, and S is any subset of X, the restriction of f to S is the function f|S from S to Y such that f|S(s) = f(s) for all s in S. The restriction f|S can also be expressed as the composition f incS,X, where incSSee also:

Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices. Under simple expansions and contractions of the coordinates, the reciprocity is exact; under affine transformations the components of a v ...

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Covariance and contravariance, Covariance and contravariance - Informal usage, Covariance and contravariance - Example: covariant basis vectors in Euclidean R³, Covariance and contravariance - What 'contravariant' means, Covariance and contravariance - Usage in tensor analysis, Covariance and contravariance - Algebra and geometry

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If e1, e2, e3 are contravariant basis vectors of R3 (not necessarily orthogonal nor of unit norm) then the covariant basis vectors of their reciprocal system are: Note that even if the ei and ei are not orthonormal, they are still by this definition mutually orthonormal: Then the contravariant coordinates of any vector v< ...

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Covariance and contravariance, Covariance and contravariance - Informal usage, Covariance and contravariance - Example: covariant basis vectors in Euclidean R³, Covariance and contravariance - What 'contravariant' means, Covariance and contravariance - Usage in tensor analysis, Covariance and contravariance - Algebra and geometry

Read more here: » Covariance and contravariance: Encyclopedia II - Covariance and contravariance - Example: covariant basis vectors in Euclidean R³

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is justified by the fact that |YX| = |Y||X|. See the article on cardinal numbers for more details. It is traditional to write f: X → Y to mean f ∈ [X → Y]; that is, "f is a function from X to Y". This statement is sometimes read "f ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Set of all functions

Function mathematics - Functions of two or more variables. The concept of function can be extended to an object that takes a combination of two (or more) argument values to a single result. This intuitive concept is formalized by a function whose domain is the Cartesian product of two or more sets. For example, consider the multiplication function that associates two integers to their product: f(x, y) = x·y. This function can be defined formally as having domain Z ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Functions with multiple inputs and outputs