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  • equality | planetmath.org
    92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A09 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit equality In any set S S S the equality denoted by is a binary relation which is reflexive symmetric transitive and antisymmetric i e it is an antisymmetric equivalence relation on S S S or which is the same thing the equality is a symmetric partial order on S S S In fact for any set S S S the smallest equivalence relation on S S S is the equality by smallest we mean that it is contained in every equivalence relation on S S S This offers a definition of equality From this it is clear that there is only one equality relation on S S S Its equivalence classes are all singletons x x x where x S x S x in S The concept of equality is essential in almost all branches of mathematics A few examples will suffice 1 1 1 1 displaystyle 1 1 displaystyle 2 2 displaystyle 2 e i π superscript e i π displaystyle e i pi displaystyle 1 1 displaystyle 1 ℝ i ℝ i displaystyle mathbb R i displaystyle ℂ ℂ displaystyle mathbb C The second example is Euler s identity Remark 1 Although the four characterising properties reflexivity symmetry transitivity and antisymmetry determine the equality on S S S uniquely they cannot be thought to form the definition of the equality since the concept of antisymmetry already contains the equality Remark 2 An equality equation in a set S S S may be true regardless to the values of the variables involved in the equality then one speaks of an identity or identic equation in this set E g x y 2 x 2 y 2 superscript x y 2 superscript x 2 superscript y 2 x y 2 x 2 y 2 is an identity in a field with characteristic 2 2 2 Defines equality relation identity identic equation Related Equation Type of Math Object Topic Major Section Reference Parent equivalence relation Groups audience World Writable Buddy List of pahio Mathematics Subject Classification 06 00 no label found Add a correction Attach a problem Ask a question Comments Equalities set theory and branch without equalities Permalink Submitted by CompositeFan on Thu 05

    Original URL path: http://www.planetmath.org/equality (2016-04-25)
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  • equality of complex numbers | planetmath.org
    ℝ X 2 1 ℝ superscript X 2 1 mathbb R X 2 1 which enables the interpretation of the complex numbers as the ordered pairs a b a b a b of real numbers and also as the sums a i b a i b a ib where i 2 1 superscript i 2 1 i 2 1 a 1 i b 1 a 2 i b 2 a 1 a 2 b 1 b 2 formulae sequence subscript a 1 i subscript b 1 subscript a 2 i subscript b 2 normal subscript a 1 subscript a 2 subscript b 1 subscript b 2 displaystyle a 1 ib 1 a 2 ib 2 quad Longleftrightarrow quad a 1 a 2 wedge b 1 b 2 1 This condition may as well be derived by using the field properties of ℂ ℂ mathbb C and the properties of the real numbers a 1 i b 1 a 2 i b 2 subscript a 1 i subscript b 1 subscript a 2 i subscript b 2 displaystyle a 1 ib 1 a 2 ib 2 a 2 a 1 i b 2 b 1 absent subscript a 2 subscript a 1 i subscript b 2 subscript b 1 displaystyle implies a 2 a 1 i b 2 b 1 a 2 a 1 2 b 2 b 1 2 absent superscript subscript a 2 subscript a 1 2 superscript subscript b 2 subscript b 1 2 displaystyle implies a 2 a 1 2 b 2 b 1 2 a 2 a 1 2 b 2 b 1 2 0 absent superscript subscript a 2 subscript a 1 2 superscript subscript b 2 subscript b 1 2 0 displaystyle implies a 2 a 1 2 b 2 b 1 2 0 a 2 a 1 0 b 2 b 1 0 formulae sequence absent subscript a 2 subscript a 1 0 subscript b 2 subscript b 1 0 displaystyle implies a 2 a 1 0 b 2 b 1 0 a 1 a 2 b 1 b 2 formulae sequence absent subscript a 1 subscript a 2 subscript b 1 subscript b 2 displaystyle implies a 1 a 2 b 1 b 2 The implication chain in the reverse direction is apparent If a i b 0 a i b 0 a ib neq 0 then at least one of the real numbers a a a and b b b differs from 0 We can set a r cos φ b r sin φ formulae sequence a r φ b r φ displaystyle a r cos varphi qquad b r sin varphi 2 where r r r is a uniquely determined positive number and φ φ varphi is an angle which is uniquely determined up to an integer multiple of 2 π 2 π 2 pi In fact the equations 2 yield a 2 b 2 r 2 cos 2 φ sin 2 φ r 2 superscript a 2 superscript b

    Original URL path: http://www.planetmath.org/equalityofcomplexnumbers (2016-04-25)
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  • equation | planetmath.org
    FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit equation Equation An equation concerns usually elements of a certain set M M M where one can say if two elements are equal In the simplest case M M M has one binary operation producing as result some elements of M M M and these can be compared Then an equation in M M M is a proposition of the form E 1 E 2 subscript E 1 subscript E 2 displaystyle E 1 E 2 1 where one has equated two expressions E 1 subscript E 1 E 1 and E 2 subscript E 2 E 2 formed with of the elements or indeterminates of M M M We call the expressions E 1 subscript E 1 E 1 and E 2 subscript E 2 E 2 respectively the left hand side and the right hand side of the equation 1 Example Let S S S be a set and 2 S superscript 2 S 2 S the set of its subsets In the groupoid 2 S superscript 2 S 2 S smallsetminus where smallsetminus is the set difference we can write the equation A B B A B A B B A B A smallsetminus B smallsetminus B A smallsetminus B which is always true Of course M M M may be equipped with more operations or be a module with some ring of multipliers then an equation 1 may contain them But one need not assume any algebraic structure for the set M M M where the expressions E 1 subscript E 1 E 1 and E 2 subscript E 2 E 2 are values or where they represent generic elements Such a situation would occur e g if one has a continuous mapping f f f from a topological space L L L to another M M M then one can consider an equation f x y f x y f x y A somewhat comparable case is the equation dim V 2 dimension V 2 dim V 2 where V V V is a certain or a generic vector space both sides represent elements of the extended real number system Root of equation If an equation 1 in M M M contains one indeterminate say x x x then a value of x x x which satisfies 1 i e makes it true is called a root or a solution of the equation Especially if we have a polynomial equation f x 0 f x 0 f x 0 we may speak of the multiplicity or the order of a root x 0 subscript x 0 x 0 it is the multiplicity of the zero x 0 subscript x 0 x 0 of the polynomial f x f x f x A multiple root has

    Original URL path: http://www.planetmath.org/equation (2016-04-25)
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  • equation $y'' = f(x)$ | planetmath.org
    y f x superscript y f x y prime prime f x A simple special case of the second order linear differential equation with constant coefficients is d 2 y d x 2 f x superscript d 2 y d superscript x 2 f x displaystyle frac d 2 y dx 2 f x 1 where f f f is continuous We obtain immediately d y d x C 1 f x d x d y d x subscript C 1 f x d x displaystyle frac dy dx C 1 int f x dx y C 1 x C 2 f x d x d x y subscript C 1 x subscript C 2 f x d x d x displaystyle y C 1 x C 2 int left int f x dx right dx 2 A particular solution y x y x y x of 1 satisfying the initial conditions y x 0 y 0 y x 0 y 0 formulae sequence y subscript x 0 subscript y 0 superscript y normal subscript x 0 superscript subscript y 0 normal y x 0 y 0 quad y prime x 0 y 0 prime is obtained more simply by integrating 1 twice between the limits x 0 subscript x 0 x 0 and x x x thus getting y x y 0 y 0 x x 0 x 0 x x 0 x f x d x d x y x subscript y 0 normal superscript subscript y 0 normal x subscript x 0 superscript subscript subscript x 0 x superscript subscript subscript x 0 x f x d x d x y x y 0 y 0 prime cdot x x 0 int x 0 x left int x 0 x f x dx right dx But here the two first addends are the first terms of the Taylor polynomial of y x y x y x expanded by the powers of x x 0 x subscript x 0 x x 0 whence the double integral is the corresponding remainder term x 0 x y x x t d t x 0 x f t x t d t superscript subscript subscript x 0 x superscript y x x t d t superscript subscript subscript x 0 x f t x t d t int x 0 x y prime prime x x t dt int x 0 x f t x t dt Hence the particular solution can be written with the simple integral as y x y 0 y 0 x x 0 x 0 x f t x t d t y x subscript y 0 normal superscript subscript y 0 normal x subscript x 0 superscript subscript subscript x 0 x f t x t d t displaystyle y x y 0 y 0 prime cdot x x 0 int x 0 x f t x t dt 3 The result may be generalised for the n th superscript n th n mathrm th

    Original URL path: http://www.planetmath.org/equationyfx (2016-04-25)
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  • equation of catenary via calculus of variations | planetmath.org
    skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit equation of catenary via calculus of variations Using the mechanical principle that the centre of mass places itself as low as possible determine the equation of the curve formed by a flexible homogeneous wire or a thin chain with length l l l when supported at its ends in the points P 1 x 1 y 1 subscript P 1 subscript x 1 subscript y 1 P 1 x 1 y 1 and P 2 x 2 y 2 subscript P 2 subscript x 2 subscript y 2 P 2 x 2 y 2 We have an isoperimetric problem to minimise P 1 P 2 y d s to minimise superscript subscript subscript P 1 subscript P 2 y d s displaystyle mbox to minimise quad int P 1 P 2 y ds 1 under the constraint P 1 P 2 d s l superscript subscript subscript P 1 subscript P 2 d s l displaystyle int P 1 P 2 ds l 2 where both the path integrals are taken along some curve c c c Using a Lagrange multiplier λ λ lambda the task changes to a free problem P 1 P 2 y λ d s x 1 x 2 y λ 1 y 2 d x min superscript subscript subscript P 1 subscript P 2 y λ d s superscript subscript subscript x 1 subscript x 2 y λ 1 superscript y normal 2 d x min displaystyle int P 1 P 2 y lambda ds int x 1 x 2 y lambda sqrt 1 y prime 2 dx mbox min 3 cf example of calculus of variations The Euler Lagrange differential equation the necessary condition for 3 to give an extremal c c c reduces to the Beltrami identity y λ 1 y 2 y y λ y 1 y 2 y λ 1 y 2 a y λ 1 superscript y normal 2 normal superscript y normal y λ superscript y normal 1 superscript y normal 2 y λ 1 superscript y normal 2 a y lambda sqrt 1 y prime 2 y prime cdot y lambda cdot frac y prime sqrt 1 y prime 2 equiv frac y lambda sqrt 1 y prime 2 a where a a a is a constant of integration After solving this equation for the derivative y superscript y normal y prime and separation of variables we get d y y λ 2 a 2 d x a plus or minus d y superscript y λ 2 superscript a 2 d x a pm frac dy sqrt y lambda 2 a 2 frac dx a which may become clearer by notating y λ u assign y λ u y lambda u then by integrating d

    Original URL path: http://www.planetmath.org/equationofcatenaryviacalculusofvariations (2016-04-25)
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  • equation of plane | planetmath.org
    2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A47A60 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A47A60 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit equation of plane The position of a plane τ τ tau can be fixed by giving the position vector O Q normal O Q overrightarrow OQ of the projection point Q Q Q of the origin on the plane Let the length of the position vector be r r r and the angles formed by the vector with the positive coordinate axes α α alpha β β beta γ γ gamma Let P x y z P x y z P x y z be an arbitrary point Then P P P is in the plane τ τ tau iff its projection on the line O Q O Q OQ coincides with Q Q Q i e iff the projection of the coordinate way of P P P is r r r This may be expressed as the equation x cos α y cos β z cos γ r x α y β z γ r x cos alpha y cos beta z cos gamma r or x cos α y cos β z cos γ r 0 x α y β z γ r 0 displaystyle x cos alpha y cos beta z cos gamma r 0 1 which thus is the equation of the plane Conversely we may show that a first degree equation A x B y C z D 0 A x B y C z D 0 displaystyle Ax By Cz D 0 2 between the variables x x x y y y z z z represents always a plane In fact we may without hurting generality suppose that D 0 D 0 D leqq 0 Now R A 2 B 2 C 2 0 assign R superscript A 2 superscript B 2

    Original URL path: http://www.planetmath.org/equationofplane (2016-04-25)
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  • equation of tangent of circle | planetmath.org
    line for a circle with radius r r r For simplicity we chose for the origin the centre of the circle when the points x y x y x y of the circle satisfy the equation x 2 y 2 r 2 superscript x 2 superscript y 2 superscript r 2 displaystyle x 2 y 2 r 2 1 Let the point of tangency be x 0 y 0 subscript x 0 subscript y 0 x 0 y 0 Then the slope of radius with end point x 0 y 0 subscript x 0 subscript y 0 x 0 y 0 is y 0 x 0 subscript y 0 subscript x 0 frac y 0 x 0 whence according to the parent entry its opposite inverse x 0 y 0 subscript x 0 subscript y 0 frac x 0 y 0 is the slope of the tangent being perpendicular to the radius Thus the equation of the tangent is written as y y 0 x 0 y 0 x x 0 y subscript y 0 subscript x 0 subscript y 0 x subscript x 0 y y 0 frac x 0 y 0 x x 0 Removing the denominator and the parentheses we obtain from this first x 0 x y 0 y x 0 2 y 0 2 subscript x 0 x subscript y 0 y superscript subscript x 0 2 superscript subscript y 0 2 x 0 x y 0 y x 0 2 y 0 2 and then x 0 x y 0 y r 2 subscript x 0 x subscript y 0 y superscript r 2 displaystyle x 0 x y 0 y r 2 2 since x 0 y 0 subscript x 0 subscript y 0 x 0 y 0 satisfies 1 Remark In the

    Original URL path: http://www.planetmath.org/equationoftangentofcircle (2016-04-25)
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  • equivalent valuations | planetmath.org
    org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit equivalent valuations Let K K K be a field The equivalence of valuations 1 fragments normal normal subscript normal 1 cdot 1 and 2 fragments normal normal subscript normal 2 cdot 2 of K K K may be defined so that 1 1 fragments normal normal subscript normal 1 cdot 1 is not the trivial valuation 2 if a 1 1 subscript a 1

    Original URL path: http://www.planetmath.org/equivalentvaluations (2016-04-25)
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