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  • example of vector potential | planetmath.org
    beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit example of vector potential If the solenoidal vector U U x y z normal U normal U x y z vec U vec U x y z is a homogeneous function of degree λ λ lambda 2 absent 2 neq 2 then it has the vector potential A 1 λ 2 U r normal A 1 λ 2 normal U normal r displaystyle vec A frac 1 lambda 2 vec U times vec r 1 where r x i y j z k normal r x normal i y normal j z normal k vec r x vec i y vec j z vec k is the position vector Proof Using the entry nabla acting on products we first may write 1 λ 2 U r 1 λ 2 r U U r U r r U normal 1 λ 2 normal U normal r 1 λ 2 normal normal r normal normal U normal normal U normal normal r normal normal normal U normal r normal normal normal r normal U nabla times frac 1 lambda 2 vec U times vec r frac 1 lambda 2 vec r cdot nabla vec U vec U cdot nabla vec r nabla cdot vec U vec r nabla cdot vec r vec U In the brackets the first product is according to Euler s theorem on homogeneous functions equal to λ U λ normal U lambda vec U The second product can be written as U x r x U y r y U z r z subscript U x normal r x subscript U y normal r y subscript U z normal r z U x frac partial vec r partial x U y frac

    Original URL path: http://www.planetmath.org/exampleofvectorpotential (2016-04-25)
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  • examples of lamellar field | planetmath.org
    Now U i j k x y z ω y ω x 0 ω x x ω y y k 0 normal normal U i j k x y z ωy ωx0 ω x x ω y y normal k normal 0 displaystyle nabla times vec U left begin matrix vec i vec j vec k frac partial partial x frac partial partial y frac partial partial z omega y omega x 0 end matrix right left frac partial omega x partial x frac partial omega y partial y right vec k vec 0 and so U normal U vec U is lamellar Therefore there exists a potential field u u u with U u normal U normal u vec U nabla u We deduce successively u x ω y u x y 0 ω x y f y u y ω x f y ω x f y 0 f y C formulae sequence formulae sequence u x ω y formulae sequence u x y 0 ω x y f y u y ω x superscript f normal y ω x formulae sequence superscript f normal y 0 f y C frac partial u partial x omega y u x y 0 omega xy f y frac partial u partial y omega x f prime y equiv omega x f prime y 0 f y C Thus we get the result u x y 0 ω x y C u x y 0 ω x y C u x y 0 omega xy C which corresponds to a particular case in ℝ 2 superscript ℝ 2 mathbb R 2 Example 3 Given U a x i b y j a b z fragments normal U assign a x normal i b y normal j fragments normal a b normal z normal displaystyle vec U ax vec i by vec j a b z vec k The rotor is now U i j k x y z a x b y a b z 0 normal normal U i j k x y z ax by abz normal 0 displaystyle nabla times vec U left begin matrix vec i vec j vec k frac partial partial x frac partial partial y frac partial partial z ax by a b z end matrix right vec 0 From u U normal u normal U nabla u vec U we obtain u x a x u a x 2 2 f y z 1 formulae sequence u x a x u a superscript x 2 2 f y z 1 frac partial u partial x ax implies u frac ax 2 2 f y z quad 1 u y b y u b y 2 2 g z x 2 formulae sequence u y b y u b superscript y 2 2 g z x 2 frac partial u partial y by implies u frac by 2 2 g z x quad 2 u z a b z u a b z 2

    Original URL path: http://www.planetmath.org/examplesoflamellarfield (2016-04-25)
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  • examples of periodic functions | planetmath.org
    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 3A26B12 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 examples of periodic functions We list common periodic functions In the parentheses there are given their period with least modulus One periodic functions with a real period sine 2 π 2 π 2 pi cosine 2 π 2 π 2 pi tangent π π pi cotangent π π pi secant 2 π 2 π 2 pi cosecant 2 π 2 π 2 pi and functions depending on them especially the triangular wave function 2 π 2 π 2 pi the mantissa function x x x x x lfloor x rfloor 1 One periodic functions with an imaginary period exponential function 2 i π 2 i π 2i pi hyperbolic sine 2 i π 2 i π 2i pi hyperbolic cosine 2 i π 2 i π 2i pi hyperbolic tangent i π i π i pi hyperbolic cotangent i π i π i pi and functions depending on them Two periodic functions elliptic functions Functions with infinitely many periods the Dirichlet s function f x normal f maps to x absent displaystyle f

    Original URL path: http://www.planetmath.org/examplesofperiodicfunctions (2016-04-25)
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  • examples using comparison test without limit | planetmath.org
    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 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 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 examples using comparison test without limit Do the following series converge n 1 1 n 2 n 1 superscript subscript n 1 1 superscript n 2 n 1 displaystyle sum n 1 infty frac 1 n 2 n 1 1 n 1 n 3 n 1 n 4 n 1 superscript subscript n 1 superscript n 3 n 1 superscript n 4 n 1 displaystyle sum n 1 infty frac n 3

    Original URL path: http://www.planetmath.org/examplesusingcomparisontestwithoutlimit (2016-04-25)
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  • exclusion of integer root | planetmath.org
    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 3A40 00 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 exclusion of integer root Theorem The equation p x a n x n a n 1 x n 1 a 0 0 assign p x subscript a n superscript x n subscript a n 1 superscript x n 1 normal subscript a 0 0 p x a n x n a n 1 x n 1 ldots a 0 0 with integer coefficients a i subscript a i a i has no integer roots if p 0 p 0 p 0 and p 1 p 1 p 1 are odd Proof Make the antithesis that there is an integer x 0 subscript x 0 x 0 such that p x 0 0 p subscript x 0 0 p x 0 0 This x 0 subscript x 0 x 0 cannot be even because else all terms of p x 0 p subscript x 0 p x 0 except a 0 subscript a 0 a 0 were even and thus the whole sum could not have the even value 0 Consequently x 0 subscript x 0 x 0 and also its powers have to be odd Since 2 0 p x 0 and 2 p 0 a 0 fragments 2 normal 0 p fragments normal subscript x 0 normal and 2 not divides p fragments normal 0 normal subscript a 0 normal 2 mid 0 p x 0 quad textrm and

    Original URL path: http://www.planetmath.org/exclusionofintegerroot (2016-04-25)
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  • expectation example | planetmath.org
    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 3A12D05 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 3A12D05 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 The contraharmonic mean of several positive numbers u 1 subscript u 1 u 1 u 2 subscript u 2 u 2 normal ldots u n subscript u n u n is defined as c u 1 2 u 2 2 u n 2 u 1 u 2 u n assign c superscript subscript u 1 2 superscript subscript u 2 2 normal superscript subscript u n 2 subscript u 1 subscript u 2 normal subscript u n c frac u 1

    Original URL path: http://www.planetmath.org/expectationexample (2016-04-25)
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  • explementary | 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 3A26E60 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 3A26E60 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 User error Socket error Could not connect to 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 3A60A05 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 3A60A05 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 explementary The explementary arc of an arc a a a of a circle is the arc forming together with a a a the full circle Two

    Original URL path: http://www.planetmath.org/explementary (2016-04-25)
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  • exponent valuation | planetmath.org
    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 3A51F20 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 exponent valuation Definition A function ν ν nu defined in a field K K K is called an exponent valuation or shortly an exponent of the field if it satisfies the following conditions 1 ν 0 ν 0 nu 0 infty and ν α ν α nu alpha runs all rational integers when α α alpha runs the nonzero elements of K K K 2 ν α β ν α ν β ν α β ν α ν β nu alpha beta nu alpha nu beta 3 ν α β min ν α ν β ν α β ν α ν β nu alpha beta geqq min nu alpha nu beta Note that because of the discrete value set ℤ ℤ mathbb Z an exponent valuation belongs to the discrete valuations and because of notational causes to the order valuations Properties ν 1 0 ν 1 0 nu 1 0 ν α ν α ν α ν α nu alpha nu alpha ν α β ν α ν β ν α β ν α ν β displaystyle nu left frac alpha beta right nu alpha nu beta ν α n n ν α ν superscript α n n ν α nu alpha n n nu alpha ν α 1 α n min ν α ν α n ν subscript α 1 normal subscript α n ν α normal ν subscript α n nu alpha 1 ldots alpha n geqq min nu alpha ldots nu alpha n ν α β min ν α ν β if ν α ν β formulae sequence ν α β ν α ν β if ν α ν β nu alpha beta min nu alpha nu beta quad mbox if nu alpha neq nu beta Example If an integral domain mathcal O has a divisor theory normal superscript mathcal O to mathfrak D then for each prime divisor mathfrak p there is an exponent valuation ν subscript ν nu mathfrak p of the quotient field K K K of mathcal O It is given by ν α when α 0 max k ℤ k α when α 0 fragments subscript ν fragments normal α normal normal when α0 max k Z pkα when α0 nu mathfrak p alpha begin cases infty quad mbox when alpha 0 max k in mathbb Z vdots mathfrak p k mid alpha mbox when alpha neq 0 end cases ν ξ ν α ν β when ξ α β with α β fragments

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