archive-org.com » ORG » P » PLANETMATH.ORG

Total: 488

Choose link from "Titles, links and description words view":

Or switch to "Titles and links view".
  • fundamental theorem of integral calculus | planetmath.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 3A11R04 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 fundamental theorem of integral calculus The derivative of a real function which has on a whole interval a constant value c c c vanishes in every point of this interval d d x c 0 d d x c 0 frac d dx c 0 The converse theorem of this is also true Ernst Lindelöf calls it the fundamental theorem of integral calculus in Finnish integraalilaskun peruslause It can be formulated as Theorem If a real function in continuous and its derivative vanishes in all points of an interval the value of this function does not change on this interval Proof We make the antithesis that there were on the interval two distinct points x 1 subscript x 1 x 1 and x 2 subscript x 2 x 2 with f x 1 f x 2 f subscript x 1 f subscript x 2 f x 1 neq f x 2 Then the mean value theorem guarantees a point ξ ξ xi between x 1 subscript x 1 x 1 and x 2 subscript x 2 x 2 such that f ξ f x 1 f x 2 x 1 x 2 superscript f normal ξ f subscript x 1 f subscript x 2 subscript x 1 subscript x 2 f prime xi frac f x 1 f x 2 x 1 x

    Original URL path: http://www.planetmath.org/fundamentaltheoremofintegralcalculus (2016-04-25)
    Open archived version from archive


  • fundamental theorem of symmetric polynomials | planetmath.org
    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 3A26A06 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 fundamental theorem of symmetric polynomials Every symmetric polynomial P x 1 x 2 x n P subscript x 1 subscript x 2 normal subscript x n P x 1 x 2 ldots x n in the indeterminates x 1 x 2 x n subscript x 1 subscript x 2 normal subscript x n x 1 x 2 ldots x n can be expressed as a polynomial Q p 1 p 2 p n Q subscript p 1 subscript p 2 normal subscript p n Q p 1 p 2 ldots p n in the elementary symmetric polynomials p 1 p 2 p n subscript p 1 subscript p 2 normal subscript p n p 1 p 2 ldots p n of x 1 x 2 x n subscript x 1 subscript x 2 normal subscript x n x 1 x 2 ldots x n The polynomial Q Q Q is unique its coefficients are elements of the ring determined by the coefficients of P P P and its degree with respect to p 1 p 2 p n subscript p 1 subscript p 2 normal subscript p n p 1 p 2 ldots p n is same as

    Original URL path: http://www.planetmath.org/fundamentaltheoremofsymmetricpolynomials (2016-04-25)
    Open archived version from archive

  • fundamental theorems in mathematics | planetmath.org
    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 3A12F10 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 3A12F10 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 fundamental theorems in mathematics

    Original URL path: http://www.planetmath.org/fundamentaltheoremsinmathematics (2016-04-25)
    Open archived version from archive

  • fundamental units | planetmath.org
    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 3A00A05 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 ring R R R of algebraic integers of any algebraic number field contains a finite set H η 1 η 2 η t H subscript η 1 subscript η 2 normal subscript η t H eta 1 eta 2 ldots eta t of so called fundamental units such that every unit ε ε varepsilon of R R R is a power product of these multiplied by a root of unity ε ζ η 1 k 1 η 2 k 2 η t k t ε normal ζ superscript subscript η 1 subscript k 1 superscript subscript η 2 subscript k 2 normal superscript subscript η t subscript k t varepsilon zeta cdot eta 1 k 1 eta 2 k 2 ldots eta t k t Conversely every such element ε ε varepsilon of the field is a unit of R R R Examples units of quadratic fields units of certain cubic fields For some algebraic number fields such as all imaginary quadratic fields the set H H H may be empty t 0 t 0 t 0 In the case of a single fundamental unit t 1 t 1 t 1 which occurs e g in all

    Original URL path: http://www.planetmath.org/fundamentalunits (2016-04-25)
    Open archived version from archive

  • Galileo's paradox | planetmath.org
    msc 3A11R27 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 3A11R04 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 3A11R04 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

    Original URL path: http://www.planetmath.org/galileosparadox (2016-04-25)
    Open archived version from archive

  • Gauss--Lucas theorem | planetmath.org
    derivative f z superscript f normal z f prime z Proof Due to the fundamental theorem of algebra the polynomial f z f z f z can be written in the form f z a 0 z z 1 z z 2 z z n f z subscript a 0 z subscript z 1 z subscript z 2 normal z subscript z n displaystyle f z a 0 z z 1 z z 2 cdots z z n 1 where a 0 subscript a 0 a 0 is the leading coefficient and z 1 subscript z 1 z 1 z 2 subscript z 2 z 2 z n subscript z n z n are the zeros of f z f z f z some of these may coincide When the derivative f z μ 1 n ν μ z z ν superscript f normal z superscript subscript μ 1 n subscript product ν μ z subscript z ν f prime z sum mu 1 n prod nu neq mu z z nu is divided by 1 we have the identic equation f z f z 1 z z 1 1 z z 2 1 z z n superscript f normal z f z 1 z subscript z 1 1 z subscript z 2 normal 1 z subscript z n frac f prime z f z frac 1 z z 1 frac 1 z z 2 ldots frac 1 z z n and therefore f z f z 1 z z 1 1 z z 2 1 z z n superscript f normal z f z 1 z subscript z 1 1 z subscript z 2 normal 1 z subscript z n displaystyle f prime z f z left frac 1 z z 1 frac 1 z z 2 ldots frac 1 z z n right 2 Since f z f z f z has no zeros outside the polygon we have according to 2 to show only that the same concerns the second factor of the right hand side of 2 Let z z z be an arbitrary point outside the polygon Because of its convexity there is a line l l l through z z z such that the polygon is completely on the other side of l l l Thus all directed line segments from z z z to the points z 1 z 2 z n subscript z 1 subscript z 2 normal subscript z n z 1 z 2 ldots z n lie on the same side of the line l l l The direction angles of those segments being the arguments of the complex numbers z 1 z z 2 z z n z subscript z 1 z subscript z 2 z normal subscript z n z z 1 z z 2 z ldots z n z are between the values α α alpha and α π α π alpha pi where α α alpha is one of the two angles which l l l

    Original URL path: http://www.planetmath.org/gausslucastheorem (2016-04-25)
    Open archived version from archive

  • Gaussian sum | planetmath.org
    n 1 e frac 2 pi i n j 2 1 where n n n is a positive integer The explicit value of S S S may be found by using the Cauchy residue theorem for determining the contour integral I γ e 2 π i n z 2 1 e 2 π i z d z assign I subscript contour integral γ superscript e 2 π i n superscript z 2 1 superscript e 2 π i z d z I oint gamma frac e frac 2 pi i n z 2 1 e 2 pi iz dz where γ γ gamma goes once anticlockwise round the set of the poles 1 2 n 1 2 1 2 normal n 1 2 1 2 ldots lfloor frac n 1 2 rfloor of the integrand γ γ gamma is formed by a rectangle where two segments are replaced by arcs of half circle psaxes Dx 13 Dy 13 0 0 0 5 3 7 12 5 3 7 1 2 r i r i ri a i a i ai r i r i ri a i a i ai n 2 r i n 2 r i frac n 2 ri n 2 a i n 2 a i frac n 2 ai n 2 r i n 2 r i frac n 2 ri n 2 a i n 2 a i frac n 2 ai γ γ gamma 0 r 1 2 a r formulae sequence 0 r 1 2 a r 0 r frac 1 2 a r The connection between S S S and I I I is S 2 I 1 when 2 n 1 i n when 2 n S 2 I 1 when 2n1 in when 2 n displaystyle S 2I begin

    Original URL path: http://www.planetmath.org/gaussiansum (2016-04-25)
    Open archived version from archive

  • Gelfand--Tornheim theorem | planetmath.org
    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 3A11L05 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 Gelfand Tornheim theorem Theorem Any normed field is isomorphic either to the field ℝ ℝ mathbb R of real numbers or to the field ℂ ℂ mathbb C of complex numbers The normed field means a field K K K having a subfield R R R isomorphic to ℝ ℝ mathbb R and satisfying the following There is a mapping fragments parallel to normal parallel to cdot from K K K to the set of non negative reals such that a 0 norm a 0 a 0 iff a 0 a 0 a 0 a b a b norm a b normal norm a norm b ab leqq a cdot b a b a b norm a b norm a norm b a b leqq a b a b a b norm a b normal a norm b ab a cdot b when a R a R a in R and b K b K b in K Using the Gelfand Tornheim theorem it can be shown that the only fields with archimedean valuation are

    Original URL path: http://www.planetmath.org/gelfandtornheimtheorem (2016-04-25)
    Open archived version from archive