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  • converse of Euler's homogeneous function theorem | planetmath.org
    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 3A03F07 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 converse of Euler s homogeneous function theorem Theorem If the function f f f of the real variables x 1 x k subscript x 1 normal subscript x k x 1 ldots x k satisfies the identity x 1 f x 1 x k f x k n f subscript x 1 f subscript x 1 normal subscript x k f subscript x k n f displaystyle x 1 frac partial f partial x 1 ldots x k frac partial f partial x k nf 1 then f f f is a homogeneous function of degree n n n Proof Let f t x 1 t x k φ t assign f t subscript x 1 normal t subscript x k φ t f tx 1 ldots tx k varphi t Differentiating with respect to t t t we obtain φ t x 1 f x 1 t x 1 t x k x k f x k t x 1 t x k 1 t t x 1 f x 1 t x 1 t x k t x k f x k t x 1 t x k superscript φ normal t subscript x 1 subscript superscript f normal subscript x 1 t subscript x 1 normal t subscript x k normal subscript x k subscript superscript f normal subscript x k t subscript x 1 normal t subscript x k 1 t t subscript x 1 subscript superscript f normal subscript x 1 t subscript x 1 normal t subscript x k normal t subscript x k subscript superscript f normal subscript x k t subscript x 1 normal t subscript x k varphi prime t x 1 f prime x 1 tx 1 ldots tx k ldots x k f prime x k tx 1 ldots tx k frac 1 t tx 1 f prime x 1 tx 1 ldots tx k ldots tx k f prime x k tx 1 ldots tx k which by 1 may

    Original URL path: http://www.planetmath.org/converseofeulershomogeneousfunctiontheorem (2016-04-25)
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  • convex angle | planetmath.org
    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 3A26A06 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 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 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 3A15 00 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 3A15 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

    Original URL path: http://www.planetmath.org/convexangle (2016-04-25)
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  • coordinate vector | planetmath.org
    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 coordinate vector Let V V V be a vector space of dimension n n n over a field K K K If b 1 b n subscript b 1 normal subscript b n b 1 ldots b n is a basis of V V V then any element v v v of V V V can be uniquely expressed in the form v ξ 1 b 1 ξ n b n v subscript ξ 1 subscript b 1 normal subscript ξ n subscript b n v xi 1 b 1 ldots xi n b n with ξ 1 ξ n K subscript ξ 1 normal subscript ξ n K xi 1 ldots xi n in K The n n n tuplet ξ 1 ξ n subscript ξ 1 normal subscript ξ n xi 1 ldots xi n is called the coordinate vector of v v v with respect to the basis in question The scalars ξ i subscript ξ i xi i are the coordinates or the components

    Original URL path: http://www.planetmath.org/coordinatevector (2016-04-25)
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  • coordinate way | planetmath.org
    ℝ 3 mathbb R 3 If N N N is the projection of P P P on the x y x y xy plane A A A the projection of N N N on the x x x axis and O O O the origin then the broken line P N A O P N A O PNAO is the coordinate way of the point P P P The lengths of the line segments forming the coordinate way are A O x N A y P N z formulae sequence A O x formulae sequence N A y P N z AO x quad NA y quad PN z z z z y y y x x x P P P N N N A A A O O O The coordinate way of P P P red Synonym coordinate broken line Type of Math Object Definition Major Section Reference Parent polygon Groups audience Buddy List of pahio Mathematics Subject Classification 51N20 no label found 51G05 no label found 51 00 no label found Add a correction Attach a problem Ask a question Comments Mysterious phenomenon Permalink Submitted by pahio on Sun 08 19 2007 15 03 I made a pstrics picture in the entry coordinate way and thereafter the entry text has mysteriously changed in page images mode to a text concerning some Gabor frames HTML functions normally Who can say what is wrongly and how the fault coud be corrected Jussi Log in to post comments Re Mysterious phenomenon Permalink Submitted by CompositeFan on Sun 08 19 2007 18 31 I ve noticed things like that in my entries too Have you tried a rerender Lisa P S I m coming back to the States this week Log in to post comments Re Mysterious phenomenon Permalink Submitted by

    Original URL path: http://www.planetmath.org/coordinateway (2016-04-25)
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  • coordinates of midpoint | planetmath.org
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    Original URL path: http://www.planetmath.org/coordinatesofmidpoint (2016-04-25)
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  • corollary of B\'ezout's lemma | planetmath.org
    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 3A51M15 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 3A51M15 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 3A51 00 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 3A51 00 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28

    Original URL path: http://www.planetmath.org/corollaryofbezoutslemma (2016-04-25)
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  • corollary of Cauchy integral theorem | planetmath.org
    If the closed holey domain between γ γ gamma and the γ j subscript γ j gamma j s is contained in a domain U U U where f f f is holomorphic then γ f z d z γ 1 f z d z γ n f z d z subscript contour integral γ f z d z subscript contour integral subscript γ 1 f z d z normal subscript contour integral subscript γ n f z d z displaystyle oint gamma f z dz oint gamma 1 f z dz ldots oint gamma n f z dz 1 where all integrals are taken with the same direction of circulation Especially in the case n 1 n 1 n 1 one has γ f z d z γ 1 f z d z subscript contour integral γ f z d z subscript contour integral subscript γ 1 f z d z displaystyle oint gamma f z dz oint gamma 1 f z dz 2 Note 1 The integrals in 1 and 2 need not necessarily vanish since inside a γ j subscript γ j gamma j there may be points not belonging to U U U Note 2 When n 0 n 0 n 0 i e the sum on the right hand side of 1 is empty it has the value 0 thus also the Cauchy integral theorem is a special case of 1 Note 3 The theorem implies easily the residue theorem Proof We prove the theorem only in the case n 1 n 1 n 1 Other cases may be handled analogously Draw two auxiliary ways connecting γ γ gamma and γ 1 subscript γ 1 gamma 1 The integral of f f f taken anticlockwise around the route consisting of the upper parts of the curves and the auxiliary ways is by the fundamental theorem of complex analysis equal zero Similarly the integral of f f f taken anticlockwise around the route consisting of the lower parts of the curves and the auxiliary ways is equal zero Thus also the sum of both equals zero But in the sum the portions taken along the auxiliary ways are run in opposite directions and so they cancel each other Therefore in the sum only the portions which are run along γ γ gamma and γ 1 subscript γ 1 gamma 1 remain then γ f z d z γ 1 f z d z 0 subscript contour integral γ f z d z subscript contour integral subscript γ 1 f z d z 0 oint gamma f z dz oint gamma 1 f z dz 0 i e γ f z d z γ 1 f z d z subscript contour integral γ f z d z subscript contour integral subscript γ 1 f z d z oint gamma f z dz oint gamma 1 f z dz However here γ γ gamma is run anticlockwise and γ 1 subscript γ 1 gamma 1 clockwise Reversing the

    Original URL path: http://www.planetmath.org/corollaryofcauchyintegraltheorem (2016-04-25)
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  • cosine at multiples of straight angle | planetmath.org
    msc 3A30E20 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 3A30E20 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 cosine at multiples of straight angle Everybody remembers the cosine values cos 0 1 cos π 1 cos 2 π 1 cos 3 π 1 formulae sequence 0 1 formulae sequence plus or minus π 1 formulae sequence plus or minus 2 π 1 plus or minus 3 π 1 normal cos 0 1 quad cos pm pi 1 quad cos pm 2 pi 1 quad cos pm 3 pi 1 The thing can be concisely expressed as the formula cos n π 1 n n π superscript 1 n cos n pi 1 n for each integer n n n The values of sine at the same angles are simply

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