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Question relate to multi variable.

  1. Jul 22, 2011 #1
    Let [itex]\;P(R,\theta,\phi)\; [/itex] be function at each point defined by [itex] R,\theta,\phi[/itex] in spherical coordinates.

    Let [itex]\;P_{max} \;[/itex] be the maximum value of [itex]\;P(R,\theta,\phi)\; [/itex] in the closed sphere S.

    [tex]\hbox {Let }\;F(R,\theta,\phi)=\frac {P(R,\theta,\phi)}{P_{max}}[/tex]

    Which is the normalized value of [itex]\;P(R,\theta,\phi)\; \hbox {where } \; F_{max} = 1[/itex].

    My question is whether:

    [tex] \frac {P(R,\theta,\phi)}{\oint_S P(R,\theta,\phi) d\;S}\; =\; \frac {F(R,\theta,\phi)}{\oint_S F(R,\theta,\phi) d\;S}[/tex]

    I thought

    [tex] \frac {\left [\frac {P(R,\theta,\phi)}{P_{max}}\right ]} {\left [\frac {\oint_S P(R,\theta,\phi) d\;S}{P_{max}}\right ]} \;\hbox { not equal to } \; \frac {F(R,\theta,\phi)}{\oint_S F(R,\theta,\phi) d\;S}[/tex]

    Unless we can consider [itex] \;P_{max}\;[/itex] is a constant and can be moved inside the integration. So the question is whether [itex] \;P_{max}\;[/itex] is a constant? I am not sure.

    Please help.

    Thanks

    Alan
     
    Last edited: Jul 23, 2011
  2. jcsd
  3. Jul 23, 2011 #2

    HallsofIvy

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    Yes, once P has been defined, Pmax is a specific number, a constant.
     
  4. Jul 23, 2011 #3
    Thanks so much for answering.

    So I can move [itex]\;P_{max}\;[/itex] inside the integral and:


    [tex] \frac {P(R,\theta,\phi)}{\oint_S P(R,\theta,\phi) d\;S}\; =\; \frac {F(R,\theta,\phi)}{\oint_S F(R,\theta,\phi) d\;S}[/tex]

    Thanks

    Alan
     
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