Let [itex]\;P(R,\theta,\phi)\; [/itex] be function at each point defined by [itex] R,\theta,\phi[/itex] in spherical coordinates.(adsbygoogle = window.adsbygoogle || []).push({});

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

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