Question relate to multi variable.

[yungman]

Let \;P(R,\theta,\phi)\; be function at each point defined by R,\theta,\phi in spherical coordinates.

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

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

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

My question is whether:

\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}

I thought

\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}

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

Please help.

Thanks

Alan

 

[HallsofIvy]

Yes, once P has been defined, P_max is a specific number, a constant.

 

[yungman]

Yes, once P has been defined, P_max is a specific number, a constant.

Thanks so much for answering.

So I can move \;P_{max}\; inside the integral and:


\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}

Thanks

Alan