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