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The question is to obtain flux through a hemisphere $x^2+y^2+z^2=25$ oriented in the direction of the positive y-axis. The field is given by $ \overrightarrow{F} = xz\hat{i} + x \hat{j}+y\hat{k}$.

I use spherical coordinates for the flux:

$flux=\iint_s \overrightarrow{F} \cdot \hat{n} dS$

where $0 < \phi < \pi$ and $0 < \theta < \pi$. I am wondering why these should be the limits, because I thought that $-\pi/2<\phi<\pi/2$ and $-\pi/2<\theta<\pi/2$ seems to be a more natural answer for me...

I need some clarifications on this, thank you!

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Regardless of whether $\varphi$ is your polar angle and $\theta$ is your azimuthal angle or vice versa (because both suggestions in your question are symmetric in $\varphi$ and $\theta$):

  • $0 < \varphi,\theta < \pi$ specifies the hemisphere whose axis of symmetry is the positive $y$ axis.
  • $-\dfrac{\pi}2<\varphi,\theta<\dfrac{\pi}2$ specifies the hemisphere whose axis of symmetry is the positive $z$ axis.

Recall that spherical coordinates look like

enter image description here

Note that the above shows $\varphi$ as the polar angle. If your book/ professor uses the other convention just switch the position of $\varphi$ and $\theta$ in the image.

To get the hemisphere whose axis of symmetry is the positive $y$ axis, you'll want the polar angle to go all the way from $0$ (the North pole) down to $\pi$ (the South pole). Then you'll want the aximuthal angle to go from $0$ (on the positive $x$ axis) to $\pi$ (on the negative $x$ axis).

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  • $\begingroup$ Ok!! That makes much more sense now! I have always had troubles with these angles, now it is much clear! Thank you! $\endgroup$ – Blackgirl5 Nov 21 '16 at 20:02

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