Spherical coordinates and triple integrals

[Rorschach]

Suppose $\displaystyle f = e^{(x^2+y^2+z^2)^{3/2}}$. We want to find the integral of $f$ in the region $R = \left\{x \ge 0, y \ge 0, z \ge 0, x^2+y^2+z^2 \le 1\right\}$.

Could someone tell me how we quickly determine that $R$ can be written as: $R = \left\{\theta \in [0, \pi/2], \phi \in [0, \pi/2], r \in [0,1]\right\}$?

I get that $r \in [0,1]$. But I don't know how to determine $\phi$ and $\theta$. I'd prefer an algebraic explanation, if possible.

 

[tkhunny]

Suppose $\displaystyle f = e^{(x^2+y^2+z^2)^{3/2}}$. We want to find the integral of $f$ in the region $R = \left\{x \ge 0, y \ge 0, z \ge 0, x^2+y^2+z^2 \le 1\right\}$.

Could someone tell me how we quickly determine that $R$ can be written as: $R = \left\{\theta \in [0, \pi/2], \phi \in [0, \pi/2], r \in [0,1]\right\}$?

I get that $r \in [0,1]$. But I don't know how to determine $\phi$ and $\theta$. I'd prefer an algebraic explanation, if possible.

1st Octant. How much is swept getting from the positive x-axis to the positive y-axis?

 

[Rorschach]

1st Octant. How much is swept getting from the positive x-axis to the positive y-axis?

I see. Is there a way to see it algebraically from the equations $x = r\sin \phi \cos \theta, ~ y= r \sin \phi \sin \theta, ~ z= r\cos \phi$?

 

[tkhunny]

I see. Is there a way to see it algebraically from the equations $x = r\sin \phi \cos \theta, ~ y= r \sin \phi \sin \theta, ~ z= r\cos \phi$?

Just off the top of my head, I would guess those three expressions are too complicated to help most folks "see it algebraically" - whatever that means.

You can sort of talk yourself into the "r" part of polar / cylindrical coordinates, but the Jacobian for this one is an entirely different animal.

I defer to the multi-dimensional musers for additional input.

 

[Rorschach]

Just off the top of my head, I would guess those three expressions are too complicated to help most folks "see it algebraically" - whatever that means.

You can sort of talk yourself into the "r" part of polar / cylindrical coordinates, but the Jacobian for this one is an entirely different animal.

I defer to the multi-dimensional musers for additional input.

I think my attempt to get this done algebraically is misguided.

The main reason I wanted to do it algebraically is because anything greater than 2 dimensions is a challenge for me to visualise. I know there are standard ways to graph it etc, but it just doesn't click. I'm probably spatially challenged. But I'll have to endure, it seems.

 

[tkhunny]

I think my attempt to get this done algebraically is misguided.

The main reason I wanted to do it algebraically is because anything greater than 2 dimensions is a challenge for me to visualise. I know there are standard ways to graph it etc, but it just doesn't click. I'm probably spatially challenged. But I'll have to endure, it seems.

This is where the bad news comes in. If you have been relying on visualization to this point, you'll have to start giving it up. That MAY work for 3D, but what's your plan for 4D and above. I have seen projected representations of 5D objects in a 4D spatial construct, but you don't want to walk around with that in your head. Learn to trust your methods, your processes, your extensions, and yourself. It does take more work without your eyes.