Spherical coordinates and triple integrals

In summary: 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.
  • #1
Rorschach
10
0
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.
 
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  • #2
Rorschach said:
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?
 
  • #3
tkhunny said:
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 $x = r\sin \phi \cos \theta, ~ y= r \sin \phi \sin \theta, ~ z= r\cos \phi$?
 
  • #4
Rorschach said:
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.
 
  • #5
tkhunny said:
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.
 
  • #6
Rorschach said:
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.
 

FAQ: Spherical coordinates and triple integrals

What are spherical coordinates and how are they different from Cartesian coordinates?

Spherical coordinates are a system of locating points in three-dimensional space using two angles and a distance from a reference point. These angles are the polar angle (θ) and the azimuthal angle (φ), and the distance is known as the radial distance (r). This system is different from Cartesian coordinates which use three perpendicular axes (x, y, z) to locate a point.

How do you convert between spherical and Cartesian coordinates?

To convert from spherical coordinates (r, θ, φ) to Cartesian coordinates (x, y, z), you can use the following formulas:
x = r * sin(θ) * cos(φ)
y = r * sin(θ) * sin(φ)
z = r * cos(θ)
To convert from Cartesian coordinates to spherical coordinates, the following formulas can be used:
r = √(x² + y² + z²)
θ = arccos(z/r)
φ = arctan(y/x).

What is a triple integral and how is it different from a double integral?

A triple integral is an integral that is computed over a three-dimensional region. It is used to calculate the volume of a solid or to find the average value of a function over a three-dimensional region. A double integral, on the other hand, is computed over a two-dimensional region and is used to find the area under a curve or the volume of a solid with a varying cross-section.

What is the order of integration for a triple integral in spherical coordinates?

The order of integration for a triple integral in spherical coordinates is typically dr dθ dφ, where r is the radial distance, θ is the polar angle, and φ is the azimuthal angle. This order can be changed depending on the region of integration and the function being integrated.

How do you set up a triple integral in spherical coordinates?

To set up a triple integral in spherical coordinates, the limits of integration must first be determined for each variable (r, θ, φ). The limits will depend on the shape and size of the region being integrated. Once the limits are determined, the integral can be written as:
∫∫∫ f(r, θ, φ) r² sin(θ) dr dθ dφ, where f(r, θ, φ) is the function being integrated and r² sin(θ) is the Jacobian determinant.

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