The sin(theta) is an interesting component in the differential as it is only there for one of the angles but not the other angle. You would assume it shouldn't be there because spheres are symmetrical. But it should be there because of the curvature of the sphere, and how longitude lines converge at the poles.
s3kim2018
I think the reason why only sin(theta) is present and not the other angle is because the flat vertical distance of the small rectangular chunk is dphi. However, when we drop a vertical line from our point (x,y,z) onto the xy plane it has a length r*sin(theta). Hence, the total area of the small chunk scales with sin(theta). I also think its interesting that phi is left out of the picture.
ArjunPalkhade
If you want to learn more about solid angles, one useful video that I saw was: https://youtube.com/watch?v=VmnkkWLwVsc
ttalati
If anyone has seen spherical coordinates and dealt with doing integrals on these sorts of surfaces, whether it is finding some quantity integrated over the surface, I believe this is the same idea of doing that change of basis and then we have to carefully reason about what we want to integrate. In this case we only care about the surface area not integrating through the volume.
JunoLee128
Is this analagous to a change of basis matrix? Or is polar coordinates not related to that (not a linear transformation?)
The sin(theta) is an interesting component in the differential as it is only there for one of the angles but not the other angle. You would assume it shouldn't be there because spheres are symmetrical. But it should be there because of the curvature of the sphere, and how longitude lines converge at the poles.
I think the reason why only sin(theta) is present and not the other angle is because the flat vertical distance of the small rectangular chunk is dphi. However, when we drop a vertical line from our point (x,y,z) onto the xy plane it has a length r*sin(theta). Hence, the total area of the small chunk scales with sin(theta). I also think its interesting that phi is left out of the picture.
If you want to learn more about solid angles, one useful video that I saw was: https://youtube.com/watch?v=VmnkkWLwVsc
If anyone has seen spherical coordinates and dealt with doing integrals on these sorts of surfaces, whether it is finding some quantity integrated over the surface, I believe this is the same idea of doing that change of basis and then we have to carefully reason about what we want to integrate. In this case we only care about the surface area not integrating through the volume.
Is this analagous to a change of basis matrix? Or is polar coordinates not related to that (not a linear transformation?)