The image provides an explanation of the differential volume element in spherical coordinates, specifically discussing how to represent a small volume element $dV$ in terms of spherical coordinates $(r, \theta, \phi)$.

Key Points from the Image

1. Spherical Coordinates Definition: In this context:

   • $u_1 = r$ (radial distance),
   • $u_2 = \theta$ (polar angle),
   • $u_3 = \phi$ (azimuthal angle).

2. Scale Factors: The differential volume element in spherical coordinates incorporates scale factors:

   • $h_1 = 1$,
   • $h_2 = r$,
   • $h_3 = r \sin \theta$.

3. Differential Volume Element: Using these scale factors, the volume element $dV$ is calculated as: $dV = h_1 h_2 h_3 \, dr \, d\theta \, d\phi = (1)(r)(r \sin \theta) \, dr \, d\theta \, d\phi = r^2 \sin \theta \, dr \, d\theta \, d\phi$

This volume element is useful in various applications, such as integrating functions over a spherical region, especially in physics and engineering.

Would you like further details or explanations on any part of this?

Related Questions:

1. What are the applications of spherical coordinates in physics?
2. How do we derive the scale factors in spherical coordinates?
3. Why is $\sin \theta$ included in the volume element?
4. What are the uses of the spherical volume element in integrals?
5. How would this change if we were in cylindrical coordinates?

Tip:

In spherical coordinates, always remember that the $r^2 \sin \theta$ term in the volume element accounts for the scaling due to the spherical geometry, helping to measure volume correctly over spherical shells.