Finding volumes from infinitesimal displacements

In summary, the infinitesimal displacement in spherical polar coordinates can be expressed as ds^2 = dr^2 + r^2 d\theta^2 + r^2 \sin\left(\theta\right)^2 d\phi^2. From this, we can derive the spherical volume-element dV = r^2 \sin\left(\theta\right)drd\theta d\phi. While a figure can be used to illustrate the geometry, there is no general method for directly finding the volume from ds. One possibility is to first find the area and then use it to calculate the volume. Trying to work with ds^3 is not a viable option since ds is a vector. No hints were provided for a
  • #1
Niles
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Homework Statement


In spherical polar coordinates, the infinitesimal displacement ds is given by:

[tex]
ds^2 = dr^2 + r^2 d\theta ^2 + r^2 \sin \left( \theta \right)^2 d\phi ^2
[/tex]

Can I find the volume of a sphere using ds?

The Attempt at a Solution


I know the spherical volume-element is given by [tex]dV = r^2 \sin \left( \theta \right)drd\theta d\phi[/tex].

I can always make a figure showing the geometry of spherical coordinates, but is there a general way of finding the volume from ds? Perhaps finding the area, and then the volume?
 
Last edited:
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  • #2
No hints at all?

I thought of ds^3, but ds is a vector, so that won't work.
 

1. What is the concept of "infinitesimal displacements"?

Infinitesimal displacements refer to very small changes or movements in a system. In the context of finding volumes, this concept is used to break down a larger volume into smaller, more manageable infinitesimal volumes that can be easily calculated.

2. How is the volume calculated from infinitesimal displacements?

The volume is calculated by integrating the infinitesimal displacements over the entire region of interest. This involves adding up all the infinitesimal volumes to find the total volume. This method is based on the fundamental principle of calculus, where the sum of infinitely small changes can lead to a finite value.

3. What is the significance of finding volumes from infinitesimal displacements?

Finding volumes from infinitesimal displacements is important in many areas of science and engineering, such as fluid mechanics, thermodynamics, and electromagnetism. It allows for the calculation of complex volumes and shapes that cannot be easily determined using traditional methods.

4. Are there any limitations to using this method?

One limitation of finding volumes from infinitesimal displacements is that it assumes the system is continuous and without any gaps or holes. In reality, there may be irregularities or discontinuities that can affect the accuracy of the calculated volume. Additionally, this method may be more difficult to apply in systems with changing or non-uniform densities.

5. Can this method be applied to three-dimensional volumes?

Yes, this method can be applied to calculate the volume of three-dimensional shapes. It involves integrating the infinitesimal displacements in three dimensions, and the resulting volume will also be in cubic units. This method can be extended to higher dimensions as well, although it becomes more complex and challenging to visualize.

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