Proof of Divergence Formula in Spherical Coordinates

In summary, the conversation discusses deriving the divergence formula for spherical coordinates by carrying out surface integrals of a piece of a sphere. The formula involves a radial coordinate, polar angle, and azimuthal angle. The volume of the little piece of a sphere is also mentioned. However, there is confusion about setting up the limits and the use of sin\theta. The difference between the polar angle (φ) and azimuthal angle (θ) is also explained.
  • #1
eliasds
4
0
Hello - I'm supposed to derive the divergence formula for spherical coordinates by carrying out the surface integrals of the surface of the volume in the figure (the figure is a piece of a sphere similar to a box but with curves). The radial coord is r. The polar angle is [tex]\varphi[/tex] and the azimuthal angle is [tex]\theta[/tex].The divergence formula is easy enought to look up: DIV(F) = [tex]\nabla\bullet[/tex]F =

[tex]\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}F_{r}[/tex]+[tex]\frac{1}{rsin\varphi}\frac{\partial}{\partial \varphi}\left( sin\varphi F_{\varphi}\right)[/tex] + [tex]\frac{1}{rsin\varphi}[/tex][tex]\frac{\partial F_{\theta}}{\partial\theta}[/tex]

And the volume of the little piece of a sphere is easy enough:
[tex]r^{2}sin\varphi \Delta r \Delta\varphi\Delta\theta[/tex]

But when I try to set up the limits for each side as the volume goes to zero I never end up with the first and second [tex]sin\varphi[/tex] in the equation. Supposedly I'm supposed to multiply by a [tex]sin\theta[/tex] but I don't see why.

What I end up with is:
[tex]\frac{\partial}{\partial r}F_{r}[/tex]+[tex]\frac{1}{r}\frac{\partial}{\partial \varphi}\left( F_{\varphi}\right)[/tex] + [tex]\frac{1}{rsin\varphi}[/tex][tex]\frac{\partial F_{\theta}}{\partial\theta}[/tex]
 
Last edited:
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  • #2
How is polar angle different from azimuth angle?
 
  • #3
Phi is the angle in the page, and the theta is the angle out of the page.
 

What is the Proof of Divergence Formula in Spherical Coordinates?

The Proof of Divergence Formula in Spherical Coordinates is a mathematical formula used in vector calculus to calculate the divergence of a vector field in three-dimensional spherical coordinates.

Why is the Proof of Divergence Formula important?

The Proof of Divergence Formula is important because it allows us to calculate the net flow of a vector field through a closed surface in three-dimensional space. This is useful in many scientific fields, such as fluid dynamics, electromagnetism, and thermodynamics.

How is the Proof of Divergence Formula derived?

The Proof of Divergence Formula is derived using the divergence theorem, which states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the enclosed volume. By converting the coordinates from Cartesian to spherical, the formula can be expressed in terms of the spherical coordinates.

What are the advantages of using the Proof of Divergence Formula in Spherical Coordinates?

The main advantage of using the Proof of Divergence Formula in Spherical Coordinates is that it simplifies calculations for problems involving spherical symmetry. It also allows for easier interpretation of the results, as the spherical coordinates are more intuitive in many physical situations.

Are there any limitations to using the Proof of Divergence Formula in Spherical Coordinates?

Like any mathematical formula, the Proof of Divergence Formula in Spherical Coordinates has its limitations. It can only be used for problems with spherical symmetry and is not applicable to other coordinate systems. It also assumes that the vector field is continuous and differentiable, and the closed surface is smooth.

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