Divergence Theorem: Volume and Surface Integral Solutions

  • Thread starter Phymath
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In summary, the conversation discusses using the divergence theorem and spherical coordinates to calculate the volume and surface integrals of a hemisphere. While the volume integral results in 10Rπ, the surface integral is incorrect due to an error in the formula used, which should be d\vec{a} = R^2 sin\theta d\theta d\phi \hat{r}, not R^3.
  • #1
Phymath
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ok well basicly use the divergence thrm on this..
[tex]
\vec{v} = rcos\theta \hat{r} + rsin\theta\hat{\theta} + rsin\theta cos \phi \hat{\phi}[/tex]

so i did (remember spherical coords) i get..
[tex] 5cos \theta - sin\phi[/tex]
taking that over the volume of a hemisphere resting on the xy-plane i get [tex]10R \pi[/tex]

however with the surface intergral i use [tex] d\vec{a} = R^3 sin\theta d\theta d\phi \hat{r}[/tex] where R is the radius of the sphere doing that intergral gives me something obvioiusly R^3 which is not what I'm getting in the volume intergral any help or explanations (if its i need to take the surface of the bottom of the hemi sphere aka the xy-plane in a circle that will suck but let me know!
 
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  • #2
however with the surface intergral i use [tex] d\vec{a} = R^3 sin\theta d\theta d\phi \hat{r}[/tex] where R is the radius of the sphere doing that intergral
Your surface integral is wrong.
[tex] d\vec{a} = R^2 sin\theta d\theta d\phi \hat{r}[/tex] not R3.
 
  • #3


The Divergence Theorem states that the volume integral of a vector field over a closed region is equal to the surface integral of the vector field over the boundary of that region. In this case, the vector field \vec{v} is given in spherical coordinates and you are trying to find the volume integral over a hemisphere resting on the xy-plane. However, your calculation for the surface integral over the hemisphere's boundary is incorrect.

To properly use the Divergence Theorem, you need to consider the entire boundary of the hemisphere, which includes both the curved surface and the flat surface on the bottom. This means you need to add the surface integral over the flat surface as well.

Using the given formula for d\vec{a}, the surface integral over the curved surface of the hemisphere will give you R^3 as you have calculated. However, the surface integral over the flat surface will give you 0, as the normal vector \hat{r} is perpendicular to the flat surface and therefore the dot product with \vec{v} will be 0.

Thus, the total surface integral over the boundary of the hemisphere will be R^3, which is equal to the volume integral you have calculated. This confirms the validity of the Divergence Theorem in this case.

In summary, to properly use the Divergence Theorem, you need to consider the entire boundary of the region, which includes both the curved surface and the flat surface. In this case, the surface integral over the flat surface will give you 0, but it is still an important part of the calculation.
 

1. What is the Divergence Theorem?

The Divergence Theorem is a mathematical theorem that relates the volume of a three-dimensional region to the surface integral of its boundary. It states that the flux of a vector field through the boundary of a closed surface is equal to the volume integral of the divergence of that vector field over the region enclosed by the surface.

2. What is the significance of the Divergence Theorem?

The Divergence Theorem is significant because it provides a way to calculate the volume of a three-dimensional region without having to directly measure it. It also allows for the conversion between surface and volume integrals, making it a valuable tool in many areas of mathematics and physics.

3. How is the Divergence Theorem applied in real-world scenarios?

The Divergence Theorem has various applications in fields such as fluid mechanics, electromagnetism, and engineering. For example, it can be used to calculate the flow rate of a fluid through a given surface or to determine the strength of an electric field using Gauss's law.

4. What are the conditions for the Divergence Theorem to hold?

The Divergence Theorem holds when the region in question is bounded by a closed surface and the vector field is continuous and differentiable within the region. Additionally, the surface must be smooth and orientable, meaning that it has a consistent normal direction at each point.

5. Are there any limitations to the Divergence Theorem?

While the Divergence Theorem is a powerful tool, it does have some limitations. It only applies to three-dimensional regions and cannot be used for regions with holes or gaps. Additionally, it assumes that the vector field is defined and continuous everywhere within the region, which may not always be the case in real-world scenarios.

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