Calculating Surface Integrals Using the Divergence Theorem

In summary, the task is to evaluate the double integral over the surface of a sphere with radius 3 centered at the origin. The divergence theorem is used to transform the double integral into a triple integral, with the integrand being 3y^2+3x^2+3z^2. The limits of integration are z=-3 to 3, y=-\sqrt{}3-z^2 to \sqrt{}3-z^2, and x=-\sqrt{}3-y^2-z^2 to \sqrt{}3-y^2-z^2. To simplify the computation, it is recommended to use spherical coordinates.
  • #1
liishii
3
0

Homework Statement


Evaluate the double integral over M (F [tex]\circ[/tex] dS) where M is the surface of the sphere of radius 3 centered around the origin. (Sorry! I couldn't figure out how to use math symbols!)


Homework Equations


double integral(F[tex]\bullet[/tex]dS)=triple integral ([tex]\nabla[/tex][tex]\bullet[/tex] F)dV due to the divergence thm.


The Attempt at a Solution


I used the divergence theorem and got triple int(3y^2+3x^2+3z^2) dx dy dz with the limits z=-3 to 3, y=-[tex]\sqrt{}3-z^2[/tex] to [tex]\sqrt{}3-z^2[/tex] , x=-[tex]\sqrt{}3-y^2-z^2[/tex] to [tex]\sqrt{}3-y^2-z^2[/tex] . I plugged this into wolfram alpha but the answer i get isn't the right answer...

THanks in advance for the help!
 
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  • #2
Greetings! Did you try using spherical coordinates? This will make the computation much easier, as well as make any mistakes easy to identify.
 

Related to Calculating Surface Integrals Using the Divergence Theorem

1. What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem, is a mathematical equation that relates the surface integral of a vector field over a closed surface to the volume integral of the divergence of the same vector field over the region enclosed by that surface. It is a fundamental theorem in vector calculus and has many practical applications in physics and engineering.

2. How is the divergence theorem used in science?

The divergence theorem is used in science to simplify the computation of certain integrals involving vector fields. It is commonly used in fluid dynamics, electromagnetism, and other fields where vector fields are present. It allows scientists to convert a difficult surface integral into a simpler volume integral, making calculations easier and more efficient.

3. What is the significance of the divergence theorem?

The divergence theorem is significant because it relates the behavior of a vector field on the surface of a region to its behavior inside the region. This allows scientists to make predictions and analyze the behavior of vector fields in a more comprehensive way. It also allows for the simplification of mathematical calculations, making it a useful tool in many scientific fields.

4. Can you give an example of how the divergence theorem is used?

Sure, one example of the use of the divergence theorem is in fluid dynamics. If we have a closed surface that encloses a volume of fluid, we can use the divergence theorem to relate the flow of the fluid through the surface to the rate of change of the fluid's density inside the volume. This allows us to make predictions about the behavior of the fluid without having to directly measure the flow at each point on the surface.

5. Are there any limitations to the divergence theorem?

Like any mathematical theorem, the divergence theorem has its limitations. It can only be used for closed surfaces and regions with continuous and differentiable vector fields. It also assumes that the vector field is "well-behaved" and does not have any singularities. In some cases, the divergence theorem may not be applicable, and other methods must be used to solve the problem at hand.

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