Calculating Flux through a Sphere using Divergence Theorem

In summary, the problem involves evaluating a surface integral over the surface of a sphere with radius 6 centered at the origin. The solution involves using polar coordinates and setting up a triple integral. However, there may be a mistake in the limits of integration, as the given answer is incorrect.
  • #1
CB4
10
0
Evaluate [URL]http://webwork.latech.edu/webwork2_files/tmp/equations/93/91cfe28c766cad38444f0213c651281.png[/URL] where [URL]http://webwork.latech.edu/webwork2_files/tmp/equations/59/a56001472f977192637ea927c607a61.png[/URL] and is the surface of the sphere of radius 6 centered at the origin.

Ok so I started by taking the divF to get 3y^2+3x^2+3z^2. Using polar coordinates I created the integral (from 0 to 2pi) of the integral (from 0 to pi/2) of the integral (from 0 to 6) (3r2sin2(theta)sin2(phi)+3r2cos2(theta)sin2(phi)+3r2cos2(phi)))r2sin(phi) d(phi)d(theta)d(r).

After all of that I simplified my answer down but I keep getting a huge number, 29314.82937, for an answer and it's incorrect. This problem seems very easy so I feel like I'm missing a very obvious step.
 
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  • #2
If it helps, instead of converting each term, convert them as a group to r^2.
 
  • #3
That is a good idea, but I still didn't come out with the right answer.
 
  • #4
Why are you taking phi from 0 to pi/2? It should be to pi.
 
  • #5
That is the million dollar question. Thank you very much my friend.
 

Related to Calculating Flux through a Sphere using Divergence Theorem

1. What is the Divergence Theorem in Calculus?

The Divergence Theorem, also known as Gauss's Theorem, is a fundamental theorem in calculus that relates the surface integral of a vector field over a closed surface to the triple integral of the divergence of the same vector field within the region enclosed by the surface. It is a powerful tool for solving problems in fluid mechanics, electromagnetism, and other areas of physics and engineering.

2. How is the Divergence Theorem used in real-world applications?

The Divergence Theorem has many practical applications, including calculating the flux of a vector field through a given surface, determining the charge enclosed by a closed surface in electromagnetism, and evaluating fluid flow and circulation in fluid mechanics. It is also used in solving problems related to conservation laws, such as the conservation of mass and energy.

3. What is the difference between the Divergence Theorem and the Fundamental Theorem of Calculus?

The Divergence Theorem is a generalization of the Fundamental Theorem of Calculus, which states that the integral of a function over a closed interval can be calculated by evaluating the antiderivative of the function at the endpoints of the interval. The Divergence Theorem extends this concept to vector fields and closed surfaces in three-dimensional space.

4. How is the Divergence Theorem related to the other fundamental theorems of calculus?

The Divergence Theorem is closely related to the other two fundamental theorems of calculus, the Fundamental Theorem of Line Integrals and the Fundamental Theorem of Surface Integrals. Together, these theorems form the basis of vector calculus and are essential in solving problems in physics, engineering, and other fields.

5. What are some common misconceptions about the Divergence Theorem?

One common misconception about the Divergence Theorem is that it only applies to specific types of vector fields, such as conservative or solenoidal fields. In reality, the theorem can be applied to any vector field, regardless of its properties. Another misconception is that the Divergence Theorem is only applicable to closed surfaces, when in fact it can also be used for open surfaces by considering a limiting case as the surface becomes closed.

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