Use Divergence Theorem to Compute the Flux Integral Just a work check

In summary, the Divergence Theorem, also known as Gauss's Theorem, is a mathematical theorem 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 enclosed volume. It is used to convert a difficult surface integral into an easier volume integral and is beneficial for calculating flux integrals. However, it has limitations such as only being applicable to closed surfaces and requiring the vector field to be continuously differentiable within the enclosed volume. Additionally, it can only be applied to vector fields that have well-defined partial derivatives at every point within the volume.
  • #1
mmmboh
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2ykdi4l.jpg


Alright so I found div F=3x2+3y2+3z2

The integral then becomes the triple integral of the divergence of F times the derivative of the volume.

Changing into spherical coordinates, the integral becomes [tex]3\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{1}p^{4}sin{\phi}dpd{\phi}d{\theta} [/tex] which ends up equaling [tex]2.4{\pi}[/tex].

Is this correct?
 
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  • #2
anyone?
 
  • #3
Yes, that's right.
 

1. What is the Divergence Theorem?

The Divergence Theorem, also known as Gauss's Theorem, is a mathematical theorem 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 enclosed volume.

2. How is the Divergence Theorem used to compute the flux integral?

The Divergence Theorem allows us to convert a difficult surface integral into an easier volume integral by using the concept of divergence. The flux integral is calculated by taking the volume integral of the divergence of the vector field over the enclosed volume.

3. What is the purpose of using the Divergence Theorem to compute the flux integral?

The Divergence Theorem simplifies the computation of flux integrals by converting a surface integral into a volume integral, which is typically easier to evaluate. It also allows for the application of the fundamental theorem of calculus, making the calculation more efficient.

4. Are there any limitations to using the Divergence Theorem to compute the flux integral?

The Divergence Theorem can only be applied to closed surfaces, meaning that the surface must form a complete boundary around a 3-dimensional volume. It also requires the vector field to be continuously differentiable within the enclosed volume.

5. Can the Divergence Theorem be applied to any vector field?

No, the Divergence Theorem can only be applied to vector fields that are continuously differentiable within the enclosed volume. This means that the vector field must have well-defined partial derivatives at every point within the volume.

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