Compute Surface Integral: Divergence Theorem & F=(xy^2,2y^2,xy^3)

In summary, the Divergence Theorem is a mathematical theorem that relates the flow of a vector field through a closed surface to the behavior of the vector field inside the surface. It states that the flux of a vector field through a closed surface is equal to the triple integral of the divergence of the vector field over the enclosed volume. This can be used to compute surface integrals by simplifying the calculation to a triple integral. The formula for computing a surface integral using the Divergence Theorem is: ∫∫(F∙n) dS = ∭(div F) dV, where F is the vector field, n is the unit normal vector to the surface, and dS and dV represent the
  • #1
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Use the divergence theorem to compute the surface integral F dot dS , where
F=(xy^2, 2y^2, xy^3) over closed cylindrical surface bounded by x^2+z^2=4 and y is from -1 to 1.

I've tried doing it and got 32pi/3 (i guess its wrong, so how to do it?)
Is it ok to compute Div F in terms of xyz and after that change into cylindrical polar coordinates to calculate volume?
 
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  • #2
Yes, that's perfectly legitimate.
 

1. What is the Divergence Theorem?

The Divergence Theorem is a mathematical theorem that relates the flow of a vector field through a closed surface to the behavior of the vector field inside the surface.

2. How is the Divergence Theorem related to Compute Surface Integrals?

The Divergence Theorem states that the flux of a vector field through a closed surface is equal to the triple integral of the divergence of the vector field over the enclosed volume. This can be used to compute surface integrals by simplifying the calculation to a triple integral.

3. What is the formula for calculating a surface integral using the Divergence Theorem?

The formula for computing a surface integral using the Divergence Theorem is: ∫∫(F∙n) dS = ∭(div F) dV, where F is the vector field, n is the unit normal vector to the surface, and dS and dV represent the surface and volume elements, respectively.

4. How do you apply the Divergence Theorem to a specific vector field, such as F=(xy^2,2y^2,xy^3)?

To apply the Divergence Theorem to a specific vector field, you must first calculate the divergence of the vector field. In this case, the divergence of F is (2xy + 2y^2). Then, you can use this value in the triple integral ∭(2xy + 2y^2) dV to compute the surface integral.

5. Can the Divergence Theorem be used to compute surface integrals for any vector field?

Yes, the Divergence Theorem can be applied to any vector field, as long as the vector field is defined and continuous within the enclosed volume. However, the calculations may become more complex for more complicated vector fields.

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