Evaluate the Flux with Divergence Theorem

In summary, the task is to evaluate the flux of F = <(e^z^2,2y+sin(x^2z),4z+(x^2+9y^2)^(1/2)> over the boundary of a region defined by x^2 + y^2 < z < 8-x^2-y^2. Using the divergence theorem, it can be shown that the flux is equal to 6 times the volume enclosed by two paraboloids. The limits of integration for z are between r^2 and 8-r^2, for r between 0 and 2, and for θ between 0 and 2∏. There was some discussion and uncertainty about the
  • #1
PsychonautQQ
784
10

Homework Statement


Evaluate the flux where F = <(e^z^2,2y+sin(x^2z),4z+(x^2+9y^2)^(1/2)> in the boundary of the region x^2 + y^2 < z < 8-x^2-y^2


Homework Equations





The Attempt at a Solution


So using the divergence Theorem,

∇ dot F = 6

∫∫∫6r dzdrdθ

where z is bounded between r^2 and 8-r^2
and r is bounded between 0 and 8^(1/2)
and θ is bounded between 0 and 2∏

is this correct?
I'm mainly worried about my limits of integration for r and z are incorrect, can anyone verify?
 
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  • #2
Okay so yeah what I wrote is probably wrong, can anyone help me and my friends along?
 
  • #3
BRAIN BLAST!
I just realized that the r bound should go from 0 to 2? right??
 
  • #4
PsychonautQQ said:
BRAIN BLAST!
I just realized that the r bound should go from 0 to 2? right??
The enclosed volume is the volume between two paraboloids. I think what you have is correct.
 

Related to Evaluate the Flux with Divergence Theorem

1. What is the Divergence Theorem?

The Divergence Theorem is a mathematical theorem that relates the flux, or flow, of a vector field through a closed surface to the divergence of the vector field within the volume bounded by the surface.

2. How is the Divergence Theorem used to evaluate flux?

The Divergence Theorem states that the flux through a closed surface is equal to the triple integral of the divergence of the vector field within the volume bounded by the surface. This allows us to evaluate the flux by finding the divergence of the vector field and integrating it over the volume.

3. What is the significance of evaluating flux with the Divergence Theorem?

Evaluating flux with the Divergence Theorem allows us to easily calculate the flow of a vector field through a closed surface, which has many applications in physics and engineering. It also provides a powerful tool for solving various types of differential equations.

4. Are there any limitations to using the Divergence Theorem for evaluating flux?

One limitation of the Divergence Theorem is that it can only be used for closed surfaces, meaning surfaces with no holes or boundaries. Additionally, the vector field must be continuous and have a well-defined divergence for the theorem to be applied.

5. Can the Divergence Theorem be applied in higher dimensions?

Yes, the Divergence Theorem can be generalized to higher dimensions through the use of the generalized Stokes' theorem. In three dimensions, it is known as the Divergence Theorem, but in higher dimensions, it is known as the Fundamental Theorem of Calculus for Gradients.

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