Help find the flux through the surface

In summary, the problem is to find the flux of a vector field A=(2x,-z^2,3xy) through a surface defined by ρ=2, 0<\phi<\pi/2, 0<z<1. The surface is not closed and the vector field does not need to be converted to cylindrical form. The easiest way to solve the problem is to integrate in cylindrical form, as the surface normal, differential area, and limits of integration will be simpler.
  • #1
Rombus
16
0

Homework Statement



Given a vector field [itex]A=(2x,-z^2,3xy)[/itex], find the flux of A through a surface defined by [itex] ρ
=2, 0<\phi<\pi/2, 0<z<1[/itex]

Homework Equations



∇[itex]\bullet[/itex]A?


The Attempt at a Solution



Can I use divergence method here?
This is a closed surface correct? A cylindrical wedge?
Also do I need to convert the vector field to cylindrical form? Or the defined surface to rectangle form?

If I used divergence do I divide my answer by 4 since the wedge is a 1/4 of the cylinder?

Thanks
 
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  • #2
The surface is not closed.
 
  • #3
I agree. I read the problem as asking for the flux through the round surface of the wedge and not the other four faces.
 
  • #4
Thanks for the replies.

This makes a lot more sense now. So knowing this I would integrate over the surfaces separately.

So it appears it would be easier to integrate in cylindrical form correct? So I would want to change the vector field from rectangular to cylindrical?
 
  • #5
Rombus said:
So knowing this I would integrate over the surfaces separately.

You should only need to integrate over the one surface that is defined by the equalities & inequalities given.

So it appears it would be easier to integrate in cylindrical form correct? So I would want to change the vector field from rectangular to cylindrical?

Yes, that would probably be the easiest way to do it since the surface normal and differential area, and limits of integration will all be much simpler in cylindrical coordinates than in Cartesian coordinates.
 

1. What is flux through a surface?

The flux through a surface is a measure of the amount of a vector field that passes through the surface. It is represented by the integral of the dot product of the vector field and the surface's normal vector over the surface.

2. How is flux through a surface calculated?

Flux through a surface is calculated using the formula: ∯ F ⋅ dA, where F is the vector field, ⋅ is the dot product, and dA is the differential area element of the surface. This is known as the flux integral.

3. What is the difference between positive and negative flux through a surface?

Positive flux through a surface indicates that the vector field is passing through the surface in the same direction as the surface's normal vector, while negative flux indicates that the vector field is passing through the surface in the opposite direction of the normal vector.

4. Why is finding the flux through a surface important?

Finding the flux through a surface is important in many scientific fields, such as fluid dynamics, electromagnetism, and heat transfer. It allows us to understand the flow of a vector field through a given surface and can help us make predictions and calculations in these fields.

5. Are there any factors that can affect the flux through a surface?

Yes, there are several factors that can affect the flux through a surface, including the orientation and shape of the surface, the strength and direction of the vector field, and the presence of any boundaries or obstacles that may alter the flow of the vector field.

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