Vector Calc Homework Help Divergence Free Vector Fields

In summary, we have found that F is undefined at the center of the ellipsoid, and the divergence of F is (1 - a)x/||r - ai||^3 + (1 - b)y/||r - ai||^3 + (1 - c)z/||r - ai||^3. Using the divergence theorem, we have also found that the flux of F through the ellipsoid is equal to (4/3)πabc * [(1 - a)x/||r - ai||^
  • #1
leext101
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Homework Statement


Let S be the ellipsoid where a,b, and c are all positive constants.
x2/(a+1)+y2/(b2)+z2/(c2) = 1

→ → → → →
Let F = (r - ai) / ||r - ai|| [* r and i are vectors = I tried inserting the arrows]

a)Where is F undefined?
b)Find divF at points where divF is defined.
c) Find ∫s F dA.


Homework Equations


Divergence theorem


The Attempt at a Solution


I know that the divF=0 making it a divergence free vector field, but I am not sure how to solve for the flux through the ellipsoid.

I really would like to understand the actual concept of divergence free theorems and a step by step explanation would really help.

Thanks for your time and help!
 
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  • #2


Thank you for your post. Let me address your questions one by one.

a) F is undefined at points where r - ai = 0, which means that r = ai. In other words, F is undefined at the center of the ellipsoid.

b) To find divF at points where it is defined, we can use the definition of divergence: divF = ∇ · F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z. In this case, F = (r - ai) / ||r - ai||, so we can write:

∂Fx/∂x = ∂(r - ai)x/∂x * 1/||r - ai||
= (1 - a)x/||r - ai||^3

Similarly, we can find ∂Fy/∂y and ∂Fz/∂z. Putting it all together, we get:

divF = ∇ · F = (1 - a)x/||r - ai||^3 + (1 - b)y/||r - ai||^3 + (1 - c)z/||r - ai||^3

c) To find ∫s F dA, we can use the divergence theorem, which states that the flux of a vector field through a closed surface S is equal to the volume integral of the divergence of the vector field over the entire volume enclosed by S. In this case, the surface S is the ellipsoid, and the volume enclosed by S is the region inside the ellipsoid.

So we have:

∫s F dA = ∫∫∫v divF dV

To evaluate this integral, we can use the fact that divF is constant over the entire volume, since it does not depend on the position inside the ellipsoid. So we can take it out of the integral:

∫s F dA = divF * ∫∫∫v dV

Now, the volume integral can be evaluated using the formula for the volume of an ellipsoid:

V = (4/3)πabc

So we have:

∫s F dA = divF * (4/3)πabc

Putting it all together, we get:

∫s F dA = (4/3)πabc * [(
 

1. What is a divergence free vector field?

A divergence free vector field is a vector field in which the divergence (or flux) at any point is equal to zero. This means that the flow of the vector field is equal into and out of the point, resulting in no net change in the magnitude of the vector field at that point. In simpler terms, it means that the vector field has no sources or sinks.

2. Why is the concept of divergence free vector fields important?

The concept of divergence free vector fields is important because it is a fundamental property of vector fields in physics and engineering. Many physical phenomena, such as fluid flow and electromagnetic fields, can be described using vector fields. Divergence free vector fields are particularly useful in these applications because they represent fields with no sources or sinks, making them easier to analyze and understand.

3. How do you determine if a vector field is divergence free?

A vector field is divergence free if its divergence is equal to zero at every point. To determine this, you can use the divergence theorem, which states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the enclosed volume. If the flux is equal to zero, then the divergence must also be equal to zero, indicating a divergence free vector field.

4. Can a vector field be both divergence free and curl free?

Yes, a vector field can be both divergence free and curl free. In fact, these two properties are related, as a vector field that is both divergence free and curl free is called a solenoidal vector field. This means that the field has no sources or sinks and is also non-rotational, making it a very special type of vector field.

5. How are divergence free vector fields used in real-world applications?

Divergence free vector fields have a wide range of applications in physics, engineering, and mathematics. Some examples include analyzing fluid flow in pipes, modeling electromagnetic fields, and studying the behavior of magnetic fields in plasma physics. Additionally, divergence free vector fields are also used in computer graphics to create realistic fluid simulations and in computer vision for object tracking and motion estimation.

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