Conservative Vector Field: Finding the Value of 'a

In summary, the vector field F(x,y,z)= 2xz i + ay^3z j + (x^2 + y^4) k is conservative for the value of a=4. This was determined by using the 3D curl test to show that all three components of the vector field have a curl of 0.
  • #1
hils0005
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Homework Statement



For what value(s) of the scalar 'a' is the vector field
F(x,y,z)= 2xz i + ay^3 j + (x^2 + y^4) k conservative



The Attempt at a Solution



F1=2xz
F2=ay^3z
F3=(x^2 + y^4)

I used 3D curl test??

1)(partial F2)/(partial dx) - (partial F1)/ (partial dy)= 0-0 = 0
2)(partial F3)/(partial dy) - (partial F2)/ (partial dz)= 4y^3 - ay^3=0 so a=4
3)(partial F1)/(partial dz) - (partial F3)/ (partial dx)= 2x-2x = 0

my answer would be a=4
 
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  • #2
If your Fy is supposed to be ay^3z then you're right. Your initial statement of the problem has Fy = ay^3.
 
  • #3
yes it should be ay^3z j, Thanks!
 

1. What is a vector field?

A vector field is a mathematical concept that associates a vector to every point in a given space. The vectors in a vector field can represent physical quantities such as velocity, force, or electric and magnetic fields.

2. What does it mean for a vector field to be conservative?

A vector field is considered conservative if the work done by the field on a particle moving along any closed path is zero. In other words, the energy of a particle moving in a conservative vector field is conserved.

3. How can one determine if a vector field is conservative?

One way to determine if a vector field is conservative is by using a mathematical property known as the curl. If the curl of a vector field is equal to zero, then the field is conservative. Additionally, a conservative vector field can be represented by the gradient of a scalar function known as the potential function.

4. What are some real-life applications of conservative vector fields?

Conservative vector fields have numerous applications in physics and engineering. Some examples include modeling fluid flow, predicting weather patterns, and calculating electric and magnetic fields in circuits and devices.

5. Can a vector field be both conservative and non-conservative?

No, a vector field cannot be both conservative and non-conservative. If a vector field is conservative, then it must satisfy certain mathematical conditions such as having a zero curl, which would make it impossible for the field to also be non-conservative. However, a vector field can be non-conservative in some regions and conservative in others.

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