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Curl problem

  • Thread starter M.M.M
  • Start date
  • #1
24
0
Hi everybody ....


I have a general question ..


if i have for example...

curl F = B

F and B are vectors and B is constant vector ..

is there any way to extract vector F?

thanks

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
901
2
Must [tex]F[/tex] be a vector field [tex]F = F_{x}\widehat{i} + F_{y}\widehat{j} + F_{z}\widehat{k}+...[/tex]? I thought [tex]\nabla \times F[/tex] only operates if [tex]F[/tex] is a vector field.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,833
955
Yes, that's true. He said "vector F" several times.

[tex]\nabla\times F= \left(\frac{\partial F_y}{\partial z}- \frac{\partial F_z}{\partial y}\right)\vec{i}- \left(\frac{\partial F_x}{\partial z}- \frac{\partial F_z}{\partial x}\right)\vec{j}+ \left(\frac{\partial F_y}{\partial x}- \frac{\partial F_x}{\partial y}\right)\vec{k}[/tex]
so that would be essentially solving the system of equations
[tex]\frac{\partial F_y}{\partial z}- \frac{\partial F_z}{\partial y}= B_x[/tex]
[tex]\frac{\partial F_x}{\partial z}- \frac{\partial F_z}{\partial x}= B_y[/tex]
[tex]\frac{\partial F_y}{\partial x}- \frac{\partial F_x}{\partial y}= B_z[/tex]

Since we can think of the cross product of two vectors as giving a vector perpendicular to both, that system, and the original equation, has a solution only if B is "perpendicular" to the "vector" [itex]\nabla[/itex]", that is if [itex]div B= \nabla\cdot B= 0[/itex].
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
Hi everybody ....


I have a general question ..


if i have for example...

curl F = B

F and B are vectors and B is constant vector ..

is there any way to extract vector F?

thanks

Homework Statement





Homework Equations





The Attempt at a Solution

There's no unique solution since there are a lot of vector fields with curl zero. If you just want any one try messing around with linear functions of the coordinates, like F=(0,a*x,b*x+c*y).
 

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