Extracting Vector F: A General Question

[M.M.M]

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

 

[Kevin_Axion]

Must $$F$$ be a vector field $$F = F_{x}\widehat{i} + F_{y}\widehat{j} + F_{z}\widehat{k}+...$$? I thought $$\nabla \times F$$ only operates if $$F$$ is a vector field.

 

[HallsofIvy]

Yes, that's true. He said "vector F" several times.

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

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" $\nabla$", that is if $div B= \nabla\cdot B= 0$.

 

[Dick]

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

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