A vector field with a vector potential has zero flux through every closed surface in its domain. it is observed that although the inverse-square radial vector field F = (e^r)/(p^2) satisfies div(F) = 0, F cannot have a vector potential on its domain {(x,y,z)does not equal (0,0,0)} because the flux of F through a sphere containing the origin is nonzero.
(a) show that F also has a vector potential on the domains obtained by removing either the x-axis or the z-axis from R^3.
(b) Does the existence of a vector on these restricted domains contradict the fact that the flux of F through a sphere containing the origin is nonzero?
To show that the vector field F = (e^r)/(r^2) has a vector potential on the domains obtained by removing either the x-axis or the z-axis from R^3, we need to find a vector field A such that curl(A) = F.
Let's first consider the domain obtained by removing the x-axis. In this case, we can write the vector field F as:
F = (e^r)/(r^2) = (1/r^2)e^r
To find a vector potential A, we need to solve the equation curl(A) = F. The curl of a vector field A = (Ax, Ay, Az) is given by the vector:
curl(A) = (dz/dy - dy/dz, dx/dz - dz/dx, dy/dx - dx/dy)
Substituting this into the equation curl(A) = F, we get the following system of equations:
dz/dy - dy/dz = (1/r^2)e^r
dx/dz - dz/dx = 0
dy/dx - dx/dy = 0
Integrating the first equation with respect to y, we obtain:
z - ydz/dy = (1/r^2)e^r y + h(x, z)
where h(x, z) is an arbitrary function of x and z. Taking the derivative of this equation with respect to z, we get:
dz/dy - y(d^2z/dy^2) = 0
Solving this differential equation, we find that z = y^2/2 + g(x) where g(x) is an arbitrary function of x.
Now, substituting z - ydz/dy = (1/r^2)e^r y + h(x, z) into the second equation dx/dz - dz/dx = 0, we get:
dx/dz - dz/dx = -dg(x)/dx = 0
Integrating this equation, we find g(x) = C, where C is an arbitrary constant.
Therefore, a possible vector potential A for the vector field F = (e^r)/(r^2) on the domain obtained by removing the x-axis is:
A = (x, y^2/2 + C, z)
Similarly, we can show that F has a vector potential on the domain obtained by removing the z-axis. In this case, a possible vector potential A is:
A = (x, y, z^2/2 + C)
To answer part (b) of the question, the existence of a vector potential on these restricted domains does not contradict the fact that the flux of F through a sphere containing the origin is nonzero. The key point here is that the original statement "a vector field with a vector potential has zero flux through every closed surface in its domain" applies specifically to closed surfaces within the complete domain of the vector field. By removing the x-axis or the z-axis, we are restricting the domain and changing the nature of the surfaces that we consider. So, while the flux of F through a sphere containing the origin may be nonzero in these restricted domains, it does not invalidate the fact that the vector potential exists for F within these modified domains.