(Surfaces of superconductors) Show that for one-dimensional...

(Surfaces of superconductors) Show that for one-dimensional problems, the Ginzburg-Landau equation for $\psi(x)$ can be rewritten as $\frac{d^2f(y)}{dy^2} - f(y) + [f(y)]^3 = 0$. (7) where $\psi(x) = \sqrt{|a|/b} f(y)$ and $y = x/\xi$ ($\xi$ is the Ginzburg-Landau correlation length), and verify that $f(y) = \tanh(y/\sqrt{2})$ is a solution with boundary condition $\psi(0) = 0.$

Transcript

00:01 We are given a set of fields, and we're asked to use stokes ' theorem to show that the circulations of these fields around the boundary of any smooth, orientable surface, and space are zero.

00:20 So in part a, field f is 2xi plus 2 .y .j plus 2zk, while it follows that the curl of f is going to be 0.

00:43 And therefore, we have that the circulation of this field around the boundary of any smooth orientable surface, say the boundary is c.

01:04 Well, by stokes ' theorem, this is equal to the double integral over the surface s of the curl of f in the direction of the outward normal vector n.

01:25 But as we previously calculated, this is zero, and therefore our whole integral is zero.

01:45 In part b, we are given the field f equals the gradient of x, y, squared, z cubed.

01:59 So we'll let little f of x, y, z, be our scalar function x, y, squared, z cubed.

02:11 Then it follows that the curl of f, the big f, this is equal to the gradient crossed with the gradient of little f.

02:33 And we know by equation 8 from this section, but this is equal to the zero vector.

02:44 The gradient crossed with the gradient of the function is zero.

02:49 And therefore we have that the curl of f is zero.

02:57 And so, by x -toxys theorem, if c is the boundary of any smooth, orientable surface in space, then the circulation of this field over c is equal to the flux of the curl of f across the surface s in the direction of the outward unit normal n...

Question

Please give Ace some feedback