Understanding Weyl Rule: A Comprehensive Guide with References and Equations

[MathematicalPhysicist]

I hope someone can help me cite the right reference that explains how to arrive at Weyl rule in the next paper, in the second page (eq. (1)).

http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry340.pdf

Thanks in advance.

 

[Avodyne]

EVerything you could ever want to know:

http://media.wiley.com/product_data/excerpt/04/35274083/3527408304.pdf

 

[MathematicalPhysicist]

Thanks.

 

[MathematicalPhysicist]

Another question I have from the same paper by Berry, in page 3032 in the section on curvatures of nodal lines, he writes that the curvature of a contour line u, at a point r is given by:
$$\kappa(r)=\frac{(u_x)^2u_{xx}+(u_y)^2 u_{yy} -2 u_x u_y u_{xy}}{|\nabla u|^3}$$

Now I know from elementary differential geometry, that for two dimensional curve, the curvature is given by:
$$\kappa=\frac{det(\gamma ' | \gamma '')}{|\gamma ' |^3}$$

Now if I apply it to the above equation then presumably, $$\nabla u =(u_x,u_y)= \gamma ' (t)=(x'(t),y'(t))$$.
which means that $$x''(t)= u_{xx} x'(t) +u_{xy} y'(t)= u_{xx} u_x +u_{xy} u_y$$
$$y''(t)= u_{yy} u_y +u_{xy} u_x$$, and unless I have mistaken somewhere the determinant becomes: $$(u_x^2 -u_y^2)u_{xy}+u_x u_y(u_{yy}-u_{xx})$$.

Anyone care to explain?

Thanks in advance.