Necessary and sufficient condition for line of curvature to be parametric curve is F=0, M=0 and EN−GL≠0
Let the line of curvature b parametric curve such that u= constant and v=constant.
So combined the differential equation of parameter curve is dudv=0.
Now differential equation of line of curvature is (EM-FL)du$^2$+(EN-GL)dudv+((FN-GM)dv$^2$ = 0
So from above equations EM-FL=0,FN-GM=0 and EN−GL≠0.
But as per my understanding since parametric curve has one parameter so other is constant so u=constant or v=constant or in another form parametric curve is r(u,c) or r(c,v) where c is constant.
Hence either du=0 or dv=0. So either EM-FL=0 or FN-GM=0. Further, Since dvdu=0 hence EN−GL≠0 .
Could you explain why book has both parameter of parametric curve u=constant and v=constant and why both term taken EM-FL and FN-GM of du$^2$ and dv$^2$ equal to 0.