Principal curvatures of a surface

[Unicorn.]

Hello !

Homework Statement

In an exercise we consider two surfaces
given by
f₁=(x,y,f(x,y)) defined in a domain D
And
f₂=f₁ in D'
And a line d with two parametrizations
d1=(f(y,z),y,z)
d2=(x,f(x,z),z)
They ask to find the principal curvatures of S1 U S2 U d

Homework Equations

The Attempt at a Solution

To find the second fundamental form of a union of surfaces, do I have to do the calculations for every surface separately ?
And we know that d is a line so the principal curvatures must be 0, right ?
So I just end up doing the work for f₁=f₂ ?
Thanks

 

[mighty2000]

for your question! Yes, in order to find the second fundamental form of a union of surfaces, you will need to calculate it separately for each surface. This is because each surface may have different curvature properties and therefore will have different second fundamental forms. As for the line d, since it is a straight line, its principal curvatures will indeed be 0. However, it is still important to include it in your calculations for the union of surfaces as it is part of the overall shape. So in your solution, you will need to do the calculations for both f1 and f2, as well as include the line d in your final answer. I hope this helps!