Vector calculus, Torricelli's Trumpet/Gabriel's Horn

[jackscholar]

Homework Statement

I've recently been completing an assessment on Torricelli's Trumpet and was told to look into the geometry aspect. I've been following this website: http://www.palmbeachstate.edu/honors/documents/jeansergejoseph.pdf
I understand all the steps but am not privvy to how they went from

Ru*Rv=-f(u)cos(v)+f(u)sin(v)+f(u)f'(u)
to ||Ru*Rv||=√(f}^2+(ff')^2)

I understand that they taken the inner product space (or at least that is what I believe it is) but I don't know how they managed to manipulate that equation to the inner product space? These are seen at the bottom of page 12 and the beginning of page 13 for the website. If someone could explain further what happened I would be very greatful.

 

[Dick]

Homework Statement

I've recently been completing an assessment on Torricelli's Trumpet and was told to look into the geometry aspect. I've been following this website: http://www.palmbeachstate.edu/honors/documents/jeansergejoseph.pdf
I understand all the steps but am not privvy to how they went from

Ru*Rv=-f(u)cos(v)+f(u)sin(v)+f(u)f'(u)
to ||Ru*Rv||=√(f}^2+(ff')^2)

I understand that they taken the inner product space (or at least that is what I believe it is) but I don't know how they managed to manipulate that equation to the inner product space? These are seen at the bottom of page 12 and the beginning of page 13 for the website. If someone could explain further what happened I would be very greatful.

It's not an inner product. It's a cross product. Ru x Rv=[-f(u)cos(v),f(u)sin(v),f(u)f'(u)]. It's a vector with three components. You left out the unit vectors. Now take the norm. || [x,y,z] ||=sqrt(x^2+y^2+z^2).

 

[jackscholar]

I was looking into norms but I tried it and it didn't work out but now I see where I went wrong. Thank you very much.