Finding the cross product of surface 
I have done part a). I attached it incase it was useful in calculating part b.
For b) I am confused because vector product is usually for vectors.
I tried writing $\sigma=(x,y+rn \cos v ,z+rb \sin v)$
Then calculated the 2 derivatives but I do not get the required answer. Maybe I need to understand what the vector should be first. 
 A: $\newcommand{\Vec}[1]{\mathbf{#1}}\newcommand{\Tgt}{\Vec{t}}\newcommand{\Nml}{\Vec{n}}\newcommand{\Bin}{\Vec{b}}\newcommand{\NML}{\Vec{N}}$Since $\gamma$ is a unit-speed parametrization, $\gamma'(s) = \Tgt$. The Frenet-Serret equations read
\begin{alignat*}{3}
  \Tgt_{s} &=              &&\quad\kappa \Nml, && \\
  \Nml_{s} &= -\kappa \Tgt &&                 &&+ \tau \Bin, \\
  \Bin_{s} &=              &&-\tau \Nml.      &&
\end{alignat*}
By definition,
$$
\sigma(s, v) = \gamma(s) + r(\Nml \cos v + \Bin \sin v),
$$
so the Frenet-Serret equations give
\begin{align*}
  \sigma_{v} &= r(-\Nml\sin v + \Bin\cos v), \\
  \sigma_{s} &= \Tgt + r(\Nml_{s}\cos v + \Bin_{s}\sin v) \\
  &= \Tgt + r(-\Tgt\kappa\cos v + \Bin\tau\cos v - \Nml\tau\sin v) \\
  &= \Tgt(1 - r\kappa\cos v) + \tau\sigma_{v}.
\end{align*}
Since $\Tgt \times \Nml = \Bin$, $\Nml \times \Bin = \Tgt$, and $\Bin \times \Tgt = \Nml$,
\begin{align*}
  \sigma_{s} \times \sigma_{v}
  &= \bigl[\Tgt(1 - r\kappa\cos v) + \tau\sigma_{v}\bigr] \times \sigma_{v}
  = \Tgt(1 - r\kappa\cos v) \times \sigma_{v} \\
  &= \Tgt(1 - r\kappa\cos v) \times r(-\Nml\sin v + \Bin\cos v) \\
  &= r(1 - r\kappa\cos v)(-\Bin\sin v - \Nml\cos v).
\end{align*}
As in the comments, $\NML = -\Bin\sin v - \Nml\cos v$ is a unit normal field to your surface, and
$$
\|\sigma_{s} \times \sigma_{v}\| = r(1 - r\kappa\cos v).
$$
Now,
\begin{align*}
  \NML_{v} &= -\Bin\cos v + \Nml\sin v, \\
  \NML_{s} &= \Nml\tau\sin v - (-\kappa\Tgt + \tau\Bin)\cos v
  = \Tgt\kappa\cos v + \tau\NML_{v}, \\
  \NML_{s} \times \NML_{v}
  &= \Tgt\kappa\cos v \times (-\Bin\cos v + \Nml\sin v) \\
  &= \kappa\cos v(\Nml\cos v + \Bin\sin v) = K\, \sigma_{s} \times \sigma_{v},
\end{align*}
from which you can extract the Gaussian curvature $K$.
