Find the equation of the tangent plane of of the following surface patch at the indicated point: $$ σ(r, θ) = (r \cosh θ, r \sinh θ, r^2), (1, 0, 1).$$
I know that the tangent space of a surface patch is the vector space of the partial derivatives of the patch. So i took $$ dσ/dθ=(r \sinh θ,r \cosh θ,0) $$ and $$ dσ/dr=(\cosh θ,\sinh θ,2r). $$ Now in order to be at the point $(1,0,1)$, $r$ must be $r=1$ and $θ=0$. So the first partial der. is $(0,1,0)$ and th second $(1,0,2)$ now I can get the normal vector from the cross product and I know how to find the plane with a normal vector and a point on the plane. But what I get is a line, what am I doing wrong?