Let S be the surface parameterised by
$\displaystyle r(u,v):=(e^ucosv,u,e^usinv),
u\in\Re, v\in[0,2\pi]$
(i) Find the equation of the plane tangent to S at r(u,v).
(ii) At what points is the tangent plane to S parallel to the x-y plane?
Let S be the surface parameterised by
$\displaystyle r(u,v):=(e^ucosv,u,e^usinv),
u\in\Re, v\in[0,2\pi]$
(i) Find the equation of the plane tangent to S at r(u,v).
(ii) At what points is the tangent plane to S parallel to the x-y plane?
$\displaystyle r_u= (e^ucos(v), 1, e^usin(v))$ and $\displaystyle r_v= (-e^usin(v), 0, e^ucos(v))$ are two vectors in the tangent plane. Their cross product will be perpendicular to the plane. The tangent plane will be parallel to the xy-plane when that normal vector is in the z direction (has both x and y components equal to 0).