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?

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- Mar 11th 2011, 04:51 AMmaximus101equation of plane tangent
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? - Mar 11th 2011, 04:57 AMHallsofIvy
$\displaystyle r_u= (e^ucos(v), 1, e^usin(v))$ 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).