Let and .
a) Find a parametrization of the hyperboloid
b) Find an expression for a unit normal to this surface.
c) Find an equation for the plane tangent to the surface at , where
For the parametrization I got
is that the correct parametrization?
Think about an auxilliary plane uz, for example, which is perpendicular to the plane and makes degress starting from x axis. In this plane, we will make each point rotate. But and will have a parametrization too thus this will nicely describe the hyperboloid. Lets see:
In the plane, you parametrize the hyperbola with and with . Notice that this parameter s will describe the contour of the hyperbola.
Now you should use as another parameter, an this would be the "revolution parameter". It will take each contour of hyperbola and rotate it degrees.
So, in the plane, we should have and with . If we make in , we would end with , which requires the in the parametrization and .
Putting all together, we must have:
For the B) you want the expression of the normal vector?
Only to ellucidate, here's the hyperboloid. We can see the mesh caused by this type of parametrization - there are circunferences parallel to the xy plane, in which we have some z coordinate fixed, and there are hyperbolas, when we have either x or y fixed.
For the B, I guess you should use the definition of gradient. I'll work on this later. I hope it helps!
Yes, if , , and , then
(Although there is no such thing as "the" parameterization. A surface has many different parameterizations. Since your x, y, and z satisfy the equation , it is a valid parameterization.)
To find a vector perpendicular to the surface think of that as a "level surface" of and use the fact that grad F is always perpendicular to level surfaces of F.
And, once you have a perpendicular, use the fact that if the vector at point in the plane, then the plane is given by .