Description of surface, vector calculus

[ohlala191785]

Homework Statement

Consider the surface parameterized by (v cos(u), v sin(u), 45 v cos(u)), where u and v both vary from 0 to 2∏.

Homework Equations

(v cos(u), v sin(u), 45 v cos(u))
I think this is supposed to be a vector function? As in r(u,v) = <v cos(u), v sin(u), 45 v cos(u)>.

The Attempt at a Solution

In the x-y plane, this is a circle. x = v cos(u) so z = 45x. This is a slanted plane? So I thought the surface would be an ellipse, since the coefficient of x is 45 and the circle would be very squashed. Does this seem to be part of a cylinder because the cross section of a cylinder, if the plane is slanted, would it be a portion of an elliptical paraboloid? Or could it be a portion of a cone or hyperboloid? One of the answer options is an ellipsoid, but I don't think that's right because when I graphed this on a computer, it showed a flat surface.

There are so many possibilities!

 

[haruspex]

Take v to be fixed, initially. Varying u should give you a curve, and yes, it's an ellipse. Describe the plane the ellipse lies in and where it is centred. Then imagine varying v over its given range.

 

[LCKurtz]

Homework Statement

Consider the surface parameterized by (v cos(u), v sin(u), 45 v cos(u)), where u and v both vary from 0 to 2∏.

Homework Equations

(v cos(u), v sin(u), 45 v cos(u))
I think this is supposed to be a vector function? As in r(u,v) = <v cos(u), v sin(u), 45 v cos(u)>.

Yes, that's exactly what it is and how you describe a surface.

The Attempt at a Solution

In the x-y plane, this is a circle.

Yes, that is correct for fixed ##v##. Its equation is ##x^2+ y^2 = v^2##. But if you let ##v## vary from ##0## to ##2\pi## what do you get?

x = v cos(u) so z = 45x. This is a slanted plane?

Yes. And ##x## and ##z## must be on that plane, no matter what ##y## is.

So I thought the surface would be an ellipse, since the coefficient of x is 45 and the circle would be very squashed. Does this seem to be part of a cylinder because the cross section of a cylinder, if the plane is slanted, would it be a portion of an elliptical paraboloid? Or could it be a portion of a cone or hyperboloid? One of the answer options is an ellipsoid, but I don't think that's right because when I graphed this on a computer, it showed a flat surface.

There are so many possibilities!

There aren't that many possibilities. You know all the points on the surface must be on the plane ##z=45x## and your computer shows the surface is flat. Don't those agree? If you project the ##xy## "shadow" in the ##xy## plane up onto the slanted plane, what do you get?

 

[ohlala191785]

So the description of the surface just an ellipse? If you project the circle onto the slanted plane, it should look like an ellipse.

 

[haruspex]

So the description of the surface just an ellipse? If you project the circle onto the slanted plane, it should look like an ellipse.

I suppose you could describe it as an elliptical disc.

 

[ohlala191785]

Oh OK.
Thank you for your help.