Parameterization of hyperbola intersecting cone

[jake2009]

Hello. I am having some trouble with the following problem and would be thankful if any of you could help me out.

Homework Statement

Let C be the hyperbola formed by intersecting the cone

$$x^2+y^2=z^2$$, $$z>0$$

with the plane $$x+y+z=1$$, and let

$$\textbf{f}(x,y,z)=<0,0,1/z^2>$$.

I am trying to calculate $$\int_C \textbf{f} \wedge d\textbf{r}$$.

Homework Equations

Here $$\wedge$$ is the cross product in 3-space.

The Attempt at a Solution

I am able to solve the same sort of problem for the cylinder
$$x^2+y^2=1$$
and the plane
$$z=2y+1$$.
In which case we let $$x = \cos t, y = \sin t$$ and $$z = 2\sin t +1$$ yeilding

$$r(t) = <\cos t,\sin t,2\sin t +1>$$

from which the result follows from integration.

I attempted to do something simular for the problem I am trying to figure out. Using $$r(t,z) = <z \cos t, z\sin t, z>$$ to represent points on the cone, I tried to move forward, but did not find a good way to combine this with the equation for the plane as I did in the above example for the cylinder. Perhaps it is because I have z in r(t,z)? Following this line of attack I can comebine this with the equation for the plane z = 1-x-y = 1-z cos t - sin t, solving for z and combining with r(t,z) yeilds r(t) as

$$r(t) = (1+\cos t+\sin t)^{-1}<\cos t,\sin t,1>$$

but then I end up with horrible singularities!

Would be most greatful if you would let me know how to solve this one!

 

[Dick]

You meant to post x^2+y^2=z^2. x^2+y^2+z^2=z^2 is a line. The problem is that the hyperbola is unbounded and sin(t) and cos(t) are bounded functions. Try using their unbounded hyperbolic cousins, sinh(t) and cosh(t), in a similar way.

 

[jake2009]

Thanks, I fixed the equation you mention. I am not sure how I would use hyperbolic functions to represent the cone in the manor done with r(t,z) = <z cos t, z sin t, z>. I guess a hyperbola can be expressed as (+/- a cosh(u),b sinh(u)) but I am not sure how that helps me. Arg, this is confusing.

 

[jake2009]

Perhaps what I did is actually correct, privided I do not integrate past the singularities? This should not be so hard, but it is.

 

[Cyosis]

Not entirely sure if this helps you but, the equation for a circle $x^2+y^2=r^2$ is parametrized by $x=r \cos \theta, y =r \sin \theta$, which you of course already know. Similarly the equation for a hyperbola x^2-y^2=r^2 is parametrized by $x=r \cosh \theta, y=r \sinh \theta$, note that $\cosh^2 \theta-\sinh^2 \theta=1$.

 

[Dick]

Thanks, I fixed the equation you mention. I am not sure how I would use hyperbolic functions to represent the cone in the manor done with r(t,z) = <z cos t, z sin t, z>. I guess a hyperbola can be expressed as (+/- a cosh(u),b sinh(u)) but I am not sure how that helps me. Arg, this is confusing.

Your posted quadratic is now a sphere. I still think you want x^2+y^2=z^2. If so try z=x*cosh(t) and y=x*sinh(t). That satisfies x^2+y^2=z^2 for any choice of x. Now substitute into the plane equation to find x in terms of t. It's really almost exactly the same thing you did with the cylinder example.