Question

Find a vector function, `r(t)`, that represents the curve of intersection of the two surfaces. The cylinder `x^2 + y^2 = 81` and the surface `z = xy`?

Answer 1

The curve of intersection may be parametrized as `(z,r) = ((81/2) sin2\theta, 9)`.

Explanation:

I am not sure what you mean by vector function. But I understand it that you seek to represent the curve of intersection between the two surfaces in the question statement.

Since the cylinder is symmetric around the `z` axis, it may be easier to express the curve in cylindrical coordinates.

Change to cylindrical coordinates:
`x = r cos\theta`
`y = r sin\theta`
`z = z`.
`r` is the distance from the `z` axis and `\theta` is the counter-clockwise angle from the `x` axis in the `x,y` plane.

Then the first surface becomes
`x^2 + y^2 = 81`
`r^2cos^2\theta + r^2sin^2\theta = 81`
`r^2=81`
`r=9`,
because of the Pythagorean trigonometric identity.

The second surface becomes
`z = xy`
`z = rcos\theta rsin\theta`
`z= r^2sin\theta\cos\theta`.
We learned from the equation of the first surface that the intersecting curve must be at a squared distance `r^2=81` from the first surface, giving that
`z = 81 sin\theta cos \theta`,
`z = (81/2) sin2\theta`,
a curve parametrized by `\theta`. The last step is a trigonometric identity and is done just from personal preference.

From this expression we see that the curve is indeed a curve, as it has one degree of freedom.

All, in all, we can write the curve as
`(z,r) = ((81/2) sin2\theta, 9)`,
which is a vector valued function of a single variable `\theta`.

Answer 2

See below.

Explanation:

Considering the intersection of

`C_1->{(x^2+y^2=r^2),(z in RR):}`

with

`C_2-> z = x y`

or `C_1 nn C_2`

we have

`{(x^2+y^2=r^2),(x^2y^2= z^2):}`

now solving for `x^2,y^2` we obtain the parametric curves

`{(x^2=1/2(r^2-sqrt(r^2-4 z^2))),(y^2=1/2(r^2+sqrt(r^2-4 z^2))):}` or

`{(x = pm sqrt(1/2(r^2-sqrt(r^2-4 z^2)))),(y= pm sqrt(1/2(r^2+sqrt(r^2-4 z^2)))):}`

which are real for

`r^2-4 z^2 ge 0 rArr z lepm( r/2)^2`

Attached a plot showing the intersection curve in red (one leaf).