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#?

2 Answers
Feb 22, 2018

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#.

Feb 22, 2018

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).

enter image source here