Let C be a curve, then we define a cylinder to be the set of
all lines through C and perpendicular to the plane that
C lies in.
Recall that the quadrics or conics are lines , hyperbolas,
parabolas, circles, and ellipses. In three dimensions, we can combine
any two of these and make a quadric surface. For example
Surface of Revolution
Let y = f(x) be a curve, then the equation of the surface
of revolution abut the x-axis is
Find the equation of the surface that is formed when the curve
y = sin x 0 < x < p/2
is revolved around the y-axis.
This uses a different formula since this time the curve is revolved around the y-axis. The circular cross section has radius sin-1 y and the circle is perpendicular to the y-axis. Hence the equation is
x2 + z2 = (sin-1 y)2
We can extend polar coordinates to three dimensions by
that the cylindrical coordinates are
r = r sin f
z = r cos f
From this we can find
We use spherical coordinates whenever the problem involves a distance from a source.
convert the surface
z = x2 + y2
to an equation in spherical coordinates.
We add z2 to both sides
z + z2 = x2 + y2 + z2
Now it is easier to convert
r cos f + r2 cos2 f = r2
Divide by r to get
cos f + r cos2 f = r
Now solve for r.