Surfaces 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.
Quadric Surfaces 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. Solution 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
Cylindrical Coordinates We can extend polar coordinates to three dimensions by
and
so
that the cylindrical coordinates are Spherical Coordinates r = r sin f and that z = r cos f From this we can find
We use spherical coordinates whenever the problem involves a distance from a source.
Example convert the surface z = x2 + y2 to an equation in spherical coordinates. Solution 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.
cos f
cos f
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