Vectors Calaulus 1.4

Surfaces Cylinders 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.           We can tell that an equation is a cylinder is it is missing one of the variables.   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         is a paraboloid since for constant z we get a circle and for constant x or y we get a parabola.  We use the suffix -oid to mean ellipse or circle.  We have:            x2          y2          z2              +           +            =  1 is an ellipsoid           a2          b2          c2                  x2         y2         z2     -          -           +            =  1 is a hyperboloid of 2 sheets while             a2         b2          c2               x2          y2          z2              +           -            =  1 is a hyperboloid of 1 sheet           a2          b2          c2                 Surface of Revolution Let y = f(x) be a curve, then the equation of the surface of revolution  abut the x-axis is         y2 + z2 = f(x)2 Example 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      x = rcosq      y = rsinq      z = z Example We can write (1,1,3) in cylindrical coordinates.  We find          and           so that the cylindrical coordinates are         (, p/4, 3) Spherical Coordinates An alternate coordinate system works on a distance and two angle method called spherical coordinates.  We let r denote the distance from the point to the origin, q represent the same q as in cylindrical coordinates, and f denote the angle from the positive z-axis to the point.  The picture tells us that         r  =  r sin f   and that          z  =  r cos f   From this we can find      x = rcosq = r sin f cosq      y = r sinq = r sin f sinq      z = r cos f Immediately we see that         x2 + y2 + z2 = r2 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           r =                            =                     =  csc f  cot f                      1 - cos2 f                sin2 f