ABSTRACT
Volume Using the metric coefficients defined in (5.2), we define 9 = 911922933·
we have the following definitions:
origin.
[0, 211']. The length of this curve is
L =12ft Va2 sin2 t + a2c082 t + b2 dt = 12• ../a2 + b2 dt = 21r../a2 + b2. Example 2
Consider a torus defined by x = (( b + a sin </>) cos 9, ( b + a sin </>) sin 9, acos</>), where 0 ~ 9 ~ 211' and 0 ~ </> ~ 211'. From (5.1), we can compute E = xe · xe = (b + asin</>)2, F = xe · X4J = 0, and G = X4J • x.q, = a2• Therefore, the surface area of the torus is
In cylindrical coordinates we have { Xt = r cos</>, x2 = r sin</>, X3 = z} so that {hr = 1, h9 = r, h. = 1}. Consider a cylinder of radius R and height H. This cylinder has three possible areas we can determine:
5. Geometric Applications 27
We can identify each of these: S9z is the area of the outside of the cylinder, S9r is the area of an end of the cylinder, and Sn is the area of a radial slice (that is, a vertical cross-section from the center of the cylinder).
