Equation:  Chapter .. 1 




The .. principal .. curvat ures  . 





 1   .   1  .. Volume .. of a thickened .. hypersurface

 
We want to consider the following problem  :  Let  Y   subset   R  to the power of  n  be an oriented hyper  hyphen 





surface, so there is a well defined unit normal vector,  nu   open parenthesis   y   closing parenthesis   ,  at each point of  Y   . 





Let  Y  sub  h  denote the set of all points of the form




 y   plus   t   nu   open parenthesis   y   closing parenthesis   ,   0   less or equal    t   less or equal    h.  




We wish to compute  V  sub  n   open parenthesis   Y  sub  h   closing parenthesis  where  V  sub  n  denotes the  n   minus  dimensional volume. We





will do this computation for small  h   ,  see the discussion after the examples.





Examples in three dimensional space.





 1   .  Suppose that  Y  is a bounded region in a plane, of area  A   .  Clearly




 V  sub  3   open parenthesis   Y  sub  h   closing parenthesis   =   hA 




in this case.





 2   .  .. Suppose that  Y  is a right circular cylinder of radius  r  and height  l  with

 
outwardly pointing normal. .. Then  Y  sub  h  is the region between the right circular





cylinders of height  l  and radii  r  and  r   plus   h  so




Line 1  V  sub  3   open parenthesis   Y  sub  h   closing parenthesis   =   pi   l   open square bracket   open parenthesis   r   plus   h   closing parenthesis  to the power of  2   minus   r  to the power of  2   closing square bracket  Line 2  =   2   pi   l   rh   plus   pi   l   h  to the power of  2  Line 3  =   hA   plus   h  to the power of  2   times   1  divided by  2   r   times   A  Line 4  =   A   parenleftbigg   h   plus   1  divided by  2   times   kh  to the power of  2   parenrightbigg   , 




where  A   =   2   pi   r   l  is the area of the cylinder and where  k   =   1   slash   r  is the curvature of

 
the generating circle of the cylinder. For small  h   ,  this formula is correct, in fact,




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