regular smooth curve - Penn Math
regular smooth curve - Penn Math
regular smooth curve - Penn Math
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The "tube" of radius � about a round sphere of radius r in<br />
3-space is just the region between the concentric spheres of<br />
radii r + � and r � � , with volume<br />
4/3 � (r + �) 3 � 4/3 � (r � �) 3 = 4/3 � (6 r 2 � + 2 � 3 )<br />
= (4 � r 2 ) 2� + 8/3 � � 3 = (area of sphere) 2� + 8/3 � � 3<br />
= 2� (area of sphere + 2�/3 �(sphere) � 2 ) ,<br />
which is exactly "Weyl's tube formula" for surfaces in R 3 .<br />
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