EQUATIONS OF ELASTIC HYPERSURFACES

2. OUTLINE OF DIFFERENTIAL GEOMETRY 27
this will replace all vectors u 2 in the determinant by the vectors v 2 without altering the determinant.
We continue the procedure, based on the formulae (2.84) and replace successively
all u 1 , . . . , u k by v 1 , . . . , v k without altering the determinant. Since v 1 , . . . , v k are orthogonal
(see (2.85)), the matrix is diagonalized and maintains the value.
Remark 2.28 If u 1 , . . . , u k are linearly independent, the system
is even orthonormal
w j := 1
|v j | v0 j j = 1, . . . , k , (2.90)
〈w j , w m 〉 = δ jk , j, m = 1, . . . , k , (2.91)
where δ jk = 0 for k ≠ j and δ kk = 1 is the Kroneker symbol.
Let
P(u 1 , . . . , u k ) =
{
}
α 1 u 1 + . . . + α k u k : 0 ≤ α j ≤ 1 , j = 1, . . . , k
denote the hyper-parallelepiped in R n spanned by the vectors u 1 , . . . , u k ∈ R n .
Lemma 2.29 Let u 1 , . . . , u n−1 ∈ R n . The Volume Vol P(u 1 , . . . , u n−1 ) of the hyper-parallelepiped
coincides with the length of the vector product
(cf. Lemma 2.25.h).
Proof: Clearly,
Vol P(u 1 , . . . , u n−1 ) = |u 1 ∧ . . . ∧ u n−1 | = √ G (u 1 , . . . , u n−1 ) (2.92)
P(u 1 ) ⊂ P(u 1 , u 2 ) ⊂ · · · ⊂ P(u 1 , . . . , u n−1 )
and Vol P(u 1 ) = |u 1 |, Vol P(u 1 , u 2 ) = |u 1 |h 2 , where h 2 = |u 2 | cos θ 2 is the projection of
the vector u 2 on the axes orthogonal to u 1 and θ 2 is the angle between the vector u 2 and this
orthogonal axes. By proceeding further we find that
Vol P(u 1 , . . . , u n−1 ) = Vol P(u 1 , . . . , u n−2 )h n−1 = |u 1 |h 2 . . . h n−1 , (2.93)
where h k = |u k | cos θ k = |v k | is the altitude of the hyper-parallelepiped P(u 1 , . . . , u k )
with the base P(u 1 , . . . , u k−1 ) and θ k is the angle between u k and v k ((see Fig. 1; due to
(2.87) the vector v k is orthogonal to the frame P(u 1 , . . . , u k−1 ) ⊂ L (u 1 , . . . , u k−1 ) for all
k = 1, . . . , n − 1). Then
|u 1 | = |v 1 |, h 2 2 = 〈u 2 , v 2 ) = |v 2 | 2 , . . . , h 2 n−1 = 〈u n−1 , v n−1 〉 = |v n−1 | 2
(cf. (2.84), (2.87)) and we get the final result
(cf. (2.86), (2.93)).
Vol P(u 1 , . . . , u n−1 ) = |v 1 | |v 2 | . . . |v n−1 | = √ G (u 1 , . . . , u n−1 )