Page 35 - Intro to Tensor Calculus

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I 26. ( Generalized Kronecker delta ) Define the generalized Kronecker delta as the n×n determinant

i i i
δ m δ n ··· δ
p
j
j δ n ··· δ j
δ
r
δ ij...k m p where δ is the Kronecker delta.
mn...p = . . . . s
. . . . . . . .

k k k
δ m δ n ··· δ p
123
(a) Show e ijk = δ
ijk
(b) Show e ijk = δ ijk
123
(c) Show δ ij ij
mn = e e mn
(d) Define δ rs = δ rsp (summation on p)
mn mnp
r
s
r s
and show δ rs = δ δ − δ δ
mn m n n m
Note that by combining the above result with the result from part (c)
r s
rs
r
s
we obtain the two dimensional form of the e − δ identity e e mn = δ δ − δ δ .
m n n m
1 rn
(e) Define δ r m = δ (summation on n)and show δ rst =2δ r p
pst
2 mn
(f) Show δ rst =3!
rst
1 1
a 1 a 2 a
1
2 2 3
i
r
I 27. Let A denote the cofactor of a in the determinant a 1 a 2 a 2 as given by equation (1.1.25).

r
i
3

a 1 a 2 a 3
3 3 3
r
j k
i
(a) Show e rst A = e ijk s t (b) Show e rst A = e ijk a a
a a
r
i
j k
s t
I 28. (a) Show that if A ijk = A jik , i, j, k =1, 2, 3 there is a total of 27 elements, but only 18 are distinct.
3
2
(b) Show that for i, j, k =1, 2,...,N there are N elements, but only N (N +1)/2 are distinct.
I 29. Let a ij = B i B j for i, j =1, 2, 3where B 1 ,B 2 ,B 3 are arbitrary constants. Calculate det(a ij )= |A|.
I 30.
(a) For A =(a ij ),i, j =1, 2, 3, show |A| = e ijk a i1 a j2 a k3 .
i j k
i
(b) For A =(a ),i, j =1, 2, 3, show |A| = e ijk a a a .
j
1 2 3
i
(c) For A =(a ),i, j =1, 2, 3, show |A| = e ijk 1 2 3
a a a .
i j k
j
i
(d) For I =(δ ),i, j =1, 2, 3, show |I| =1.
j
I 31. Let |A| = e ijk a i1 a j2 a k3 and define A im as the cofactor of a im . Show the determinant can be
expressed in any of the forms:
(a) |A| = A i1 a i1 where A i1 = e ijk a j2 a k3
(b) |A| = A j2 a j2 where A i2 = e jik a j1 a k3
(c) |A| = A k3 a k3 where A i3 = e jki a j1 a k2