Page 36 - Intro to Tensor Calculus

32

I 32. Show the results in problem 31 can be written in the forms:

1 1 1 1
A i1 = e 1st e ijk a js a kt , A i2 = e 2st e ijk a js a kt , A i3 = e 3st e ijk a js a kt , or A im = e mst e ijk a js a kt
2! 2! 2! 2!

I 33. Use the results in problems 31 and 32 to prove that a pm A im = |A|δ ip .

 
1 2 1
I 34. Let (a ij )=  1 0 3  and calculate C = a ij a ij ,i, j =1, 2, 3.
2 3 2

I 35. Let
a 111 = −1, a 112 =3, a 121 =4, a 122 =2

a 211 =1, a 212 =5, a 221 =2, a 222 = −2
and calculate the quantity C = a ijk a ijk , i,j,k =1, 2.

I 36. Let
a 1111 =2, a 1112 =1, a 1121 =3, a 1122 =1

a 1211 =5, a 1212 = −2, a 1221 =4, a 1222 = −2
a 2111 =1, a 2112 =0, a 2121 = −2, a 2122 = −1

a 2211 = −2, a 2212 =1, a 2221 =2, a 2222 =2
and calculate the quantity C = a ijkl a ijkl , i,j,k,l =1, 2.

I 37. Simplify the expressions:

∂x i
(c)
(a)(A ijkl + A jkli + A klij + A lijk )x i x j x k x l j
∂x
2 i
2 m
i j k
(b)(P ijk + P jki + P kij )x x x ∂ x ∂x j ∂ x ∂x i
(d) a ij t s r − a mi s t r
∂x ∂x ∂x ∂x ∂x ∂x
I 38. Let g denote the determinant of the matrix having the components g ij ,i, j =1, 2, 3. Show that

g 1r g 1s g 1t g ir g is g it

(a) ge rst = g 2r g 2s g 2t (b) ge rst e ijk = g jr g js g jt

g 3r g 3s g 3t g kr g ks g kt
i i i
δ δ
p
m n δ

j
I 39. Show that e ijk e mnp = δ ijk = δ δ j δ j
mnp m n p
k δ k δ k
δ
m n p
I 40. Show that e ijk e mnp A mnp = A ijk − A ikj + A kij − A jik + A jki − A kji
Hint: Use the results from problem 39.
I 41. Show that
ij
(a) e e ij =2! (c) e ijkl e ijkl =4!
(b) e ijk e ijk =3! (d) Guess at the result e i 1 i 2 ...i n e i 1 i 2 ...i n