Topic 1: Index Notation

Question 1

Express d=ABc in index notation.

Answer 1

di = A_ik B_kj c_j

Question 2

Express B=AAAA in index notation

Answer 2

Bij=A_il A_lk A_kp A_pj

Question 3

Index equation for transformation of coordinates for a Rank 1 tensor.

Answer 3

t’_i = R_ia t_a

Question 4

Index equation for transformation of coordinates for a Rank 2 tensor.

Answer 4

C’_ij = RiaRjbCab

Question 5

Index equation for transformation of coordinates for a Rank 3 tensor.

Answer 5

E’_ijk = RiaRjbRkcEabc

Question 6

Index equation for transformation of coordinates for a Rank 4 tensor.

Answer 6

S’_ijkl = RiaRjbRkcRldSabcd

Question 7

Kronecker Delta

Answer 7

delta_ij:
1 if i=j
0 if i no= j

This is also the identity matrix.

Question 8

Permutation Tensor (epsilon)

Answer 8

epsilon_ijk:
1 if ijk form positive permutation
-1 if ijk form negative permutation
0 otherwise

Question 9

3 possibilities for positive permutation

Answer 9

123, 231, 312

Question 10

3 possibilities for negative permutation

Answer 10

321, 213, 132

Question 11

Index notation for cross product of C = A x B

Answer 11

Ci = eps_ijk Aj Bk

Question 12

Index notation for outer product: W = t outercross u

Answer 12

Wij = ti uj

Question 13

What does skew symmetric mean?

Answer 13

All of the elements in the diagonal are equal to zero, and the remaining elements are equal to the negative on the elements of the other side of the diagonal.

Question 14

If A is a skew symmetric matrix, what is transpose(A) equal to?

Answer 14

-A

Question 15

if A is a symmetric matrix, what is transpose(A) equal to?

Answer 15

A

Question 16

epsilon - delta identity

Answer 16

e_miqe_jkq = d_mjd_ik - d_mk*d_ij

Question 17

delta_ij*vj = ?

Answer 17

vi

Question 18

e_jkq*e_mkq = ?

Answer 18

2d_jm

Question 19

det(A)

Answer 19

det(A) = (1/6)e_ijke_qrs*AiqAjrAks

Question 20

Alternate form of det(A)

Answer 20

det(A) = e_ijk*Ai1Aj2Ak3

Question 21

What is det(A)*e_pqr equal to?

Answer 21

e_ijk*AipAjqAkr