Homework 1
1. Express the vector x as a sum in index notation. Be sure to make use of e.
2. Express a  b in both vector and index notation. (In index notation, express a  b both in
short-hand (summation notation) and as a sum of 3 terms).
3. Express the inner product, P, of A and B, where both A and B are second-order tensors. (Hint:
see equations 2.9, 2.10, and 2.11 of Kundu, Cohen, and Ira, pg. 44).
4. Study figure (1.4) in Kundu, Cohe, and Ira (pg. 46). Then discuss the second-order stress
tensor, τ , particularly explaining why it is second order. Finally, what does the element  32
represent?
5. Write out the elements of the force vector f i in terms of the stress tensor τ and the unitnormal vector n j . (Hint: See equation 2.15, pg. 48. We will revisit this way of expressing force
when we return to Newton’s second law, F  ma . The f i you are working with in this problem
is the same F in Newton’s second law. Kundu, Cohen, and Ira choose to mix capital and lowercase letters, but they are the same force.)
3
6. Simplify the following:
 u
i 1
3
7. Simplify the following:

k 1
ij
j

.
ijk klm
.
8. Show, using index notation, why u  v  v  u .
9. Explain equation (2.21) in Kundu, Cohen, and Ira (pg. 52).