Math/Phys 266E
Homework #2
Assigned: Tu Jan. 30 Due: Th Feb. 22
1. Prove each of the following in direct notation and indicate whether each is a tensor, a
vector, or a scalar.
a). () Tensor, vector, scalar ? (circle one)
+โ=โ+โab c a cb c
b). Tensor, vector, scalar ? (circle one)
()โ+=โ+โabcabac
โฌ
:โT๎๎
u
c). () Tensor, vector, scalar ? (circle one)
()โ=โabv avb
d). () Tensor, vector, scalar ? (circle one)
()()()โโ= โabcd bcadi
e). () Tensor, vector, scalar ? (circle one)
( )
()
iij j
ii
if i j
if i j
โ
โงโซ
โโ=
โจ
โ=
โฉโญ
0
eee e ee
f). Tensor, vector, scalar ? (circle one)
()()โ= โTa b Ta b
g). ( Tensor, vector, scalar ? (circle one)
)
ii
โ=Te e T
h). Tensor, vector, scalar ? (circle one)
( ) _______?tr =I
Fill in the blank and prove.
2. Is the tensor product of two vectors a commutative product? Justify your answer
without appealing to components.
3. For each of the following, the action ofTis as defined below where , is
an arbitrary vector, is a fixed, given vector, and let k is an arbitrary scalar. Determine
whether is a tensor in each case below.
33
n
T
(a)
=Tu n
(b)
k=Tu u
(c)
()=Tu u u ui