On the left, change i to s and j
to n, and on the right change i to n and j to s, then
e
k
ε
snk
∂
∂
x
s
∂
∂
x
n
φ
k
∑
n
∑
s
∑
=
−
e
k
ε
snk
∂
∂
x
s
∂
∂
x
n
φ
k
∑
s
∑
n
∑
The order of summation does not matter, so
e
k
ε
snk
∂
∂
x
s
∂
∂
x
n
φ
k
∑
n
∑
s
∑
=
0
which shows that
∇ × ∇φ
=
0
.
5.
Show that a tensor plus its transpose,
A
+
A
t
,
is always a symmetric tensor.

6.
Is the velocity field v
that of an incompressible fluid if
v
=
xe
x
+
ze
y
−
ye
z
?

so the flow is compressible.
Ungraded exercise
Write the following expressions, which are given in Gibbs notation, using summation notation.
Include
the unit vectors e
.
Label each as a scalar, vector or tensor.
Gibbs notation
B
⋅
a
A
⋅
B
A
:B
a
×
b
∇⋅
v
v
(
)
Solution
Write the following expressions, which are given in Gibbs notation, using summation notation.
Include
the unit vectors.
Label each expression as a scalar, vector or tensor.
Gibbs notation
Summation notation
B
⋅
a
B
ij
a
j
e
i
vector
A
⋅
B
A
ij
B
jk
e
i
e
k
tensor

a
×
b
ε
ijk
a
i
b
j
e
k
vector
∇⋅
v
v
(
)
∂
∂
x
i
v
i
v
j
(
)
e
j
vector