Since the question asks for the force due to hydrostatic pressure, and atmospheric pressure is in
the hydrostatic pressure, I think it is best to include it in this result.
If the question asked for “the net
force,” then the pressure inside the dome would have to be included.
If the pressure inside the dome
were maintained at atmospheric, which would be a reasonable assumption if people are inside, then that
force would cancel out the atmospheric term in the result above, and only the weight of the water would
remain.
We will accept either answer, since we are teaching engineers, not law students.
In general, if a
problem asks for the “net force” and atmospheric pressure acts on both sides of a surface, then changing
the atmospheric pressure makes no difference, and it should not appear in the answer.
3.
Show by writing out the sums that
δ
ij
∂
v
j
∂
x
i
=
∂
v
i
∂
x
i
.
Express this term in Gibbs notation, using the gradient operator.
1

4.
Show that, for any function
φ
∇ × ∇φ
=
0
.
,

but the order of differentiation does not affect the final result, so
e
k
ε
ijk
∂
∂
x
i
∂
∂
x
j
φ
k
∑
j
∑
i
∑
=
−
e
k
ε
jik
∂
∂
x
j
∂
∂
x
i
φ
k
∑
j
∑
i
∑
.
Since i and j are just dummy indices, they can be changed to any index.