Now we seek to expand the range of usefulness of Bernoulli's equation by
seeking another set of conditions in which it applies. We now have the
advantage of knowing the result that we seek. The derivation is lengthy, so the
result is presented at the start, lest the anxious anticipation become
overwhelming: Bernoulli's equation applies to a fluid as long as for the
velocity field
;
that is, the curl of the velocity is zero.
We start again with the time-independent Euler's equation, with no "other" forces:
We used the streamline integration previously in order to make the integral of the left side easy. This time we ask if there is another condition that causes the left-hand side to integrate to
.
This will (almost) happen if the
term can be rewritten as
Note that the term 
is a
vector, even though the 
part is
a scalar. This condition is not quite enough, because it still has partial
derivatives to integrate; a problem we postpone. We cannot, however postpone a
minor paroxysm of algebra.
Let us consider the x component of
. We want it to take
the form
If we write out the x component of
, we have
Only one term has the correct form:
We can add (and then subtract) terms to push this towards the form we want:
Rearranging, we have
or
.
since 
is what we want on the
right side, our condition for the integration must be that
.
Return to index.
Return to Euler's equation applications survey.
Go forward to zero-curl denouement of derivation of Bernoulli's equation.