In order to obtain Bernoulli's equation, one needs to integrate the above
equation along a streamline. Hence, take the dot product of Euler's equation
with a differential length ds along the streamline as
follows:
Recognize that the differential length ds can be expressed in terms of
the Cartesian coordinates as
ds = dx i + dy j + dz k
Each term on the left hand side of Euler's equation can be evaluated as
follows:
Since streamlines are constructed such that they are tangent to the velocity
field, the differential length ds is parallel to the velocity V.
The vector is
then perpendicular to V, which gives
Euler's equation thus becomes
which can be readily integrated to yield
For constant density, the above equation reduces to the famous Bernoulli's equation
