In the examples that follow, show that the given vector field
is lamellar everywhere in
and determine its scalar potential
.
Example 1. Given
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Thus we can write
where
Accordingly,
where
Differentiating this result with respect to
This means that
expresses the required potential function.
Example 2. This is a particular case in
:
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Therefore there exists a potential field
with
. We deduce successively:
Thus we get the result
which corresponds to a particular case in
Example 3. Given
![]() |
Differentiating (1) and (2) with respect to
We substitute
putting
whence, by comparing,
Unlike Example 1, the last two examples are also solenoidal, i.e.
, which physically may be interpreted as the continuity equation
of an incompressible fluid flow.
Example 4. An additional example of a lamellar field would be
with a differentiable function
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