examples of lamellar field
Example 1. Given
For the rotor (http://planetmath.org/NablaNabla) (curl) of the we obtain
which is identically for all , , . Thus, by the definition given in the parent (http://planetmath.org/LaminarField) entry, is lamellar.
Since , the scalar potential must satisfy the conditions
Thus we can write
where may depend on or . Differentiating this result with respect to and comparing to the second condition, we get
where may depend on . So
Differentiating this result with respect to and comparing to the third condition yields
This means that is an arbitrary . Thus the form
expresses the required potential function.
Example 2. This is a particular case in :
Now, , and so is lamellar.
Therefore there exists a potential with . We deduce successively:
Thus we get the result
which corresponds to a particular case in .
Example 3. Given
The rotor is now From we obtain
Differentiating (1) and (2) with respect to and using (3) give
We substitute and again into (1) and (2) and deduce as follows:
putting , into (1), (2) then gives us
whence, by comparing, , so that by (3), the expression and itself have been found, that is,
Example 4. An additional example of a lamellar field would be
with a differentiable function ; if is a constant, then is also solenoidal.
|Title||examples of lamellar field|
|Date of creation||2013-03-22 17:39:25|
|Last modified on||2013-03-22 17:39:25|
|Last modified by||pahio (2872)|
|Synonym||example of scalar potential|
|Synonym||determining the scalar potential|