In each case, find the potential function for the vector
field,
(or prove that no such potential function exists).
We are seeking a potential function
f (x, y, z)
such that
It is often helpful to start with the most complicated component,
in this case the x component.
f must include a term
Checking the other two partial derivatives,
which is completely consistent. Therefore the potential
function must be
which matches the components of F exactly.
Therefore the potential function is
We have an inconsistency, which suggests that no potential
function exists.
[g (y, z)
must be a function of y and z
only, not of x.]
Checking for the existence of a potential function:
which confirms that, for this vector field F,
no potential function exists. |
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and the potential is zero on all coordinate planes.
which matches the components of F exactly.
The potential function is
When any one or more of x, y, z is zero,
f = C.
The condition that the potential be zero on the coordinate
planes leads to
Therefore the potential function is