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We want to prove that for a given the expression

(2) |

First,
we prove that expression (A-1) vanishes for
a volume of parallel, planes.
Applying the chain rule to the first component
yields
(*p*_{y} *g*' *p*_{z} - *p*_{z} *g*' *p*_{y}) = 0.
An analogue result
for the two additional components
of expression (A-1)
demonstrates
that the entire expression
vanishes
for a volume
that consists of parallel planes.

Next, we prove that if all three components of expression (A-1) vanish, than the function is a volume of parallel planes. We set the expression (A-1) to zero and express it as a vector product of and the gradient of the field, :

(3) |

The vector product of and is zero if and
only if one of the vectors is the null vector,
or if the vectors are parallel.
Ignoring the trivial cases, we conclude
that when the outputs of the three finite difference filters vanish
everywhere, than the gradient of *g* is parallel to the constant
vector . The constant is normal to *g* at
all locations and
therefore *g* has to be a volume of parallel planes.

11/12/1997