Standard six-sided dice have their dots arranged so that the opposite faces add up to 7.
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is conservative if we can find a scalar function such that
. This would require
Integrating both sides of the first equation wrt gives
Differentiating both sides of this wrt gives
but we assumed was a function of
and
, independent of
. So there is no such
and
is not conservative.
To find the work, first parameterize the path (call it ) by
for . Then
and the work is given by the line integral,