What is the slope of a line PARALLEL to the graph? A) -3 B) 0 C) 3 D) -1/3
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A three-dimensional vector field is conservative if it is also irrotational, i.e. its curl is 
. We have
so this vector field is not conservative.
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Another way of determining the same result: We want to find a scalar function
such that its gradient is equal to the given vector field, 
:
For this to happen, we need to satisfy
From the first equation, integrating with respect to
yields
Note that
*must* be a function of 
only.
Now differentiate with respect to
and we have
but this contradicts the assumption that
is independent of . So, the scalar potential function does not exist, and therefore the vector field is not conservative.
so this vector field is not conservative.
- - -
Another way of determining the same result: We want to find a scalar function
For this to happen, we need to satisfy
From the first equation, integrating with respect to
Note that
Now differentiate with respect to
but this contradicts the assumption that