Determine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y, z) = ye−xi + e−xj + 2zk

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.