Question: Suppose a vector field is defined by \(\mathbf{F}(x, y, z)=\left\langle y^{2} z, 2 x y z, x y^{2}+4 z\right\rangle\)

1. Determine whether there is a scalar function \(P(x, y, z)\) defined everywhere in space such that \(\nabla P=\mathbf{F}\). If there is such a \(P\), find it; if there is not explain why not.
2. Compute the integral \(\int_{W} \mathbf{F} \cdot \mathbf{T} d s\), where \(W\) is the circular helix whose position vector is given by \(\mathbf{R}(t)=\langle\cos t, \sin t, t\rangle\) for \(0 \leq t \leq 2 \pi\)
3. Compute the curvature of \(\mathbf{R}(t)\) and that of \(\mathbf{R}(2 t)\).

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