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Flowing through all space is an electric field E(x, y, z) = αyz(x hat) + αxz(y...

Flowing through all space is an electric field E(x, y, z) = αyz(x hat) + αxz(y hat) + αxy(z hat). Show that the curl of the electric field vanishes, ∇ × E = 0. Use the definition of electric potential to find the potential difference between the origin and r = x(x hat) + y(y hat) + z(z hat), V (r) − V (0) = −(integral from 0 to r of (E · dl)). As the line integral is independent of path, choose whatever path you find to be convenient. Taking V(0) = 0, what is V(r)? Finally, confirm that taking the gradient of the potential recovers our original electric field, E = −∇V.

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