In Example 4, we showed that div F = z + xz and therefore div F≠ 0.can’t be written as the curl of another vector field, that is, F≠ curl G.Show that the vector field F(x, y, z) = xzi + xyzj – y2k.Note the analogy with the scalar triple.The terms cancel in pairs by Clairaut’s Theorem.By the definitions of divergence and curl,.If F = Pi + Qj + Rk is a vector field on and P, Q, and R have continuous second-order partial derivatives, then div curl F = 0.The next theorem shows that the result is 0.As such, we can compute its divergence.If F is a vector field on, then curl F is also a vector field on.By the definition of divergence (Equation 9 or 10) we have:.If F(x, y, z) = xzi + xyzj – y2k find div F.the divergence of F can be written symbolically as the dot product of and F:.If F = Pi + Qj + Rk is a vector field on and ∂P/∂x, ∂Q/∂y, and ∂R/∂zexist, the divergence of F is the function of three variables defined by:.We give a more detailed explanation in Section 16.8 as a consequence of Stokes’ Theorem.
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