Suposse $\nabla \cdot \vec{u} = 0$
Does that imply that $\Delta \vec{u} = \vec{0}$
Thank you!
2026-03-25 10:57:28.1774436248
If the gradient of a vector is zero, does that imply that the laplacian of the vector is a null vector?
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To show that it is not true (note that the Laplacian has to be taken component wise for this question to even make sense), take a look at the vector field $$\vec{u}(x,y,z)= \begin{pmatrix} y^2\\0\\0 \end{pmatrix}$$ and show that the divergence vanishes but the Laplacian does not.