I guess no, since every vector is not the gradient of a scalar - I assume ! Please confirm or guide in this regard .
2026-03-26 10:57:52.1774522672
Is every tensor the gradient of a vector?
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Consider the second order tensor $\boldsymbol{T}$ whose matrix representation under the Cartesian basis $\{\boldsymbol{e}_{1},\boldsymbol{e}_{2}\}$ is $$\left[\boldsymbol{T}\right]=\left(\begin{array}{cc} y & 0\\ 0 & 0\end{array}\right).$$ Suppose $\boldsymbol{T}(x,y) = \nabla\boldsymbol{u}(x,y)$ for some vector field $\boldsymbol{u}(x,y)$. Do you see the issue?