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$$(\boldsymbol{\nabla}\alpha)\wedge(\boldsymbol{\nabla} \wedge \boldsymbol{x} )$$

In all the examples in lecture, it has always been a $$\boldsymbol{\nabla}$$ on the left hand side. Does this give rise to a legitimate answer as I suppose you can replace the row of the partial derivatives with the components of another vector field in the determinant matrix?

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You have a cross product of two vectors: the gradient of a scalar field $\alpha$ and the curl of a vector field $x$. This is perfectly legitimate.

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