Consider the vector field $$X=(y-10x)\partial_x-x\partial_y$$ it is not a gradient vector field with respect to the standard Riemannian metric of $\mathbb{R}^2$ but it is a gradient vector field with respect to the Riemanni8an metric $$g=5dx\otimes dx-dx\otimes dy -dy\otimes dx +5dy\otimes dy$$
Ali Taghavi
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