# the gradient of a scalar field is a covariant vector

The gradient of a scalar is a covector (1-form): its components change under coordinate transforms as $$\alpha^{\prime}\_a = \frac{\partial x^b}{\partial y^a}\alpha\_b$$. Despite the changing components, the underlying geometric gradient (its action on vectors / reconstructed physical vectors) is invariant - components change to preserve the same geometry.

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