I learned gradient in 3D space. And gradients where always vectors, pointing in the direction of steepest ... and normal to the surface where the functions is constant. But reading one-forms , a gradient of a function is not always a vector and it has something to do with metric... Can you proof this mathematically? Or an example which disproves that a gradient is a vector? Or visualise it? If it is not a vector a change of coordinate base is does not change the coordinates in the "correct way"?