We encounter expressions for vectors and tensors in Euclidean space, such as
$$\vec{F}=\vec{A}+\nabla\phi,$$ or
$$\vec{H} = \nabla\vec{u}\cdot\vec{n}+\nabla\times(\nabla\times\vec{B}) + \frac{\partial \vec{B}}{\partial t},$$ or
$$\nabla\cdot\vec{E}=\vec{C}\cdot\hat\Phi\cdot[(A\cdot\nabla)\vec{B}]-\vec{A}\cdot\vec{B}.$$
and other similar expressions, which include in themselves vectors, tensors and differential operators. Then we are told that these expressions "are independent of coordinates". Does this mean these expressions are the same in ANY coordinate system in Euclidean space? What about non-orthogonal coordinates? Curvilinear coordinates?
Can someone explain why are they the same. If changing the basis, what does one need to know to arrive at the "same" expression in the new coordinates?
If one writes the expression in the new coordinates, and it is not the same, what went wrong?