Tensor equations / nonlinear transformations

[JustinLevy]

I have some fairly basic (hopefully) questions about tensor equations. Hopefully someone here can help out.

Let us say I have a tensor equation, (I will use this as the example for discussion: $$A^{u} = b C^{uv}D_{v}$$).

If this is true in one coordinate system, it will be true in all of them correct?

Now if I know the components of A,C,D in one frame (as well as the metric in this frame), then I can find the components of A,C,D in any other frame by transforming them, correct?

Can I get the metric in this new frame by transforming the metric like a normal tensor as well (it does not seem to work in general... maybe I am making mistakes)?

How do nonlinear transformations work? (For instance if I wanted to find A,C,D in an accelerated frame.) Since the transformation tensor just gives a linear transformation ... it seems to suggest that I need the metric to no longer be a constant in the frame!? Or does it work some other way?

 

[HallsofIvy]

If A is the tensor in x,y coordinates and A' is the same tensor in x',y' coordinates, then
$$A'= \frac{\partial x'}{\partial x}\frac{\partial y'}{\partial y}A$$
That works for linear or non-linear transformations. For a linear transformation the partial derivatives would be represented by a matrix having the coefficients as entries. For a non-linear transformation, the entries in the matrix will be functions of position.