I've just started reading Arfken's book on mathematical methods for physics, and one of the very first sections is really confusing me. He is discussing the rotation of coordinates, and defining a vector as an object whose components transform in the same way as the coordinates do under a rotation. All of the "vectors" he talks about are things like (-y,x) which I recognize as vector fields, and he even notes that they are often referred to as vector fields. This definition sort of makes sense, but the examples just confused me more. He ends up showing that, under this definition, (-y,x) is a "vector", but (x,-y) is not. I just can't wrap my head around this. Obviously I can draw both of these fields, and rotating the coordinates doesn't seem to have any adverse affects on either one. It seems like he's just defining the former as a vector because it's symmetric. As in, after the rotation, the vectors assigned to each point all satisfy (-y',x'), because it was radially symmetrical to begin with. Whereas the latter field hasn't changed, but the component functions aren't the same in the primed coordinates as in the original coordinates. At least, that's how I'm following it. Let me know if that's way off. I feel like I'm missing something, because that seems like an awful selective way to define a vector field. Surely there are plenty of interesting fields in physics that aren't don't behave like that? I mean, it's not like the field is changing in either case, it's just that the first happens to be symmetric so that it's indistinguishable under rotations. I always thought a vector was something that maintained it's direction relative to everything else no matter the frame of reference? Both of these fields do that, right? I don't know, I'm just not really sure why this is a useful way to define a vector (field?), or what the big idea is behind it.