I have edited my comment in response to your edit, but I guess it all works out now. :D Thanks for clarifying

OK, but where do these 'v'-segments come in, and what you meant by "these crazy shapes"? Can you expand a bit on that? You specifically referred to the picture posted - this is why I was wondering if you're misinterpreting the image.

Sorry, but can you elaborate a bit further - I don't quite understand where you see these v-s (what you mean by that)? Are you interpreting the image as 2D? To me, this looks like a 3D picture, with 3 faces of a cube joined at one corner, and the yellow contour traces out the "open" edges where the rest of the cube is missing. The inset on the right is the same thing, from a different perspective - looking at the opening, towards the point M.

@Lorenzo - they are not at the same temperature. The piston chamber is not in thermal contact with both reservoirs at all times, but alternates (or I guess, ping-pongs) between being in contact with one, none, or the other. The gas starts out hot, flowed by isothermal expansion, but after that phase, it continues to expand, without being in thermal contact with either reservoir, and the temperature drops to nearly the temperature of the cold reservoir. (I suppose Chet Miller's reply got auto corrected or something).

A related interesting thing is that objects that we can see can have features that are invisible to us, but are revealed by cameras/sensors that can record frequencies outside of the visible range (like ultraviolet (UV) or infrared (IR), and beyond). Some examples are layering in some sedimentary rocks that otherwise look uniform (no visible layers), coloration of some flowers that look plain to us but appear vibrant/multicolored to insects, heat "glowing" from the human body in infrared, different surface layers of the Sun revealed by observing it in various frequencies, etc.

It's OK to ask here, but you should be comfortable asking the professor for clarification - that's the whole point of having a professor (as opposed to, say, studying on your own from a book).

"But obviously, if there is not magnetic field, there will not be any momentum." - but, there always is a magnetic field, otherwise you don't have any light - it's what light is made of. Self propagating self-inducing electric and magnetic fields. It's an electromagnetic wave.

Please note that (even under idealized conditions when there's no wind or air drag), this is only true if the boat/train is not accelerating (uniform motion is required - constant speed in a straight line).

The presentation in this book is a pedagogical disaster (at least when it comes to this derivation).

The assumption you make is that the Earth will rotate under flying/hovering objects. But consider this: depending on where you are on the planet, your sideways velocity due to rotation is something like 1000 km/h (or some 600 mph). As you must have noticed, if you throw a ball in the air, it doesn't fly off westward at incredible speed. It's not that different when it comes to planes.

Just wanted to add - you wrote "the most natural worldline is the one parametrized by proper time": note that parametrizations are just a mathematical way to describe a curve; the worldline itself is independent of that and, for a given object, is one and the same for all observers. It's just that relative velocity causes what each observer considers to be "now" (spacelike simultaneous hyperplane) to be tilted in spacetime (a kind of "rotation" in the 4D spacetime), so they observe the same object at different points along its worldline (as characterized by the Lorentz transformation).

"x' is not a quantity that exists on its own" - well, in a way, it isn't, and neither is x - these depend on the chosen reference frame and units. What's more fundamental than coordinates is the wordlines of the objects, and where the simultaneous hyperplane of each observer intersects them.

From what you've written it looks like you're thinking that the uncertainty is caused by the act of measurement, in the sense that performing the measurement somehow disturbs the system, and that this makes things uncertain. But it's more fundamental than that. The uncertainty is embedded into the underlying physics - in a certain sense, the particle (the underlying wavefunction) doesn't simultaneously have precisely defined position and momentum, weather you measure it or not. See: youtube.com/watch?v=MBnnXbOM5S4

See this diagram. Rays originating from the same point on the object will hit the same point in the image; we just use a parallel ray and a ray that goes through the focus to find the location of that point.

@lylehunder: "But we want the work in bringing unit charge to where it is zero units away from P." - the integral is set up so that the charge itself is at zero, and P is where you stop integrating (see comment by peek-a-boo above for details).