Tension only seems to reveal itself when you have an end to the rope because otherwise you have a static equilibrium situation at each position along the rope the net force is zero.
The magnitudes of all these forces are the same and so when the rope was whole before it was cut one can say the force on the left hand part of the rope due to the right hand part of the rope was equal in magnitude and opposite in direction to force on the right hand part of the rope due to the left hand part of the rope and these are the forces that are called the tension. In general, a tensor in 3D space has $3 \times 3=9$ components (though stress tensors are always symmetric, so there are only 6 independent components) and it can be represented as a $3 \times 3$ matrix like $$\vec F = \int_S \sigma \cdot \vec n \ dA$$ a cut through the rope) you take the dot-product of the stress tensor and the unit vector normal to the surface, integrated over the surface area: ) and to find the force on the surface of a body (e.g. In general, stress is a tensor (the next step up in the hierarchy of scalars, vectors. Stress has the dimensions of force/area - the same as pressure, but "stress" and "pressure" are different concepts. The "correct" way to deal with all this to replace the idea of "tension in the rope" with the notion of stress. If you imagine that you cut the rope somewhere along its length, you then have two more "ends" which have equal and opposite forces acting on them. You can only talk about forces acting on the ends of the rope. If you interpret "tension" as "some kind of force" then the idea of "some kind of force in the rope" doesn't really make sense. I'm not sure if UK A-level maths includes an introduction to matrices, but if it does the following might help a mathematician understand what's really going on here. The problem here is trying to convert a "common sense" idea about what "tension in a rope" means, into mathematics which is simple enough for high-school-level students to understand.
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