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What is the exact definition of the tension of a rope?

Here http://web.mit.edu/8.01t/www/materials/modules/chapter08.pdf the following definition is given

The tension $$T(x)$$ in a rope at a distance  $$x$$ from one end of the rope is is the magnitude of the action -reaction pair of forces acting at the point point $$x$$.

But I think that this is valid only in the case that the tension is the same, throughout all the rope. Is that definition still valid in that case?

What is the exact definition of the tension of a rope?

Here http://web.mit.edu/8.01t/www/materials/modules/chapter08.pdf the following definition is given

The tension $$T(x)$$ in a rope at a distance  $$x$$ from one end of the rope is the magnitude of the action -reaction pair of forces acting at the point $$x$$

But I think that this is valid only in the case that the tension is the same, throughout all the rope. Is that definition still valid in that case?

What is the exact definition of the tension of a rope?

Here http://web.mit.edu/8.01t/www/materials/modules/chapter08.pdf the following definition is given

The tension $$T(x)$$ in a rope at a distance $$x$$ from one end of the rope is the magnitude of the action -reaction pair of forces acting at the point $$x$$.

But I think that this is valid only in the case that the tension is the same, throughout all the rope. Is that definition still valid in that case?

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# Rope tension definition

What is the exact definition of the tension of a rope?

Here http://web.mit.edu/8.01t/www/materials/modules/chapter08.pdf the following definition is given

The tension $$T(x)$$ in a rope at a distance $$x$$ from one end of the rope is the magnitude of the action -reaction pair of forces acting at the point $$x$$

But I think that this is valid only in the case that the tension is the same, throughout all the rope. Is that definition still valid in that case?