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Mark_Eichenlaub comments on What is the Mantra of Polya? - Less Wrong
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I'm not really sure what you mean by "upward tension", sorry. Tension in one dimension is just a scalar. The very bottom of the spring is under no tension at all, and the tension increases as the square root of the height for a stationary hanging slinky.
The tension gradient is upward, indicating an upward force per length.
By "upward," I just meant to emphasize that it was opposing gravity, e.g., positive. But of course, now that I think about it for a minute, I see that I was wrong, it is under no tension at all. Oops.
I think I see what you mean. To clarify, though, tension doesn't have a direction. In a rope, you can assign a value to the tension at each point. This means that if you cut the rope at that point, you'd have to apply that much force to both ends of the cut to hold the rope together. It's not upward or downward, though. Instead, the net force on a section of rope depends on the change in the tension from the bottom of that piece to the top. The derivative of the tension is what tells you if the net force is upward or downward. This derivative is a force per unit length.
In general, tension is a rank-two tensor, and is just a name for when the pressure is negative.