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.
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.