Embryology Biology 441 Spring 2012 Albert Harris
A few more notes related to tensors:The directional elongation of cartilages and plant shoots can be driven by a scalar variable (osmotic pressure). But the directionality cannot be produced by osmotic pressure, because scalar properties don't have any directionality.
That may seem pretty simple, once you know it. But I am not sure the textbook knows it. Lots of good scientists have wasted hours of discussion trying to make sense of situations like these, trying to measure a tensor variable with a single number. Many researchers have used "atomic force microscope" probes to measure the inward force needed to produce a dent in a cell, and got single-number "measurements" of tension.
Actually, tension can be zero in one direction, but large in directions perpendicular to that. Without knowing it they averaged tensions of all directions. This is not too good when the phenomenon they are trying to understand is caused by directional differences in tension. (Such as cell division, or elongation of embryos.)
For example, consider the shaping of the lobes of the brain. Water pressure inside the neural tube pushes outward everywhere. This water pressure is not capable of being different in one direction than another direction. (Impossible for any scalar variable) Water pressure is also not capable of being different in one location than another. (although many scalars can vary from place to place: temperature, chemical concentration, etc.)
The osmotic pressure inside a plant tissue can differ from one place to another. The electroosmotic pressure inside a cartilage can vary from place to place, But neither can differ in one direction as compared with another direction. So when a stem elongated in one direction, or a cartilage becomes any shape but a sphere, we should look for some directional property that is channeling the pressure into some directions, and or at some locations.
Collagen fibers running through the interior of a cartilage, or wrapped around parts of the outside, can cause stiffness to be much different in some directions than others.
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