More on D'Arcy Thompson and tensor variables
Pressure difference equals Curvature multiplied times Tension.
dP = C x T
dP = T/R where R is the radius of curvature
Because both curvature and tension often vary with direction
dP = C x T + c x t c and t are curvature and
dP = T/R + t/r R and r are radii of curvature in two perpendicular directions
Unfortunately, D'Arcy Thompson and many others mistakenly used this equation
dP = T x (1/R + 1/r) Which assumes that tension is the same in all directions.
For soap bubbles, tension really is the same in all directions.
But for cell sheets and surfaces of living, actively contractile cells,
This and other simple mistakes have led biophysics down a blind alley.
How to think about quantitative variables that vary with direction?
(like curvature, tension, permeability, strain, and others).
Variables that don't have any directionality are called scalars.Chemical concentrations, density, temperature, osmotic pressure, & hydrostatic pressure are scalars.
Scalars have an amount at each location, which can differ with location; but has no direction.
Vectors have an amount and a direction at each location.
Electric fields are an example of a vector variable.
Curvature, tension (equals "stress", which includes tensile stress and compressive stress)
Variables that differ with direction are called tensors, and unfortunately are
All you need to know about tensors are the following:
1) Curvature, stress (tension) and strain (% distortion produced by stress)
2) (most) Vectors are considered to be first order tensors.
3) Elastic moduli are fourth order symmetrical tensors.
4) The word "Symmetric" applied to tensors means that
[Don't worry about it, but axial vectors are really second order anti-symmetric tensors.]
5) For second order tensors, if they vary with direction (which they CAN, but don't have to),
D'Arcy Thompson's biggest mistake was to treat mechanical tension as if it could not
It is a tragedy that so many people have such strong positive or negative responses
Incidentally, tensors played a major role in the escape of physics from the
D'Arcy Thompson mistakenly persuaded himself and his readers that self-building shapes