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Re: definitions of "linear" (with application to Sig Figures)



Also a linear differential equation involving a function y(x) is one
that involves only the first power of y (and its derivatives) but often
involves higher powers of x. See, e.g., Ince. The same definition
extends to partial differential equations.
Linear differential equations permit superposition of solutions.
Regards,
Jack

Adam was by constitution and proclivity a scientist; I was the same, and
we loved to call ourselves by that great name...Our first memorable
scientific discovery was the law that water and like fluids run downhill,
not up.
Mark Twain, <Extract from Eve's Autobiography>

On Wed, 2 Feb 2000, John Denker wrote:

At 10:51 AM 2/1/00 -0500, Richard Bowman wrote in part:
>
4. The usual "rules" only apply to linear functions.

I would like to remind everybody that there exist two grossly inconsistent
definitions of "linear".

Definition 1: A "linear transformation" (A) obeys the rules
A(x + y) = A(x) + A(y)
and (for any scalar t)
A(t x) = t A(x)

Definition 2: A "linear equation" is anything of the form
A(x) = m x + b

--------

In particular note a linear equation represents a linear transformation if
and ONLY if the y-intercept (b) happens to be zero.

Constructive suggestions: To avoid ambiguity:
*) A linear equation can be called a first-order polynomial, or
(preferably) a first-degree polynomial. It can also be called an affine
transformation.
*) A linear transformation can be called a proportionality relationship.

-----------

Also at 10:51 AM 2/1/00 -0500, Richard Bowman wrote in part:
>
I let my students simply
treat trig functions as if they are linear. Thus three digits in angle
implies three digits in sin or cos, etc.

That version of the rule applies only to proportionality laws. It doesn't
even apply to first-degree polynomials. It certainly isn't a good way to
think about trig functions. Example: suppose we know the angle
theta = 89.3(1) [roughly 3 sig digs]
then we can calculate
tan(theta) = 82(13) [roughly 1 sig dig]
sin(theta) = 0.999925(23) [roughly 5 sig digs]

This example does _not_ depend on nonlinearity. Nonlinearity is rarely
more than a minor contribution to the error propagation. A more relevant
thing to consider is the magnitude of the first derivative [dA/dx] and its
relationship to the chordal slope [A(x)/x]. Tan(theta) is very steep
compared to its chordal slope, while sin(theta) is very nearly horizontal
(given the aforementioned value of theta).