In his excellent paper, ‘

*A Missing Piece: Early Elementary Plane Rotations’*,.__Notices of the American Mathematical Society__, Vol. 61, No. 2) Bob Palais uses a simple plane geometry of the unit circle to arrive at very concise representations of plane rotation, including in the complex plane. (For example, a quarter turn rotation behaves as the square root of (-1) or the complex number, i)
(X,Y) plus the multiple y of (-Y, X) or the image of (0,1).

Readers can check this is so by careful inspection of the diagram and following
the arc from (1,0) counter-clockwise to the end of the arrow at (X,Y). In so
doing they will also see that:

(x,0) or x is now identified at (xX, xY)

(x,y) is now at: (xX – yX, xY + yX)

(0,y) or y is now at (-yY, yX) and

(0,0)remains invariant (center of the plane rotation)

The image of (0,1) then is obtained from*:

x [X,Y] + y [-Y,X] = [xX- yX, yX + xY]

“

*The key to the plane rotation formula is the observation that when you turn your head 90 degrees clockwise what used to look like (X, Y) now looks like (-Y, X). Applying this twice, the math maven should be able to show that the image of [-Y, X] is [-X, -Y] )so a quarter turn rotation behaves as i)*
[x,y] = (cos s, sin
s), [X,Y] = [cos t, sin t]

In which case the original rotation formula becomes:

[cos (s + t), sin (s + t)] =

[cos s cos t – sin s sin t, sin s cos t + cos s sin t]

[cos s cos t – sin s sin t, sin s cos t + cos s sin t]

Palais most notable observation and point probably is:

“

*Any two rectangular coordinate systems may be related by a combination of shift of origin, scaling rotation and reflection, corresponding to complex addition, multiplication and conjugation, respectively*.”
He goes on to write that “

*it seems a shame for our students to be deprived of the one missing transformation – rotation – when providing access to it is so elementary and permits so much utility and insight*.”
This may be so but when I taught calculus physics and
astronomy I often found that circular motion gave the most grief to students,
and that included visualizing rotation even in two dimensions! When one teaches
simple harmonic motion, for example, one generally employs the unit circle via
projection to illustrate how the oscillating object appears to behave.

This approach can find further application in calculus
physics, for example, by noting that the quarter turn rotation formula, [-y, x]
expresses the physics of harmonic motion in Hooke’s law, e.g. F = -kx as well
as uniform circular motion. In that case the velocity v = dx/dt is perpendicular
to displacement so: [x, y]’ = [-y, x].
Acceleration would then be [x,y]’ applied again or [x, y]” = [-x, -y].
(From the differential equations perspective, [-y, x] denotes a change or
variables while the linear combination:

X [x,y] + Y[-y, x] would be a linear combination of
solutions

See e.g.

__Problem for Math Mavens__:

x [X,Y] + y [-Y,X] = [xX- yX, yX + xY]

is “

*also the origin of the Cauchy –Riemann equations that characterize an analytic function of a complex variable*”
That is:

¶
u/ ¶
x = ¶ v/ ¶ y And: ¶
v/ ¶
x = - ¶ u/ ¶ y

Can you show that this is so?

and:

(* Notation change note: Because the blogger field

*will not accept arrow brackets*, 'square' ones, [], have been substituted.)

## No comments:

Post a Comment