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Re: GA in high schools?



Hello,

I was asked (off-list) to provide an outline from the introductory lecture on
GA for high school students given by Mr. Risto Paju (e-mail: teknohog@iki.fi;
homepage: <http://www.iki.fi/teknohog/>). The outline is written by Risto
himself. He uses GA in his PhD work in physics. If you have any questions
related to the outline, Risto is happy to answer them.

The teaching method was ordinary lecture (derivations and figures were
presented on the whiteboard). The aim was to provide some insights and
inspirations, not to actually teach GA.

RISTO'S LECTURE OUTLINE:

- The complex :-) relation between mathematics and natural sciences

- The ideal that physics should be understandable independent of the
mathematics

- The loss/gain of physical intuition due to the mathematics used

- Different tools for different jobs; GA seems the best so far, for most
areas, and the most intuitive.

- Problems with vector and tensor algebras:

* Vectors limited to 3D; especially via cross product. Rotation as a
vector works only in 3D. A directed plane element (bivector) describes
rotation universally in any dimension; for instance 2D. In 1D rotation is
not possible => it is fundamentally a 2D entity.

* Tensor algebra works in arbitrary dimensions, and has objects of higher
grade than vectors (like bivectors etc.). Unfortunately the geometric
picture is completely lost. This goes against the above ideal of physics.

- Historically, GA developed along with vectors & tensors but lost in the
late 19th century

- Complex numbers known at least since the 16th C. Powerful methods that
are not possible with 2D vectors: e.g. conformal transforms; contour
integration; potentials/fields as analytic functions. The quest to
generalize these into N dimensions leads to GA eventually.

- The complex product zw* has the form
("dot product") + i("cross product").

[The second variable is conjugated, so that the operation of z with itself
would give zz* = |z|^2 = dot product only.]

- Hamilton's quaternion algebra (1844), the first attempt to generalize
complex numbers into 3D. Defined by i^2 = j^2 = k^2 = ijk = -1. A product
of two quaternions also has the form "dot product" + "cross product". The
two are separated, because it works out easier for 19th C. physics.

- Grassmann (1844) invents the exterior (outer/wedge/anticommutative)
product between two vectors, which is a directed plane element. a/\b = -
b/\a. The magnitude is equal to a x b, but a/\b works in any dimensions.

- Clifford (1878) follows the idea of complex product, and invents the
geometric product: ab = a.b + a/\b; axiomatic basis where the dot and
wedge products are special cases. Not limited to vectors.

- The geometric product is invertible, which enables conformal
transforms and other useful results of complex analysis.

- Reason why geom. product is invertible while dot and wedge products
are not:

* Given a constant vector a, the dot product a.b only depends on the
projection of b on a (i.e. |b|cos delta). Therefore, a scalar divided by
the vector a does not give a unique b. It only fixes that component of b.

* The same goes for a cross product, replace cos by sin. The cross/outer
product also defines the plane which gives some directional info.

* The combination of these projections defines b uniquely. The geometric
product combines the two kinds of product. Thus the "geometric division"
is well defined.

- Back to history: By the beginning of the 20th C, physics needed
higher-dimensional mathematics. GA had been forgotten and scientists
resort to tensor and matrix techniques.

- In the 1960s David Hestenes realizes the geometric meaning in the
matrix algebra of quantum mechanics (Pauli/Dirac spinors). The rest is
history :-). Hestenes develops geometric calculus, and convinces people
that GA is not limited to QM, but is a general language of physics.

- Basic operations of GA: reflection and rotation in compact forms.
Reflection is defined via a normal vector, which is independent of
dimensionality.

- Reflection formula is different for vectors and bivectors. This is the
source of endless confusion in vector algebra because rotation bivectors
are represented by vectors (polar/axial vector distinction).

- Rotation formula is the same for all objects. Key difference to tensor
algebra, where rotation is increasingly complicated for higher-grade
objects.

- Emphasize the advantages of GA: not only conceptual power, but also
computational simplicity and stability.

- Relation between complex algebra and 2D geometric algebra.

* A geometric object (unit bivector) that squares to -1.

* Beware of imaginary numbers in physics. There are many different objects
that square to -1, and the complex i can be any one of these.

- www.mrao.cam.ac.uk/~clifford for more information, tutorials,
publications, and other links.


Regards,

Antti

Antti Savinainen
Senior Lecturer in Physics and Mathematics
Kuopion Lyseo High School
Puijonkatu 18
70110 Kuopio, FINLAND
E-mail: antti.savinainen@kuopio.fi
Personal web page: http://kotisivu.mtv3.fi/oma/physics/

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.