"Lecture" for Weeks 13 and 14

Position papers

• I fear that the graded book reviews may have given the misimpression that the only things I care about are syntax and spelling. In fact, the book review grades were based in large part on subjective evaluations of the depth and competence of your remarks, and those judgments are hard to put into words (for me, at least). This course is very different from the test-problem grading with which I have 30 years experience (and a welcome relief from it).
• All the grades will be very high (above 95). I see this assignment literally as an "exercise" -- it was intended to provoke thought, rather than to prove competence. There was a wide variety of approaches (a very desirable thing when one needs to read 15 papers in a row). Some people interpreted the instructions as calling for a more typical research paper (i.e., full of footnotes). Others made a more personal statement, which was closer to what I had in mind. I gave due recognition to hard work in the library or on Google (and I'll keep the papers for future reference), but I was more impressed by original observations illuminated by personal experiences. In any case, only one or two points out of 100 were at stake here.

I'll make a discussion list for posting your essays and debating them. You might want to wait till you see my critique, so that you can make minor revisions.

Good-bye for awhile

As you see, I have set up all the rest of the course. The last 3 chapters are short, and it would be reasonable to have the final homework assignment due soon after Thanksgiving, leaving you a week with nothing to do except finish your term papers. However, in the interest of flexibility, I've divided the assignment into two parts; the exercises on the 20th century may be delayed till Dec. 4. I'm sure it will help Valentina if you get them in earlier, however.

Please look over the exercises soon and try to get any questions of interpretation (like the one this week abour what identities we can assume in differentiating a^x) to me by noon Saturday Nov. 25 at latest. I've tried to help on 19.11 and 20.4 by the little essay below.

Quaternions and Grassmann units

1. 3-dimensional vectors equipped with the vector cross product (see p. 3 of the linear algebra reading);
2. Grassmann's "extensive quantities": what Grassmann and Katz (p. 486 and Ex. 19.11) call b epsilon1 + c epsilon2 + d epsilon3;
3. 3 x 3 antisymmetric matrices (see p. 4 of the linear algebra reading)

• The 2 x 2 matrices {-i sigma} satisfy the quaternion multiplication rules, ij = k and i² = - 1, etc. I think that this suggests the easiest way to work Ex. 20.4.
• The quaternion product

  (bi + cj + dk)(b'i + c'j + d')

  contains the vector (cross) and scalar (dot) products of the vectors (b,c,d) and (b',c',d'), as you can see by working it out.

• The Grassmann product of two of these things is a Grassmann "quantity of the second order" (p. 486). It represents a parallelogram (with an orientation -- a positive and a negative side), rather than a vector (oriented line segment). A special property of the number 3 is that in 3-dimensional space oriented parallelograms can be "identified" with certain vectors perpendicular to them. (The overall sign of this identification is an arbitrary convention, called "right-hand rule".) The vector associated with the Grassmann product of two vectors is just their standard Gibbs cross product. Furthermore, the Grassmann product of a first-order unit with itself is 0, not -1 as for quaternions. This reflects the fact that the scalar (sigma0, dot product) part of the quaternion product is just thrown out of the definition of the cross product.