We started by writing the antisymmetric tensor as a "three-dimensional" matrix [Transparancy 1]. Of course, it's hard to draw a 3-D matrix, so we looked at cross-sections along the "i" axis. It is mostly zeros.
We looked at a very important identity [Transparancy 2], involving expressing the product of two antisymmetric tensors in terms of kronecker deltas. (Note that there is an implied summation here; we are assuming Einstein summation notation.) A sketch of how to prove the identity was given, but you are encouraged to spend some time convincing yourself that it is true. It is also shown in section 1.15 of the textbook.
At this point we broke to discuss the date of the mid-term exam (Feb 17th), which some people were unhappy with. Alternatives were looked at, but it turned out that even more people would be unhappy. It was decided that the minimum overall unhappiness would be obtained by keeping the date of the mid-term exam to be Feb 17th.
Back to the lecture. We started looking at the vector identities in section 1.14 of the textbook. The first and most important one (from which all others can be derived) is:
A x (B x C) = (A · C) B - (A · B) C
A geometrical argument was given [Transparancy 3] showing why A x (B x C) should be a linear combination of vectors B and C. The vector identity was then proven [Transparancies 3 and 4] using antisymmetric tensors.
Next week, we will be starting with Chapter 2 of the textbook. You are encouraged to read ahead.
Here are the scanned-in transparancies, in full-colour JPEG format:
