However, we first discussed the mid-term exam which is happening on Wednesday. The exam will be closed-book, and no calculators will be allowed. You will be expected to know everything up to sec. 3.9 of the textbook -- basically everything you've had asignments on, except that you won't have to know any Maple. You won't be expected to memorize all of the vector identities; however you may be asked to derive one of them using tensor notation.
A collection of vector identities was listed [Transparancies 1 and 2]. They fell into various categories as follows:
The chain rule identity was quite easy to prove [Transparancy 2], by expanding the gradient in terms of its components. An example of the usefulness of the chain rule identity was presented, where it simplified the calculation of the gradient of 1/R.
Tensor notation was presented [Transparancy 3] and its use was demonstrated in the simple derivation of one of the product rule identities. Next, a couple of the complicated identities were derived [Transparancy 4]. The example involving the gradient of a scalar product turned out to be rather tricky [Transparancy 5], and entailed the cross-products of curls.
Finally, proofs of the identities involving mixed second-derivatives were given [Transparancies 5 and 6]. These expressions all give zero because the derivatives commute (the product is symmetric) whereas the anti-symmetric tensor is, guess what? -- anti-symmetric! The terms of the summation will therefore all cancel off.
What use is all this stuff? In order to demonstrate the usefulness of these vector identities, I introduced a set of vector equations which is probably one of the most fundamental and profound theories you'll ever run across: Maxwell's equations. These are the vector equations which completely describe all the phenomenology of electro-magnetism, including electrostatics, magnetostatics, and electrodynamic phenomena such as electromagnetic waves, ie. light! We'll talk more about this on Friday.
(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 60 and 75 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 225 K, which takes a bit longer to download.)
