PHYSICS 405

Spring 2005     Daniel Finley

Notes on    Introduction to Electrodynamics,   by David J. Griffiths

Chapter 1: Vector Analysis

  1. Very elementary vector analysis: addition, multiplication by scalars, dot and cross products; triple products. No discussion of tensor products of vectors! Claims that "vectors have no location," which is either not (quite) true, or silly. See, in particular, his footnote on p. 39, where he points out (only) some of the difficulties with this notion. Therefore he makes no real suggestions as to where vectors live, how they behave when moved from point to point, etc., etc. At least a couple of comments about (vector space) fibers over the space have to be made, although we will surely not use them in any real fashion, pretending that field lines are sufficient.
  2. Basis vectors: uses a caret over a symbol to indicate a "unit vector" in the direction as the coordinate denoted by that symbol (only) changes. One should note that here and again later on, more than once but an example is in the footnote on p. 62, that he says you cannot integrate a non-constant basis vector. Of course you can; however, you have to pay good attention to what you're doing!
  3. Discussion of transformation laws for vectors---with statement that it could be skipped! Nonetheless, Section 1.1.5 on tranformation laws is useful, and should be understood. Vague, or very brief, discussion that there exist things called tensors. Discussion about differentials and gradients, with very good discussion of physical and geometrical meaning, but, of course, nothing about 1-forms.
  4. Appropriate discussion of vector operators, such as the gradient, but nothing about, for instance the tensor . Has divergence and curl, BUT discussion in some other text should also certainly be considered about now. He SAYS that one should also use Cartesian basis vectors when using such operators, so as to not have to differentiate things like ; however, probably we should insert , etc., in (at least) a Problem Session!
  5. [Section 1.3] Integral calculus for vectors: line, surface, and volume integrals. Fundamental theorem of calculus also for line integrals! Puts here various versions of Gauss' theorem for divergences, curls, etc. An especially good job on physical meanings for integrals of curls! Need to pull out Green's theorem from the last homework problems, and insert it into the lectures; it will be used in important ways later.
  6. I would like more coordinate-free definitions of the important operators, div, and curl. These are given very well in Purcell's book, beginning about p. 55 and continuing until the end of that chapter, i.e., Ch. 2.
  7. Curvilinear coordinate section: ONLY does spherical and cylindrical coordinates. DOES give expressions for the appropriate basis vectors in these coordinates in terms of Cartesian ones, BUT not inverses---although he does ask for this in a problem---nor notions that they are orthonormal matrices.
  8. The Dirac Delta: Begins with discussion of /r2. BUT regularly refers to this quantity as a "function" when it is NOT. Then goes forward from the 1-dimensional to the 3-dimensional one, finding that
    2(1/r) = -43 ().
  9. [Section 1.6] Introduces potential functions, and the important (2) versions of the Poincaré Lemma, referring to them as the Helmholtz Theorem. Have to be careful with his "bald" statements concerning potentials that are actually ONLY TRUE in static situations! Also, no discussion of necessities concerning region of definition, simply-connected regions, etc. Does, however, make perfectly good, and helpful, statements about the notion of "textbook problems," where various quantities "extend to infinity," when one knows that they don't really do any such thing.

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  Last updated/modified: 16 January, 2005