The vector calculus --- "div, grad, curl and all that" --- is one of the most basic tools of physicists and engineers, so important that undergraduates are sometimes taught it in three or four classes at once. It allows for exceedingly compact expression of classical mechanics and electromagnetism, and it is almost unbelievable that the notation developed

Coordinate-independence is important, because physicists want, as far as
possible, to talk about Nature and not our conventions. As Russell says somewhere, it is a remarkable fact that
there are twelve inches to the foot even in the farthest reaches of space, but
it doesn't tell us much about the Universe. Coordinate systems being very
plainly human conventions (what *difference* could it possibly make
whether *we* use rectangular or spherical or hyperbolic?), we want to
minimize the part they play in our physics. The ideal physics would be stated
in a notation where the choice of coordinates is irrelevant.

Now just as beavers have an instinct to build dams, and will cover with mud and sticks a speaker from which comes the sound of rushing water, mathematicians have a primal urge to generalize, to abstract, to identify the components of a problem and see what happens when some of these are taken away. (Pedagogically one goes in the opposite direction, and speaks of "adding structure".) It so happens that some of the peculiar structures of every-day Euclidean space mesh very nicely with Cartesian coordinates, so that the tools necessary to dispense with the one turn out to be those needed to dispense with the other. It's easier to write down things like the Pythagorean formula, or parallel transport of vectors, in rectangular coordinates than any other; a formalism general enough to accommodate (say) spherical coordinates is also general enough to handle non-Euclidean spaces.

This process of generalization began with the description of curved surfaces
in Euclidean space, to some degree even before the standard vector notation
settled down. (Thus for instance on a sphere both Pythagoras and parallelism break down.) It occupied the
talents of some of the great names of mathematics --- Lie, Riemann,
Poincaré, Levi-Civita, Cartan --- and here the generalizing urge was not
mis-placed. The theory they developed, known as differential geometry, has
become a nearly universal tool of theoretical physics. It is *the* way
of stating of general relativity; it is a key to quantum mechanics and still
more quantum field theory; more recently it has invaded fluid mechanics,
thermodynamics and classical mechanics; dynamics in general owes a great deal
to it.

There is then a need for a good exposition of the subject for physicists,
and Prof. Schutz's book fills it admirably. He is clear and readable --- great
and rare merits in a textbook author; he is concise, and reasonably
comprehensive, but doesn't move so fast that the reader is left behind,
bewildered; he makes extensive use of geometrical intuition and well-drawn
diagrams. (It is surprising how *opaque* a diagram can be, if one
hasn't seen its construction, and how many of these make it into textbooks!)
Above all, he manages to convey both a sense of the physical relevance of the
theory (without falling into a mere catalog of applications or
formula-mongering) and of the importance of the mathematics as such (without
demanding a degree of rigor and abstraction more appropriate to mathematicians
than physical scientists).

The book begins with a review of some of the mathematical background
expected of the reader --- a smattering of topology and analysis, groups,
vector spaces, linear and matrix algebra. (The reader is also expected to be
familiar with vector calculus, of course, and "a little operator theory of the
sort one learns in elementary quantum mechanics"; many undergraduates would
find it heavy going, but beginning graduate students should have no trouble.)
It then turns to *manifolds,* which were what was left of Euclidean
space after Messrs. Lie, Riemann *et cie* were done with it.
Essentially, a manifold is just a set where one can assign coordinates to every
element (i.e. label it by some numbers) in a regular, continuous way. (There doesn't even have to be one consistent labeling
scheme for the whole manifold.) This is exceedingly vague; almost all of
the paraphernalia of high school geometry, even distances and angles, is
missing (indeed, they will not enter until the very last chapter). Almost
anything which can be described by continuous variables can be represented by a
manifold, but it seems at first that we've paid for this generality by being
unable to *do* anything further.

There's enough structure in a manifold, however, that one can define curves,
as a set of functions from a single parameter to the coordinates of the
manifold. Taking the derivative with respect to this parameter gives us
tangents to the curve, which are vectors. Now we're in business: we can set up
vector fields and basis vectors, construct families of curves from the vector
fields tangent to them, apply vector fields to each other and take their
commutator. Fiber bundles, which prove to be so useful later in physics, and
seem so mysterious, prove to be little more than giving each point on the
manifold its own vector-space. *One-forms* are defined as linear
functions which map a single vector to a real number, and diagrammed as sets of
contour lines --- the gradient of the old vector calculus is a one-form. (If
one imagines vectors to be column matrices, one-forms are
row-matrices.) *Tensors* are functions from vectors and one-forms to
the real numbers, linear in all their arguments. Older works defined vectors,
forms and tensors by the way their representations in particular coordinate
systems changed when the coordinate system changed. To my mind, this is about
as illuminating as saying that a length is what increases by a factor of twelve
when going from feet to inches. The new approach focuses instead on the
underlying geometrical object, and the coordinate-transformation rules are
easily derived from the definitions. The *metric tensor* is introduced
at the end of the chapter. This is a symmetric, invertible tensor taking two
vectors into a real number, and we use it to define the inner product of two
vectors, as well as a way of converting vectors and one-forms into each other.
Almost as an after-thought, Schutz shows how the metric tensor lets us
calculate the lengths of curves, and so the distances between points.

The third chapter introduces the very important subject of Lie groups and
Lie algebras, which essentially have to do with how things (functions, vectors,
tensors) change when moved along vector fields --- "Lie dragging". A Lie group
is, more or less, a group which is also a manifold; and almost anything which
is continuous is a manifold. This means that some rather abstract operations,
like rotations, have nice geometrical representations, and rotations and
angular momentum receive special attention. Physicists are particularly
interested in *invariance* under various operations, including Lie
dragging --- vector fields which leave the metric tensor invariant receive the
wonderful name of "Killing vectors." The discussion
of the relationship between symmetries and invariants is particularly fine (and
continued in chapter five, when mechanics itself is considered).

Chapter four introduces N-forms (tensors which take N vectors, and are anti-symmetric in all their arguments), which correspond to volumes and sets of surfaces in the manifold. We see how to integrate and differentiate them in coordinate-free form, see the generalizations of the familiar cross-product, curl and divergence, and are given a glimpse of the applications to solving differential equations. This is fine, as far as it goes, but that is probably not far enough; fortunately Michael Spivak's nearly classic book on Calculus on Manifolds (New York: W. A. Benjamin, 1965) complements this chapter nicely.

Following this there is a chapter of applications, showing off the power of the geometry. I found the sections on Hamiltonian mechanics and electromagnetism the most interesting, but doubtless this is a matter of taste.

Schutz concludes with a short chapter on *connections* --- rules for
the parallel transport of vectors --- and their links to curved spaces,
non-Euclidean geometry, and so-called gauge field theories. This is the least
satisfying chapter in the book, since it breaks off just at the point where one
would like to see these new beasts in their natural habitat, general
relativity; as Prof. Schutz is also the author of a text on GR, this may be intentional.

Problems are placed throughout the text, and are both doable and informative; the selected solutions in the back are also, for the most part, enlightening, though one detects a certain note of pedagogical sadism in the number of entries there reading "Trivial" or "Algebra". The books in the "suggested readings" at the end of each chapter with which I am familiar are uniformly excellent, and I've put the others on my to-read list. There is a helpful index of notation, but the general subject index is a bit sparse.

xii + 250 pp., diagrams, selective bibliography, indices of notation and subjects

Mathematics / Physics

Currently in print as a paperback, US$32.50 as of July 1995, ISBN 0-521-29887-3 [Buy from Powell's]

12 December 1995