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Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

Hung Nguyen-Schäfer and Jan-Philip Schmidt
Publication Date: 
Number of Pages: 
Mathematical Engineering
[Reviewed by
P. N. Ruane
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Not long before I began this review, I had reason to consult another book written for engineers at the postgraduate level. That was the 1962 edition of Differential Forms with Applications for the Physical Sciences, by Harley Flanders, who began by questioning the predominant use of tensor analysis around that time. Most of his argument is paraphrased as follows:

  1. Tensor analysis per se seems to consist only of techniques for calculations with indexed quantities. It lacks a body of deep results that are transferrable to other contexts. The exterior calculus does have such a body of results, and it provides a more efficient derivation of results in, for example, Riemannian geometry.
  2. In classical tensor analysis, one never knows the range of applicability because one never knows what the space is. Everything seems to work in a particular coordinate patch, which is inadequate for most applications.
  3. Tensor fields do not behave reliably under mappings.
  4. In tensor analysis, the maze of indices disguises the great differences between various quantities.
  5. Using tensor methods, it can be difficult to discern the invariants in geometric and physical situations — even the local ones.
  6. In Riemannian geometry (for example), tensors restrict one to the natural frames associated with a local coordinate system.

My personal aversion tensor analysis stemmed mainly from points 4 and 5 above but also from the fact that I’ve found most introductions to it to be as dry as dust. However, having enjoyed this book, I may now be converted to a line of thinking that began with Ricci-Curbastro and Levi-Civita over a hundred years ago.

But what kind of work is undertaken by engineers who invoke use of subjects such as differential (or algebraic) geometry? In this case, Nguyen-Schӓfer is concerned with the development of electric machines for hybrid and electric vehicles, and a chapter in one of his recent books is called ‘Applied Tribology in Oil-Film Bearings’ (well, that’s just one facet of the ‘real world’ in which engineers reside!). Jan-Philip Schmidt is the mathematical half of this joint authorship and, judging by his brief (but impressive) CV, he seems to spread his mathematical talents far and wide.

As with Flanders, the readership is expected to consist of graduate students in physics and engineering, research scientists and practicing (sic) engineers. However, with the aim of preventing rigor mortis in the reader, the approach is said to be non-rigorous — and assumptions regarding prior knowledge are (optimistically) restricted to vector analysis in n-dimensional spaces with general bases in curvilinear coordinates.

The book begins by introducing the concepts general basis and tensor types for curvilinear coordinates, followed by Dirac’s first and second order tensors \(\langle \text{bra}\mid\text{ket}\rangle\). This leads to transformations of kets with applications to eigenvalue problems. Chapter two (the heart of the book) concerns itself with tensor analysis leading to covariant derivative, the Riemann-Christoffel and Ricci tensors etc. The most salient application to be introduced by this stage concerns the eigenvalue problem of linear oscillators.

But what can be achieved within 42-pages worth of differential geometry? Mathematically, quite a lot, because this is really a condensed resumé of more material than usually occupies a one-semester undergraduate course. Matters of pedagogy are a different matter, since, within the first 20 pages of chapter 3, the reader is taken from the basic notions of arc-length and surface area to the Gauss-Bonnet theorem and the Gauss-Codazzi equations. Subsequent explanation of Lie bracket, Lie dragging, torsion and curvature and Killing vector fields leads nicely to thoughts on invariant time derivatives on moving surfaces — but all in a very short time.

Applications of tensor analysis are really what the authors have in mind here, and these are both mathematical and physical. As mentioned, differential geometry is treated by tensors methods, and so is the nabla operator and much of vector analysis. Fluid dynamics, continuum mechanics and electrodynamics are the earthly applications, while the Einstein field equations and Schwarzchild’s black hole take us into ethereal realms.

In saying that they have avoided a rigorous treatment of this subject, the authors may mean that they have omitted foundational themes (existence of geodesics, for example). But the mathematics is presented with clarity and precision. In particular, I like the way in which concepts are illustrated in the context of low dimensional cases, and the narrative is interspersed with many informative illustrations. In other words, it’s the sort of book that attracts one’s attention on a first perusal.

Finally, I couldn’t help comparing this book with another recently reviewed text on differential geometry. That was Differential Geometry, by Marcelo Epstein, which is also published within the Springer series on Mathematical Engineering. The target audience is the same as this book, but its approach is decidedly non-tensorial (if there is such a phrase). Both books are excellent in their own way.

Peter Ruane has taught mathematics to people between the ages of 5 and 55 — that is, from basic school arithmetic to transfinite arithmetic.