Introduction to Vectors and Tensors, Vol. 2: Vector and Tensor Analysis (Bowen & Wang)

ray-bowen (mechanical engineering, texas-am-university) and c-c-wang (mathematical sciences, rice-university) wrote a two-volume text on vectors and tensors. Volume 1 is the algebra; this is Volume 2, the geometry. The held copy is 246 pages of the PDF, ISBN 0-306-37509-5 (v. 2), carrying only the line "Copyright Ray M. Bowen and C.-C. Wang". The title page gives no year, so no published: is recorded. A web search for the book finds it in the Texas A&M repository (OAKTrust) as a free download from the first author's page, and finds a 1976 Plenum Press original and a 2009 Dover second edition. That is outside evidence, not the held file's own statement. Arrived by Telegram on 2026-10-03, routed by the hub.

What it is for

The preface says Volume 2 is about fields defined on domains in a Euclidean manifold, whose values are vectors or tensors, and about what the distribution of those values reveals. The authors say outright that they skip general differentiable manifolds, then include a chapter on fields on hypersurfaces in a Euclidean manifold. A student with "a modest background in linear algebra" can use it alone, and for a one-semester course the authors taught Chapters 9, 10 and 11 plus Chapters 0, 3, 4, 5 and 8 of Volume 1. The National Science Foundation is thanked for support. Dr Kurt Reinicke is credited with checking the whole manuscript.

The five chapters

The Part III numbering (chapters 9 to 13, sections 43 to 72) continues from Volume 1.

Euclidean manifolds (§43–48). The point space, charts and coordinate systems, the transformation rules for vector and tensor fields, anholonomic and physical components, Christoffel symbols and covariant differentiation, covariant derivatives along curves. See euclidean-manifold and tensor-analysis.
Vector fields and differential forms (§49–54). Lie derivatives, the Frobenius theorem, differential forms and the exterior derivative, the Poincaré lemma, then classical three-dimensional vector analysis re-derived. See differential-forms.
Hypersurfaces in a Euclidean manifold (§55–62). Normal and tangent plane, surface covariant derivatives, geodesics and the exponential map, the Weingarten and Gauss formulas, the Riemann-Christoffel tensor, the Gauss and Codazzi equations, area and minimal surfaces, and the two-dimensional case in three-space.
Classical continuous groups (§63–67). The general linear group and its subgroups, Cartan's parallelism, one-parameter groups and the exponential map, subgroups and subalgebras, maximal abelian ones. Links to group-theory.
Integration (§68–72). Arc length, area and volume; integrating vector and tensor fields and differential forms; the generalized Stokes theorem; invariant integrals on groups.

Main results

Frobenius. A smooth distribution is integrable if and only if it is closed under the Lie bracket (Thm 50.1). Section 52 gives the dual statement in forms and proves the Poincaré lemma (Thm 52.2): on a star-shaped domain a closed form is exact.
Gauss and Codazzi as integrability conditions. Given a positive-definite symmetric a and a symmetric b that satisfy the two equations, a hypersurface exists locally with a and b as its first and second fundamental forms, unique up to a rigid motion (Thm 60.1).
Extrinsic curvature is the stronger notion. The Gauss equations fix the Riemann-Christoffel tensor from the second fundamental form, but not the reverse. A cylinder has zero intrinsic curvature and is still not a plane (§60).
Intrinsic curvature as a departure. Curvature on a surface is where the Levi-Civita parallelism differs from Euclidean parallelism (§59).
Classical theorems fall out of one. Stokes' theorem and the divergence theorem are the formulas (71.28) and (71.30), special cases of the generalized Stokes theorem (§71).

Level, audience, rigour

The preface sets the entry bar at "a modest background in linear algebra" and the use at one semester, so the audience is a student past a first linear-algebra course. My reading of the level is advanced undergraduate or first-year graduate; the authors do not say so. The text is theorem and proof throughout, with exercise sets at the end of sections, and the authors make no claim about their own rigour. The selected readings for Part III run from Bishop and Goldberg's Tensor Analysis on Manifolds and Eisenhart's Riemannian Geometry to Chevalley on Lie groups, Flanders on differential forms and Ericksen's tensor-fields appendix to the Encyclopedia of Physics. No software is assumed.

Tier and limits

T2. A textbook by named, credentialed authors, thanked for NSF support, with a reviewer credited. Not T1: this is a 246-page PDF copy of a book without a year on its title page, and the surviving scan text mangles the equations.
How it was read. The PDF's extracted text keeps prose and theorem statements but garbles most formulas (sub- and superscripts run together). This summary rests on chapter and section openings, stated theorems, and the index. Proofs and exercise solutions were not checked.
Half a book. Volume 1 (algebra, tensors, exterior algebra, chapters 0 to 8) is not held. The book cites it constantly for the algebra of tensors. The tensor algebra it relies on is covered elsewhere only inside gallier-quaintance-math-deep (Part IV).
Notation is its own. The volume writes the gradient of a vector field as a tensor, grad v, and the Lie bracket as (grad v)u − (grad u)v. That is the coordinate-free style, not the index-first style of older tensor texts.

Cross-spoke context

The hub's router named two runner-ups. ../physics-wiki could take the applications, but the text here is theory throughout: the only physical wording is the section on "physical components" (§46), which means components measured in unit-length directions, not a physical claim. The text is not pedagogy about teaching, so ../engineering-education-wiki is a weaker fit still. It stays here on this spoke's rule: the maths is the subject.

Connections

authors ray-bowen, c-c-wang
builds on linear-algebra; extends vector-calculus; sits beside topology and matrix-calculus as the other route to "the derivative as a linear map"
groups chapter: group-theory