Vector Calculus (Corral)

michael-corral‘s one-semester textbook for the course variously called Vector Calculus, Multivariable Calculus or Calculus III. 222 pages, January 2008, released under the GNU Free Documentation License — a free textbook in the copyleft sense, not merely a free download. Typeset in LaTeX with KOMA-Script, written in Emacs on Fedora, figures in MetaPost, PGF and Gnuplot; the colophon is the book’s own statement of allegiance.

Parked in the hub _inbox on 2026-07-29 under the tag mathematics after no spoke would take it, and one of the three sources this spoke was created from.

What it covers

Four chapters, following the standard Calculus III arc. See vector-calculus.

Vectors in R² and R³ — dot and cross products, lines and planes, surfaces, curvilinear coordinates, vector-valued functions, arc length.
Functions of several variables — partial derivatives, the gradient and directional derivatives, maxima and minima, unconstrained numerical minimization, Lagrange multipliers.
Multiple integrals — change of variables, centre of mass, applications to probability.
Line and surface integrals — ending where the course always ends, at Green’s, Stokes’ and the Divergence theorems.

The author on his own rigour

The preface makes a statement most textbooks leave implicit, and it is the reason this spoke’s CLAUDE.md carries a textbook caveat:

“If I were to rate the level of rigor in the book on a scale of 1 to 10, with 1 being completely informal and 10 being completely rigorous, I would rate it as a 5.”

His reasoning is that students need to see full proofs, “since that is how mathematics works,” but that too much of it “can impede the flow of learning for the vast majority of the audience at this level.” A source that tells you where it sits on that scale is more useful than one that doesn’t, and a claim taken from a 5-out-of-10 text needs checking before it carries weight elsewhere.

Exercises and the computational tool

420 exercises, graded A / B / C — roughly easy, moderate and challenging, with Corral noting that “many of the B exercises are easy and not all the C exercises are difficult.” A handful ask the student to write a program: Monte Carlo approximation of multiple integrals in §3.4, Newton’s method for numerical minimization. The code samples are Java, which dates the book more sharply than any of its mathematics; Corral says students may use any language and adds that a scripting language would have been simpler, having chosen Java anyway.

Tier and limits

T2. A carefully made, openly licensed textbook by a named mathematics faculty member with stated qualifications (Berkeley BA, Michigan MA and MS), but self-published with no press, no editorial board and no peer review. The author’s own rigour rating of 5/10 is a caveat he supplies rather than one imposed here. Not T1: nothing external vouches for it.
Eighteen years old. The mathematics is permanent; the Java exercises and the Fedora/Emacs toolchain are period detail.
Adjunct faculty at a community college is an unusual provenance for a text this careful, and the free licence is why it circulates. Neither fact bears on whether the theorems are right.
Held at raw/corral-vector-calculus.pdf. The site returns HTTP 406 to a default curl user-agent and serves the file normally to a browser one, which is worth knowing before a future refetch is called a dead link.

Cross-spoke context

The _inbox record that parked this book worked through the near-misses, and the reasoning became this spoke’s scope boundary:

../optimization-algorithms-wiki — the closest miss. §2.6 (unconstrained numerical minimization) and §2.7 (Lagrange multipliers) are optimization by any reading. But that spoke owns metaheuristic, population-based global optimizers, the family that exists because the calculus route needs derivatives and settles for local optima. Two sections out of thirty are not a membership claim; the classical baseline stays here.
../loudspeaker-design-wiki, ../music-tech-wiki, ../embedded-iot-wiki — divergence, curl and surface integrals underlie the field theory in all three, at a distance of several textbooks.
../machine-learning-wiki — gradients and directional derivatives are what gradient descent descends. That spoke owns the fitting procedure; the derivative is here.

vector-calculus · mathematics · michael-corral · schoolcraft-college · mcmullen-probability-theory · thibos-fourier-analysis · synthesis