Vector calculus
Calculus of functions of several variables and of vector fields in two and three dimensions. Known in course catalogues as Vector Calculus, Multivariable Calculus, or just Calculus III. The corpus holds two accounts: corral-vector-calculus, a one-semester course with single-variable calculus assumed, and chapters 10–13 of keisler-elementary-calculus, which reaches the same material at the end of a full sequence built on infinitesimals.
The arc
The subject has a standard shape, and Corral follows it:
- Vectors in R² and R³ — dot and cross products, lines, planes, surfaces, curvilinear coordinates, vector-valued functions, arc length.
- Functions of several variables — partial derivatives, the gradient and directional derivatives, maxima and minima, Lagrange multipliers.
- Multiple integrals — change of variables, centre of mass, and probability applications.
- Line and surface integrals — Green’s, Stokes’ and the Divergence theorems.
The arc is not arbitrary. Everything before chapter 4 is machinery for the three theorems at the end, which all say a version of the same thing: an integral over a region equals an integral over its boundary.
The two exports
Most of the hub touches this subject through two of its results.
The gradient. A vector pointing along steepest ascent of a scalar field. It is why
gradient-descent in ../optimization-algorithms-wiki is called that, and why fitting a model in
../machine-learning-wiki is a differentiable problem at all. It is also, per that spoke’s standing
homonym warning, exactly what metaheuristic optimizers do not have — they exist for the cases
where no usable gradient is available.
Divergence and curl, and the theorems joining them. Field theory in
../loudspeaker-design-wiki and ../embedded-iot-wiki runs on these, at a distance of several
textbooks.
Optimization sits inside calculus, and that is the confusing part
Corral’s §2.6 covers unconstrained numerical minimization (Newton’s method, with Java exercises) and
§2.7 covers constrained optimization by Lagrange multipliers. Both are optimization in the plain
sense of the word, and both are here rather than in ../optimization-algorithms-wiki, because that
spoke owns the metaheuristic, population-based family (CMA-ES, PSO, genetic algorithms), which
came into existence to handle the problems this chapter’s methods can’t: no derivative available,
or many local optima and only one wanted. The calculus route is the baseline the metaheuristics are
benchmarked against. Cross-link the two, never merge them.
Two accounts, and the fork between them
keisler-elementary-calculus covers this subject too — vectors in chapter 10, partial differentiation in 11, multiple integrals in 12, and chapter 13 running directional derivatives and gradients, line integrals, independence of path, Green’s theorem, surface integrals, and Stokes and Gauss. That is Corral’s book, re-covered.
The overlap is real corroboration on the content and a genuine disagreement about the foundation. Corral assumes the standard limit-based Calculus I and II as prerequisite. Keisler replaces that prerequisite, building derivative and integral from infinitesimals and deferring limits to his chapter 5. Same theorems, same notation, same worked methods; two different answers to what a student should meet first. See infinitesimal-calculus and synthesis.
What the corpus can and can’t say
Corral is self-rated 5 out of 10 for rigour by its own author, with proofs present but selective; Keisler is a commercially published text whose held copy is an unsearchable 1986 scan. Where the two agree on a result, the corpus is on firm ground. Where only one covers something — Corral’s curvilinear coordinates, Keisler’s hyperreal vectors — it is one author’s account.
Related
corral-vector-calculus · mathematics · probability-theory · synthesis