Multivariable Calculus and Differential Equations (KPI, 2013–2014)
A complete source map of 333 pages of KPI lecture notes and practicum on multivariable differentiation, multiple, line, and surface integrals, vector fields and integral theorems, and ordinary differential equations. GTK keeps the curricular spine while supplying independently verified hypotheses, orientation rules, and solution methods.
This archive concatenates the 2013 lecture notes and 2014 practicum by Viktor
O. Haidei, Lidiia B. Fedorova, Iryna V. Alieksieieva, and Oleksandr O.
Dykhovychnyi. It contains 333 physical pages: pp. 1–144 are the lecture volume
and pp. 145–333 are the practicum. Every page was text-read and visually
inspected; blank separator pages were recorded rather than silently skipped.
The source hashes are
fe63c41efabbea0a8aee5dae620e7bf9f1da19030aa6ab5492b65abcc73719fb
and c3f48ef5c0612d454c6b79b43450bd8e41320aa7534ba331bbafa691eaaa8cca.
Coverage and conceptual spine #
| Territory | Durable questions retained in GTK |
|---|---|
| multivariable differential calculus | what the derivative is as a linear map; how gradient, Jacobian, Hessian, tangent plane, chain rule, and constrained extrema relate |
| multiple integration | how regions are described; why iterated bounds work; how the Jacobian measures local volume scaling |
| line and surface integration | which geometric measure is being accumulated and when orientation changes the sign |
| vector calculus | how gradient, divergence, and curl connect local behavior to Green, Stokes, and Gauss boundary identities |
| ordinary differential equations | how order, initial data, existence and uniqueness, solution families, linear systems, and qualitative behavior fit together |
The lecture notes supply definitions and derivations; the practicum supplies worked geometry and calculation. They substantially overlap, so GTK maps them to four topics rather than duplicating theory and exercises as parallel chapters.
Reliability boundary #
Every load-bearing statement was checked against MIT OpenCourseWare and NIST DLMF. The source contains matrix and repeated-eigenvalue errors, missing topological hypotheses for potential fields, misoriented or mislabeled regions, a wrong spherical-volume setup, incomplete flux combination, answer-number damage, and an initial-value example that omits a constant solution and therefore misstates uniqueness. GTK does not cite those defective derivations. It states the required regularity, connectedness or simple connectedness, orientation, and uniqueness hypotheses where they matter.
Visual audit #
All 333 page renders were inspected. Coordinate regions, projection diagrams, oriented curves and surfaces, vector fields, and direction-field-like sketches were treated as claims. No source visual is copied: independent local diagrams make coordinate bounds, Jacobian scaling, theorem orientation, and ODE flow legible at article width without inheriting the source's caption and region errors.