Engineering Mathematics

Engineering Mathematics13 Aug 2026 · 9 min read

The Tensor Product of Modules

The tensor product by universal property and construction, right exactness, flat modules, and the tensor-hom adjunction.

Engineering Mathematics13 Aug 2026 · 9 min read

The Tor Functor

Tor defined by projective resolutions, symmetry in its two variables, vanishing criteria, and the standard computations over the integers.

The Two Long Exact Sequences of Ext

The long exact sequences of Ext in the first and second variables, their connecting homomorphisms, naturality, and the direction reversal caused by contravariance.

Three Finite Basis Theorems

The finite basis problem, Baker's theorem for congruence-distributive varieties, and the further finite basis results in the source, with the proof strategy via bounded subdirec…

Ultraproducts and Jonsson's Lemma

Ultraproducts in the algebraic setting, congruence-distributive varieties, Jónsson's lemma and its proof strategy, and the consequences for finitely generated varieties.

Ultraproducts and Los's Theorem

Ultraproducts as reduced products over an ultrafilter, Łoś's theorem and its proof, ultrapowers and the elementary embedding, and the applications to compactness and non-standar…

Universal Algebra: Computation and Sources

The sourcing policy for this collection: what is written here, what is deliberately not transcribed, the reasoning behind the split, and where to obtain current computational re…

Universal Algebra: Discipline Overview

Structural overview of universal algebra: what an algebra of arbitrary type is, why congruences rather than substructures carry the quotient theory, how Birkhoff's two theorems …

Universal Coefficients and the Dual Künneth Formula

The universal coefficient theorems for homology and cohomology, the Ext and Tor correction terms, naturality and splitting, and the dual Kunneth formula.

Yoneda Ext and n-Fold Extensions

The Yoneda description of Ext^n by n-fold extensions, the equivalence relation, splicing as composition, and the Yoneda product.

Algebra and Equations

Rearranging formulas, essential identities, linear and simultaneous equations, quadratics and cubics, Newton–Raphson root finding, arithmetic and geometric progressions, binomia…

Engineering Economics

Money over time for engineering decisions: simple and compound interest, nominal and effective rates, cash-flow diagrams, equivalence factors with a computed 8% table, net prese…

Engineering Mathematics13 Aug 2026 · 9 min read

Geometry

Analytical and mensuration geometry for engineering: lines and coordinates, polar conversion, the circle and conic sections, areas of plane figures, regular polygon data, circul…

Engineering Mathematics13 Aug 2026 · 6 min read

Manufacturing Data Analysis

Statistics for the shop floor: samples and populations, mean and standard deviation, distribution curves, the normal curve and z-values with a computed table, estimating the fra…

Engineering Mathematics13 Aug 2026 · 6 min read

Matrices

Matrix operations for engineering: addition and scalar multiples, matrix multiplication, transpose, determinants, minors, cofactors and the adjoint, rank and singularity, the in…

Numbers, Fractions, and Decimals

Working arithmetic for engineering: signed numbers, ratio and proportion, percentage, fractions and reciprocals, the inch–millimetre bridge, continued fractions and change-gear …