This repository contains the developing manuscript and computational materials for the textbook Calculus as the Geometry of Relationship and Controlled Change. The project is currently in progress and consists of parallel textual and computational components that are being integrated chapter by chapter.
Calculus is the study of how relationships deform under perturbation. The central thesis of the work is that calculus is best understood not as a collection of symbolic procedures, but as a geometric language for describing structured transformation under constraint. The derivative is interpreted as a sensitivity operator governing local deformation. The integral is treated as structured accumulation. Curvature determines stability. Control modifies vector fields. Entropy characterizes dispersion relative to invariant structure. Across physiology, engineering, and inference, the same mathematical architecture recurs.
The repository presently includes a draft of the main text, a separate file containing diagrams and geometric constructions, and a Jupyter notebook that generates computational illustrations used throughout the manuscript.
The project is organized as follows:
Calculus - main text.tex / .pdf The primary manuscript containing the theoretical development.
Calculus - diagrams.tex / .pdf Standalone diagram constructions and geometric visualizations corresponding to sections of the text.
calculus_unified_geometry.ipynb A Jupyter notebook containing numerical simulations, vector field visualizations, curvature illustrations, and dynamical system models.
The manuscript and diagrams are presently maintained as separate LaTeX documents. Figures generated in the notebook are being progressively aligned with chapters of the text.
The book develops calculus as a sequence of structural unifications. Limits formalize stability under refinement. Derivatives encode local deformation. Integrals reconstruct global structure from infinitesimal contributions. Differential equations describe flow on manifolds. Curvature determines stability and instability. Control theory formalizes directed modification of trajectories. Information geometry links statistical inference to Riemannian structure. Entropy and attractors explain persistence of coherence amid noise.
Physiology, engineered systems, and inference are treated as mathematically equivalent forms of constrained dynamical flow. Each domain is modeled as a system evolving on a state manifold, governed by vector fields, projection operators, curvature, and stability criteria.
The text does not present these parallels as metaphorical analogies, but as structural identities expressed in differential form.
The accompanying Jupyter notebook contains visual and numerical demonstrations of the following themes:
Vector fields and phase portraits in low-dimensional systems. Tangent planes and local linearization on curved manifolds. Gradient descent interpreted as geometric flow. Fisher information as curvature in parameter space. Nonlinear instability illustrated through Lorenz dynamics. Comparison of sparse and dense Jacobian structure. Simple models of retinal adaptation dynamics. Mechanical analogues inspired by biological membrane elasticity. Entropy decay toward attractor states.
These notebooks are intended as geometric illustrations of theoretical claims rather than independent computational tutorials. Their purpose is to visualize structural principles described in the manuscript.
The project is intended for advanced undergraduate and graduate readers seeking a structurally unified treatment of calculus. Familiarity with linear algebra and elementary analysis is assumed. The text progressively develops manifold theory, dynamical systems, geometric integration, and control.
The framework may be relevant to mathematics, physics, engineering, computational neuroscience, systems biology, and advanced statistical modeling.
The manuscript is a work in progress. The main text and diagram file are currently maintained separately and are being systematically integrated. Section endings are being revised for formal consistency. Citations are being expanded. Figures from the computational notebook are being aligned with corresponding theoretical chapters.
The repository therefore reflects an evolving synthesis rather than a finalized edition.