The first part corresponds to Calculus I and, most notably, begins with differentiation rather than limits. Students work with derivatives immediately, appealing to differentials as infinitesimal changes, much as the early developers of calculus did. To reduce cognitive overload at first, the initial functions are restricted to power functions with rational coefficients, so that students can internalize the structural rules instead of immediately being presented with a large catalog of derivatives. Applications such as related rates and optimization appear shortly after, along with higher-order derivatives so that concavity can be used naturally toward those applications. Integration is then introduced as accumulation of change, followed by the Fundamental Theorem of Calculus and an explicit connection between average value and average rates of change. In the chapter on differential equations, exponential and trigonometric functions first emerge as solutions to physical models, and logarithms are introduced shortly thereafter. Only after all of these tools are in place are limits introduced formally as a way to clarify and justify earlier reasoning, so that students better see what purpose they serve. The capstone uses the full development of the course to prove the Fundamental Theorem of Calculus in a deliberate and comprehensive way.

The second part, corresponding to Calculus II, begins with analytic geometry and motion in the plane, providing continuity with earlier material while expanding students’ visual intuition, especially about conic sections. A subsequent chapter on area and volume emphasizes slicing and accumulation, moving students away from the narrow view that integrals simply are ‘about’ area alone. Integration techniques are introduced with deliberate attention to structure — substitution and integration by parts are presented as reversals of differentiation rules, and more advanced strategy appears later in a section devoted to reasoning about the form of an integrand. A separate chapter addresses situations in which closed-form antiderivatives do not exist and numerical methods become necessary, including applications such as to probability. The series chapter departs most sharply from tradition — rather than beginning with sequences and convergence tests, it begins immediately with Taylor series as practical tools for approximation and integration, and questions of convergence arise naturally when needed to explain unexpected behavior. This approach creates space for hyperbolic functions and even generating functions, and it culminates in a capstone on Fourier analysis.

The third part, corresponding to Calculus III, expands calculus into higher dimensions. It begins by examining what the idea of ‘dimension’ means and how familiar ideas extend into unfamiliar settings. After functions of multiple variables and their continuity have been discussed, partial derivatives are introduced through a number of applications, and multivariable Taylor approximations and foundational optimization techniques are included. The chapter on vectors moves past thinking them as lists of numbers or functions and toward treating them as mathematical objects in their own right. These assist with introducing gradients and directional derivatives to show how multivariable functions change in different ways. Constrained optimization and Lagrange multipliers are more carefully developed, and the Lagrangian function is interpreted in terms of sensitivity, giving meaning to Lagrange multipliers instead of throwing them out as is often done in standard calculus texts. Multiple integrals are developed exclusively in Cartesian coordinates to reinforce the idea of reducing complex regions to iterated problems, with applications such as center of mass and probability. The capstone introduces calculus of variations, where optimization occurs over infinite-dimensional spaces of functions.

The fourth part expands the usual treatment of vector calculus by interleaving complex variables at natural points. It begins with complex functions, focusing on building intuition through both algebraic manipulations and domain coloring. The next chapter introduces vector fields and dynamical systems, where divergence and curl are introduced as new kinds of derivatives, and complex differentiability is linked to physical interpretations such as incompressible and irrotational flow. Coordinate transformations revisit complex functions as conformal mappings of the plane, using the Jacobian matrix as a unifying concept that describes how curves and regions are reshaped, which also leads naturally into changes of variables in integration. Line and surface integrals are developed carefully, with scalar and vector cases each given their own space, and contour integrals incorporated into the same framework. Finally, the fundamental theorems of Green, Gauss, and Stokes are explored side by side, and Cauchy’s integral theorem leads to the powerful calculus of residues. The capstone introduces differential forms and exterior calculus as a unified language for all of these results.

The final part of this book turns to real analysis. Students have already worked extensively with derivatives, integrals, and series, so here, the aim is to revisit those tools with careful attention to foundations. The opening chapter motivates mathematical rigor through counterintuitive examples from Fourier analysis and begins to develop structural properties of the real numbers. Subsequent chapters treat sequences and limits, differentiability, and conditions for integrability. Compactness, uniform continuity, and Lebesgue measure appear where they are needed to clarify earlier ideas. The final chapter studies series and the interchange of limiting processes, addressing the foundational issues that originally prompted the development of analysis. The capstone introduces non-standard analysis as an alternate framework for justifying calculus, showing that the infinitesimal language from the beginning of the project can be made precise in a modern setting, while also pointing out the sophistication of the machinery needed to establish these results.