Calculus by the Spark

An Algebraic Approach to Differentiation

Calculus by the Spark presents a different way of learning differential calculus, beginning with elementary algebra and gradually developing the mathematical ideas behind differentiation.

The book introduces the Spark, a symbol ε satisfying ε² = 0, and explores how dual-number algebra can be used to calculate derivatives through familiar algebraic operations.

The approach begins with geometric tangency and double roots, then develops into a systematic treatment of differentiation.

Inside the book:

• Geometric tangency and Kapanadze's double-root criterion.
• Dual numbers and algebraic differentiation.
• Products, quotients, compositions, and elementary functions.
• Linear approximation, optimization, and Newton's method.
• Directional derivatives, gradients, and Jacobians.
• Higher-order derivatives, Hessians, and jets.
• Worked examples, exercises, and illustrated mathematical workshops.
• A Rigor Notebook containing mathematical foundations and proofs.

Across 188 pages, the book combines intuitive explanations, algebraic calculations, geometric diagrams, and formal mathematics.

The underlying mathematical structures are established results. The book's contribution is pedagogical: organizing these ideas into a progressive approach intended to make differential calculus easier to understand and work with.

Written for students, independent learners, and mathematically curious readers who want to explore differentiation from a different perspective.