“Cum Deus calculat et cogitationem exercet, fit mundus.”
“When God calculates and thinks things through, the world is made.”— Gottfried Wilhelm Leibniz, co-inventor of calculus
A Conceptual Introduction in Thirteen Ideas
God's Calculus
From change and accumulation to motion in many dimensions
Calculus asks two questions: How fast is something changing? How much has accumulated? Everything else — limits, derivatives, integrals, series, and multivariable calculus — grows from those two ideas. This course is built around thirteen conceptual atoms, not a fixed number of days; some take five minutes, a few take longer, and each carries its own optional toolkit of formulas and practice to go deeper if you want.
The Thirteen Ideas
Thirteen conceptual atoms, organized into three movements that move from foundations to many dimensions. Each is sized to the idea, not to a clock — a handful take five minutes, a few take longer, and every one carries an optional toolkit and practice problem to go deeper if you want. Each links straight to its own lesson — there's no separate track to follow.
Before you begin: what you should already know
No previous calculus is required. The lessons do assume comfort with algebra, function notation, graphing, exponents, and basic trigonometry. Can you confidently simplify (x² − 4)/(x − 2), evaluate f(x + h), recognize the graphs of x², eˣ, and sin x, and interpret slope as rise over run?
If any of that felt shaky, spend twenty minutes reviewing function notation and graphing before Idea 1 — Functions and Graphs is a fast refresher, but it moves quickly into the vocabulary the rest of the course relies on.
How Calculus Fits Together
Three questions define the subject — how fast something is changing, how much has accumulated, and how several changing quantities interact — and this map shows how the branches that answer them connect.
The Architecture of Calculus
Functions provide the language, limits make instantaneous reasoning possible, and the major branches then reconnect through the Fundamental Theorem.
Glossary
Search for any term used in this guide.
The Big Picture
- Functions describe relationships.
- Limits describe approaching behavior.
- Derivatives describe local change.
- Integrals describe accumulated change.
- Series approximate functions using infinite sums.
- Multivariable calculus studies change and accumulation across several dimensions.
- Vector calculus connects local derivatives with global behavior over curves, surfaces, and volumes.
Calculus is humanity’s language of continuous change. It describes the motion of planets, the flow of electricity, the behavior of markets, and the learning of neural networks — one connected way of understanding a universe that never stands still.
“If I have seen further, it is by standing on the shoulders of giants.”— Isaac Newton, 1675, co-inventor of calculus
This course builds conceptual fluency and introductory problem-solving ability in about thirty focused sessions. It is a foundation for deeper study, not a substitute for a full university calculus sequence.
Further Study
This course favors conceptual understanding over exhaustive technique. These are the places to go next for full proofs, more worked problems, and complete university-level coverage.
- Steven Strogatz — Infinite Powers. Best for history and intuition — how these ideas were discovered and why they matter.
- Gilbert Strang — MIT OpenCourseWare 18.01 & 18.02. Best for full university instruction — free complete lecture sequences, single- and multivariable.
- James Stewart — Calculus. Best for practice and technique — the standard textbook, thorough on methods and applications.
- Michael Spivak — Calculus. Best for proofs and rigor — the theory built carefully from the ground up.
- Grant Sanderson (3Blue1Brown) — Essence of Calculus. Best for visual understanding — a free video series close in spirit to this course.