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Line-art diagram showing a curve with upper and lower sum rectangles from a partition, illustrating the rigorous definition of the integral

What Is an Integral, Really? (It's Not Just "Ar...

A rigorous look at how Spivak's Calculus actually defines the integral through upper and lower sums over partitions — and why continuity is what makes the definition work.

What Is an Integral, Really? (It's Not Just "Ar...

A rigorous look at how Spivak's Calculus actually defines the integral through upper and lower sums over partitions — and why continuity is what makes the definition work.

Line art showing an increasing function f and its inverse f-inverse reflected across the diagonal line y=x, illustrating invertibility in calculus

How Do You Know a Function Has an Inverse? The ...

The horizontal line test is just a picture. Here's the rigorous calculus theorem — tying a derivative's sign to strict monotonicity — that actually proves when a function has an...

How Do You Know a Function Has an Inverse? The ...

The horizontal line test is just a picture. Here's the rigorous calculus theorem — tying a derivative's sign to strict monotonicity — that actually proves when a function has an...

Diagram of a convex curve with a chord connecting two points on it lying above the graph between them, illustrating the definition of convexity from Spivak's Calculus

Convexity & Concavity, Precisely: What Makes a ...

The rigorous definition of a convex function—chords above the graph, tangent lines below it, and the second-derivative test—worked out through Spivak's example f(x) = 1/(1+x^2).

Convexity & Concavity, Precisely: What Makes a ...

The rigorous definition of a convex function—chords above the graph, tangent lines below it, and the second-derivative test—worked out through Spivak's example f(x) = 1/(1+x^2).

Line-art diagram illustrating the Mean Value Theorem: a curve from point a to point b with a straight secant line connecting the endpoints and a dashed tangent line parallel to it touching the curve at an interior point x

The Mean Value Theorem: Why Average Speed Guara...

How Rolle's Theorem and the Mean Value Theorem let you prove, rigorously, that an average rate of change is always achieved exactly at some instant — with the classic driving...

The Mean Value Theorem: Why Average Speed Guara...

How Rolle's Theorem and the Mean Value Theorem let you prove, rigorously, that an average rate of change is always achieved exactly at some instant — with the classic driving...

Diagram of a curve with a tangent line touching at one point (instantaneous slope) contrasted with a dashed secant line spanning a wider interval to a second point (average slope), illustrating the definition of the derivative.

What Is a Derivative, Really? The Precise Meani...

The derivative isn't just "rise over run" — it's a limit of secant slopes, precisely defined so it can fail even at points where a function is continuous. We walk...

What Is a Derivative, Really? The Precise Meani...

The derivative isn't just "rise over run" — it's a limit of secant slopes, precisely defined so it can fail even at points where a function is continuous. We walk...

Newton's F = ma with an arrow to Hamilton's symplectic form, illustrating the arc of Spivak's Physics for Mathematicians

Physics for Mathematicians: Who It's For, What ...

Spivak's Mechanics I is classical mechanics written for people who think in proofs. Who it suits, what to read first, what it covers, and how it sits next to Arnold.

Physics for Mathematicians: Who It's For, What ...

Spivak's Mechanics I is classical mechanics written for people who think in proofs. Who it suits, what to read first, what it covers, and how it sits next to Arnold.