Calculus

calculus is the mathematics of change: it assigns exact meaning to two questions that stumped everyone from zeno to the seventeenth century — how fast is a quantity changing right now? and how much of it has accumulated so far? 𐃏 both questions are answered by the same primitive operation, the limit, applied in two different directions. everything on the pages below is a variation on that one move.

the one idea: limits#

every object in calculus is a limit in disguise. informally, \(\lim_{x \to a} f(x) = L\) means: you can force \(f(x)\) as close to \(L\) as you please by taking \(x\) close enough to \(a\). the precise \(\varepsilon\)–\(\delta\) formulation (and a worked proof) lives on the single-variable page; the full logical foundations — why the real numbers support limits at all — belong to real analysis.

from this primitive, two derived operations:

• the derivative is the limit of secant slopes:

  \begin{equation} f’(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}, \end{equation}

  the instantaneous rate of change — geometrically, the slope of the tangent line that the secants converge to.

• the integral is the limit of rectangle sums as the mesh (the widest \(\Delta x_i\)) shrinks to zero:

  \begin{equation} \int_a^b f(x)\,dx = \lim_{\max_i \Delta x_i \to 0} \sum_{i=1}^{n} f(x_i^*)\,\Delta x_i, \end{equation}

  the accumulated total — geometrically, signed area under the curve.

two operations, one theorem#

differentiation and integration look unrelated — one is local (a slope at a point), the other global (an area over an interval). the fundamental theorem of calculus says they are inverse operations, and it is the reason calculus is a calculus — a system of mechanical rules — rather than a bag of ad-hoc area tricks.

for \(f\) continuous on \([a,b]\):

• part 1 (differentiation undoes accumulation). the accumulation function \(F(x) = \int_a^x f(t)\,dt\) is differentiable, and

  \begin{equation} F’(x) = f(x). \end{equation}

• part 2 (antiderivatives compute integrals). if \(\Phi\) is any antiderivative of \(f\) (i.e. \(\Phi’ = f\)), then

  \begin{equation} \int_a^b f(x)\,dx = \Phi(b) - \Phi(a). \end{equation}

the intuition for part 1 fits in one line: over a tiny window \([x, x+h]\) a continuous \(f\) is nearly constant, so the extra accumulated area is nearly a rectangle, \(F(x+h) - F(x) \approx f(x)\,h\); divide by \(h\) and let \(h \to 0\). 𐃏 part 2 is why the tables of derivatives you memorise pay double: read backwards, they are tables of integrals. proofs and the riemann-sum machinery behind the integral are developed on the single-variable page.

the theorem also scales up: in higher dimensions it reappears as the theorems of green, stokes and gauss — “the integral of a derivative over a region equals the data on the boundary” — the capstone of the multivariable page.

the map#

the child pages, in the order you should read them:

• single-variable calculus — limits made precise (\(\varepsilon\)–\(\delta\)), continuity and its consequences (ivt, evt), differentiation rules with proofs, mean value theorem, taylor’s theorem with error control, l’hopital, riemann integration, both parts of the ftc, integration techniques, improper integrals.
• multivariable calculus — partial derivatives and the gradient, jacobians and hessians, taylor in several variables, unconstrained and constrained optimisation (lagrange multipliers), multiple integrals and change of variables, and vector calculus: div, curl, and the big three integral theorems.
• differential equations — where calculus earns its keep: equations whose unknowns are functions and whose vocabulary is derivatives. odes, solution methods, series solutions, and the frobenius machinery.

adjacent rooms in the wiki:

• real analysis is calculus with the safety off — the same theorems, but proved from the construction of the real numbers, plus the pathologies (nowhere-differentiable continuous functions) that show why the rigour is not optional.
• complex analysis is calculus over \(\mathbb{C}\), where differentiability once means differentiability infinitely often and integrals compute themselves via residues.
• optimisation consumes the derivative wholesale: every gradient method is the linear approximation of this page applied iteratively.

a short history#

newton had the essential ideas — his “method of fluxions” — by 1665–66, working privately during the plague years at woolsthorpe, but published almost nothing for decades. leibniz developed the subject independently around 1675 and published first (1684), contributing the notation that won: \(\tfrac{dy}{dx}\) for the derivative and the elongated-s integral sign \(\int\) for a sum of infinitesimal pieces. 𐃏 both men computed with “infinitesimals” — quantities smaller than any positive number yet not zero — which worked spectacularly and made no logical sense, as bishop berkeley gleefully pointed out in 1734, mocking them as “the ghosts of departed quantities”. the repair took a century and a half: cauchy (1821) reorganised the subject around limits, and weierstrass finally nailed the definition down in the \(\varepsilon\)–\(\delta\) form used today. courant’s classic treatment (Courant, Richard and Robbins, Herbert, 1941) is still one of the best places to watch intuition and rigour being balanced honestly; his two-volume calculus (Courant, Richard, 1996) carries the same programme through several variables.

see also#

• single-variable calculus — the core course
• multivariable calculus — the same ideas in \(\mathbb{R}^n\)
• differential equations — calculus applied to unknown functions
• real analysis — the rigorous foundations
• functions — the raw material

References#

Courant, Richard (1996). Differential and Integral Calculus, Springer.

Courant, Richard and Robbins, Herbert (1941). What is Mathematics?, Oxford University Press.