Calculus

Recall (doc/postdoc). Not a first course — single- and multi-variable toolkit as used in papers.

Job: limits, derivatives, integrals, and series as the working language of real analysis, differential geometry, and PDE — without \(\varepsilon\)-theater; hypotheses for interchange of limits made explicit.

Core

Calculus is local linear approximation (\(Df\)) plus global integral reconstruction (FTC / Stokes). Multivariable calculus is the finite-dimensional case of Banach calculus; series and uniform convergence control when one may pass limits inside derivatives and integrals.

Layer Content
Limits / continuity sequential and topological; uniform continuity on compacts
Differentiation \(1\)D derivative; Fréchet \(Df\) in \(\mathbb{R}^n\); Taylor with remainder
Integration Riemann \(\to\) Lebesgue; FTC; change of variables
Series tests; power series; radius; termwise differentiation
Vector calc \(\nabla\), \(\operatorname{div}\), \(\operatorname{curl}\); classical Stokes package
DE exit ODE/PDE as calculus + structure \(\to\) dedicated domains

Reading map

Page Focus
Limits sequential/topological limits; continuity
Derivatives Fréchet derivative in \(1\)D; Taylor
Integrals Riemann/Lebesgue FTC; techniques
Sequences & series convergence tests; power series
Multivariable \(\nabla\), \(Df\), Stokes package
Differential equations dedicated ODE domain (first/second order, Laplace)
Real analysis Completeness, uniform convergence, interchange
Measure theory Lebesgue integral; DCT/MCT

Dependency order

  1. Limits \(\to\) Derivatives / Sequences & series.
  2. Integrals (FTC links derivative and integral).
  3. Multivariable \(\to\) differential geometry / PDE.
  4. Upgrade measure-theoretic integration via measure theory.

Interfaces

Area Contact
Topology / real analysis Metric completeness; uniform limits
Differential geometry Charts; \(d\), exterior calculus; geodesics
Complex analysis Holomorphic \(\Rightarrow\) \(C^\omega\); Cauchy integral
Optimization Gradients, Hessians, constrained extrema
Fourier analysis Series/transform as integral operators

Notation (paper-standard)

  • \(f'\), \(Df\), \(\nabla f\), \(D^2f\) (Hessian); \(o(\|h\|)\), \(O(\|h\|^k)\)
  • \(\int_a^b\), \(\int_\Omega\), change of variables \(|\det D\Phi|\)
  • \(\sum a_n\), power series \(\sum c_n(z-a)^n\), radius \(R\)
  • \(d\omega\), \(\int_M\), Stokes \(\int_{\partial M}\omega=\int_M d\omega\)

Density

\[ f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h},\quad \int_a^b f=F(b)-F(a),\quad \sum a_n\ \text{tests},\quad \nabla\times\nabla f=0,\quad \int_{\partial M}\omega=\int_M d\omega \]

Inverse / implicit function theorems (one-liner): if \(Df_{x_0}\) is invertible (resp. surjective with split kernel), \(f\) is a local \(C^k\) diffeomorphism (resp. local \(C^k\) submersion) — manifold charts and Lagrange multipliers sit here.

See it

Interchange of limit and derivative. Pointwise \(f_n\to f\) and pointwise \(f_n'\to g\) does not imply \(f'=g\). Standard fix: uniform convergence of \(f_n'\) on compact intervals (or dominated convergence in the Lebesgue setting). Counterexamples are routine in Fourier series (differentiated series) and in approximation theory — papers always name the mode of convergence.

Check

Which hypothesis makes \(\lim f_n' = (\lim f_n)'\) fail in general?

Uniform convergence of \(f_n'\) (or an \(L^1\) domination) is the usual sufficient condition; pointwise convergence of derivatives alone is not enough.

State the change-of-variables formula for a \(C^1\) diffeomorphism \(\Phi:U\to V\subset\mathbb{R}^n\).

\(\int_V f=\int_U (f\circ\Phi)\,|\det D\Phi|\) for integrable \(f\) (Lebesgue; absolute value of Jacobian determinant). Orientation-sensitive forms drop the absolute value and use \(\Phi^*\mathrm{vol}\).

Pitfalls

  • Pathwise partials exist \(\not\Rightarrow\) differentiable as a map \(\mathbb{R}^n\to\mathbb{R}^m\); need continuous partials or the full Fréchet limit.
  • Riemann vs Lebesgue: improper Riemann integrals and conditional convergence behave badly under rearrangement; Fubini needs absolute integrability or positive integrands.
  • Stokes requires orientation and boundary regularity (or currents / GMT for nonsmooth sets).

Refs

  • Rudin, Principles of Mathematical Analysis
  • Spivak, Calculus on Manifolds (Stokes package)
  • Apostol, Mathematical Analysis
  • Zorich, Mathematical Analysis I–II

Next

Limits · Derivatives · Multivariable · Real analysis · Measure theory · DE foundations