Calculus glossary

Clear, one-line definitions of the Calculus terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

accumulation-function

The function F(x) = ∫ₐˣ f(t) dt that accumulates the signed area under f from a fixed lower limit a up to x.

arc length

The length of a curve γ: [a, b] → ℝⁿ, computed as ∫_a^b ‖γ'(t)‖ dt.

See: Radians and Arc Length

boundary value problem

A differential equation together with conditions prescribed at the boundary of the domain, rather than at a single initial point.

coordinate function

The i-th projection πᵢ: ℝⁿ → ℝ sending (x₁, ..., xₙ) ↦ xᵢ; the components of a vector-valued function.

derivative

The instantaneous rate of change f'(a) = lim_{h→0} [f(a+h) − f(a)] / h; the slope of the tangent line to f at a.

See: The Derivative, The Exterior Derivative

difference-quotient

The ratio [f(a+h) − f(a)] / h whose limit as h → 0 defines the derivative f'(a).

geometric series

A series whose successive terms share a common ratio r; with first term a, it converges to a/(1−r) when |r|<1.

See: Geometric Series

harmonic

Satisfying Laplace's equation Δf = 0; harmonic functions are smooth and obey the mean value property.

See: The Laplacian and Harmonic Functions

harmonic conjugate

A function v harmonic on the same domain as a harmonic u such that u + iv is holomorphic; v is unique up to a constant.

harmonic function

A twice-differentiable function f with Δf = 0; equals its average over every ball in its domain.

See: The Laplacian and Harmonic Functions

integral-linearity

Integration is linear: ∫(αf + βg) = α∫f + β∫g for any constants α, β and integrable functions f, g.

integral-monotonicity

If f(x) ≤ g(x) on [a, b] and both are integrable, then ∫ₐᵇ f ≤ ∫ₐᵇ g.

jacobian matrix

For f: ℝⁿ → ℝᵐ, the m×n matrix of first partial derivatives [∂fᵢ/∂xⱼ]; the best linear approximation of f at a point.

mean-value-theorem

If f is continuous on [a, b] and differentiable on (a, b), then f'(c) = [f(b) − f(a)] / (b − a) for some c ∈ (a, b).

partial differential equation

An equation relating an unknown function of several variables to its partial derivatives, abbreviated PDE.

vector-valued function

A function r: I → ℝⁿ whose output is a vector; its components are scalar coordinate functions r(t) = (x₁(t), ..., xₙ(t)).