Differentiation, Integration & Limits

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Abstractions about derivatives, antiderivatives, limits, differential operators, integral curves, convergence, and generalized calculi.

15 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Antiderivative — A function whose derivative equals a given function on a declared domain, with all solutions differing by constants on each connected component under standard assumptions.
• Curl (mathematics) — The vector differential operator measuring the local infinitesimal circulation and rotation axis of a three-dimensional vector field.
• Del — The vector differential operator whose combinations with scalar or vector fields denote gradient, divergence and curl.
• Differential of a function — The linear map giving the first-order change of a differentiable function at a point, written df and represented in one variable by dy=f′(x)dx.
• Differential operator — An operator built from derivatives and coefficient functions that maps functions or sections to new functions or sections according to a declared finite order.
• Differentiation of trigonometric functions — The calculus rule family that maps trigonometric functions and their compositions to derivatives through their periodic identities, limit behavior and the chain rule.
• Indeterminate form — A limiting-value pattern whose component limits do not by themselves determine the limit of their arithmetic combination.
• Inflection point — A point on a sufficiently smooth curve where signed curvature changes sign, or on a function graph where local concavity changes from one side to the other.
• Integral curve — A parametrized curve whose tangent at every point equals a specified vector field, representing a solution trajectory of an ordinary differential equation.
• Inverse tangent integral — The special function Ti₂(x)=∫₀ˣ arctan(t)/t dt, equivalently an odd dilogarithmic combination with a characteristic alternating odd-power series.
• Liouville's theorem (differential algebra) — A differential-algebra theorem restricting the form of an elementary antiderivative and thereby proving many elementary functions have no elementary primitive.
• Liouvillian function — A function obtainable through a finite tower of algebraic extensions, exponentials, logarithms and antiderivatives over a differential field.
• One-sided limit — The limiting value approached by a function when its argument tends to a point only through values on one specified side.
• Quantum calculus — Calculus built from finite q-ratio or h-shift difference operators and corresponding sums, recovering ordinary differential and integral calculus as q→1 or h→0 rather than using limits internally.
• Γ-convergence — A variational convergence of functionals defined by a liminf inequality and recovery sequences, designed so minimizers and minimum values behave stably under suitable compactness.