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## Calculus Definitions in Alphabetical Order

**Click on a term to go to an article and full definition:**

Jump to A B C D E F G H I J K L M N O P Q R S T U V W Y-Z

## A

- Abel’s Inequality: Definition & Proof
- Abel’s Test
- Absolute Minimum
- Absolutely Convergent
- Absolute Value Function
- Acceleration
- Accumulation Point: Definition, Examples
- Algebraic Curve: Definition, Examples
- Algebraic Limit Theorem: Definition, Examples
- Alternating Harmonic Series
- Alternating Series Test
- Alternating Series Remainder
- Amplitude of a Function: Definition, Formula, Example
- Anisotropic
- Annulus
- Ansatz
- Arc Length Formula
- Arcsin
- Asymptote
- Attracting Fixed Point
- Axis of Rotation

## B

- Base Number
- Basin of Attraction
- Bessel Function
- Big O Notation (Landau’s Symbol)
- Bijective Function
- Bisection Method
- Bolzano’s Theorem (Intermediate Zero Theorem)
- Bolzano Weierstrass Theorem
- Boundary Point
- Bounded Function
- Bounded Interval
- Bullet Nose Curve

## C

- Calculus of Variations: Simple Definition
- Cartesian Form
- Cartesian Plane
- Cauchy Principal Value: Definition
- Cavalieri’s Principle
- Centroid
- Chain Rule
- Chebyshev’s Sum Inequality
- Circle of Convergence
- Closed Form Solution
- Closed Interval
- Closed Set
- Closed Surface
- Codomain
- Coefficients
- Cofunction
- Collider Variable
- Common Ratio & Common Difference
- Compact Space: Simple Definition, Examples
- Complex Numbers / Plane
- Composite Function
- Concave Up and Down
- Constant Acceleration
- Constant of Integration
- Constant Term
- Contextual Domain
- Continued Sum
- Conditional Convergence
- Continuous Function, Data
- Converge
- Convergence of Random Variables
- Cost Function
- Critical Numbers
- Cubic Function
- Cumulant Generating Function
- Curly d
- Curvilinear
- Cylindrical Coordinates
- Cylindrical Shell Formula
- Cusps and corners in graphs.

## D

- Damped Sine Wave
- Darboux’s Theorem
- Decimal Expansion: Definition, Examples
- Del Operator (Nabla operator)
- Delta x / Delta y: Definition, Examples
- Definite Integral
- Deleted Neighborhood
- Dependent Variable
- Difference Quotient
- Differentiable
- Differential Approximation
- Differential Forms
- Differential Operator
- Differentiate Definition (“Take the Derivative”)
- Discontinuous Function
- Divergence Theorem: Definition, Examples
- Divergent Series
- Domain of a function
- Double Integral
- Double Points (Math)
- DX

## E

- Einstein Summation (Notation)
- Ellipsoid
- Empirical Rule & Research
- End Behavior
- Endpoint
- Epsilon
- Euclidean Space
- Euler–Maclaurin Summation Formula: Definition
- Euler’s Number (e)
- Even and Odd Functions
- Explicit Function
- Exponential Models, Decay & Growth
- Exponential Function
- Exterior Calculus
- External Secant Segment: Example, Proof
- Explicit Solution
- Extrema of a Function
- Extreme Values of a Polynomial
- Extreme Value Theorem

## F

- Falling Power: Definition
- Family of Functions
- Fibonacci Sequence
- Finite and Infinite Sets
- Finite Calculus (Calculus of Finite Differences)
- First Derivative Test
- Fluxion
- Frustrum
- Fourier Analysis
- Fourth Derivative
- Fractional Calculus
- Frenet Frame: Simple Definition
- Function
- Functional / Higher-Order Functions

## G

- Gamma Function, Multivariate Gamma Function
- Gaussian Distribution, Gaussian Quadrature
- General Solution (Diffeq)
- Geometric Series
- Global Minimum
- Global Maximum
- Gradient
- Gregory–Newton Interpolation Formula

## H

- Half Closed Interval (Half Open)
- Harmonic Mean & Harmonic Progression
- Hermite Polynomials
- Holomorphic Function
- Homogeneous Polynomial
- Horizontal Asymptote
- Horizontal Tangent Line
- Hyperfactorial

## Calculus Definitions: I to R

## I

- Identity Function
- Ill-conditioned
- Imaginary Numbers
- Implied Domain
- Implicit Differentiation
- Independent Variable
- Index Calculus
- Infinite Product
- Infinite Slope Example
- Integral Function
- Integral Transform: Overview & Definition
- Interval Domain
- Index Number
- Inscribed Rectangle & Circumscribed Rectangle
- Indeterminate Expression
- Infinitesimal
- Inflection Point
- Initial Value / Condition
- Injective Function
- Inner Function
- Instantaneous acceleration
- Instantaneous Velocity
- Integer & Non Integer
- Integral Bounds
- Integral Kernel (Symbol)
- Integrand
- Interval of Convergence
- Inverse Functions
- Intermediate Value Theorem
- Intermediate Variable
- Is Infinity a Number?
- Isometry

## J

## K

## L

- Lagrange Interpolating Polynomial
- Lambda Calculus
- Least Upper Bound (Supremum)
- Leminiscate
- Liebniz Notation
- Limiting Process
- Line Segment, Equivalent
- Linear Equation
- Linear Form: Definition, Examples
- Linear Operator
- Linear Term
- Local Behavior
- Local Maximum
- Local Minimum
- Logistic Growth
- Lower Bound, Greatest Lower Bound (GLB) — Infimum
- Linearization & Linear Approximation
- Linearity of Differentiation
- Linearly Independent Solutions
- Logarithm

## M

- MacLaurin Series
- Malliavin Calculus: An Overview
- Manifold
- Many to One
- Map (Mathematics)
- Mean Value Theorem
- Mellin Transform: Definition, Examples, List/Table
- Midpoint Rule
- Min-Max Theorem
- Mode
- Modulo Function
- Modulus of Continuity
- Moment
- Monomial Function
- Monotonic Sequence, Series, Function
- Multivariate

## N

- Natural Exponential Function
- Neighborhood
- Newton Notation
- Non empty set
- Non-Newtonian Calculus
- n-tuple
- Nullcline
- Numerical Integration
- Normal Line
- Normal Order
- nth Degree Taylor Polynomial

## O

- Oblique Asymptote
- One Sided Limit
- Open Interval
- Open Set
- Open Unit Disk
- Ordinary Derivative
- Ornstein-Uhlenbeck Process
- Oscillating Series

## P

- Parabola
- Paraboloid
- Parameterize a Function
- Partial Integration
- Particular Solution (Diffeq)
- Phase Lag Definition
- Piecewise Function
- Plane Curve: Definition, Examples
- Plane Region: Definition, Finding Area: Type I & II
- Polar Coordinates
- Polynomial Functions
- Position Function
- Power Rule
- Preimage & Image
- Prime Notation in Differentiation
- Product Notation (Pi Notation)
- Product Rule
- Profit Function
- Propositional Calculus
- Pseudodifferential Operators
- Punctured Disk

## Q

## R

- Radian
- Radical Function
- Range of a Function
- Ratio Test
- Rational Function
- Real Analysis
- Real Numbers
- Rectifiable Curve
- Regula-Falsi method: Definition, Example
- Relatively Prime (Coprime, Mutually Prime)
- Relative Rate of Change
- Restrictions of a Function: Definition, Examples
- Revenue Function
- Ricci Calculus & Notation
- Riemann Sums
- Rolle’s Theorem
- RTH Moment of a Distribution

## Calculus Definitions: S to W

## S

- Saddle Point
- Scalar Function, Field
- Secant Method
- Second Derivative Test
- Sequence
- Sequence of Partial Sums
- Shear Mapping
- How to Make a Sign Diagram
- Sign Function
- Simple Closed Curve
- Simple Harmonic Motion
- Simpson’s Rule
- Single Variable Calculus
- Singular Point
- Slope / Slope Field
- Smooth Curve: Definitions
- Spherical Coordinates
- Standard Form
- Step Discontinuity
- Subfactorial
- Sufficiently Large
- Summability Theory
- Summation Notation
- Surface of Revolution
- Surjective Function

## T

- Tangent Line Tangent Space
- Tangent Vector (Velocity Vector)
- Taylor’s Inequality
- Taylor Series
- Tensor Definition
- Tetration Function
- Topological Space
- Torus
- Transfinite Numbers
- Transformations
- Trapezoid Rule

## U

## V

- Vector Calculus: Definition
- Vector Function
- Vertex
- Vertical Asymptote
- Vertical Line Test
- Vertical Tangent
- Volume

## W

## X

## Y

## Z

## Sufficiently Large

Loosely speaking, **sufficiently large** means “large enough” or “sufficiently large numbers”. In mathematics though, we want to define things a little more precisely. Exactly what makes a constant, term or other quantity “large enough” really depends where you’re using the expression. It could be very well defined (for example, a quantity greater than 10) or it could be an estimate. In some cases, it might be theoretically possible but not calculable.

## Examples of Sufficiently Large

**Weakly complete sequences: **A weakly complete sequence is one where every “sufficiently large” natural number is a sum of a sequence’s terms [1]. In other words, it’s a sequence that doesn’t seem to be complete at first, but as you travel down the number line (i.e. as the numbers get “large enough”), the sequence meets the definition of completeness. There are an infinite number of possible sequences; What numbers are sufficiently large depends on the specific sequence.

**Hardy-Littlewood conjecture:** This famous theory states that every sufficiently large number (i.e. numbers beyond a certain point) can be expressed as a sum of a square and a prime and every large enough number is the sum of a cube and a prime. This theory was later dropped when Hooley [2] & Linnick [3] proved that a sufficiently large enough integer is the sum of two squares and a prime (assuming the extended Riemann hypothesis) [4]. The important thing here is that it happens at some point; the exact numerical value is largely irrelevant.

A **Haken-manifold **is manifold containing a properly embedded 2-sided incompressible surface; If a 3-manifold meets this property, it’s called sufficiently large [5].

In **statistics**, we’re often concerned with getting a sufficiently large sample: one that’s big enough to represent some aspect of the population (like the mean, for example). See: Large Enough Sample Condition (StatisticsHowTo.com).

## Umbral Calculus

**Umbral calculus** (also called *Blissard Calculus* or *Symbolic Calculus*) is a modern way to do algebra on polynomials. It is a set of exploratory “rules” or a proof technique where indices of polynomial sequences are treated as exponents; Generally speaking, it’s a way to discover and prove combinatorial identities, but it can also be viewed as a theory of polynomials that count combinatorial objects [1].

The name *Umbral Calculus* was invented by Sylvester, “that great inventor of unsuccessful terminology” [2]. The calculus is based around an *umbra*, symbol *B*, which comes from the Latin *umbral*. Although it is a “shady” way to approach problems, it actually works!

## Umbral Calculus Derivation of Bernoulli Numbers

A well known example of umbral notation is the representation of Bernoulli numbers by

(*B* + 1)^{2} – *B*^{n} = 0. After binomial theorem expansion, the B^{k} is replaced with the B_{k} to get a recursive formula for the Bernoulli numbers [2]:

The reason why lowering the index “works” has its roots in expressing an infinite sequence of numbers by a transform [2]. In other words, a linear transform B can be defined as

*Bx*^{n} = *B*_{n}.

The “lowering of the index” uses the relationship (*X* – 1)^{n} = *X*^{n} and adding B to both sides to get *B*(*X* – 1)^{n} = *B*(*X*^{n}).

## Development of Umbral Calculus

Umbral calculus is becoming more well known as it heads towards maturity, with applications in several mathematical areas [3]. For example, umbral calculus has been used to solve martingale problems [4] and recurrences as well as counting lattice paths [5].

Despite its simplicity, the early development of umbral calculus was not without its problems. For example, the following “rule” is what Roman & Rota [3] call “baffling” as seemed to imply that a + a ≠ 2:

## Umbral Calculus: References

[1] Bucchianico, A. (1998). An introduction to Umbral Calculus. Retrieved May 4, 2021 from: https://www.researchgate.net/publication/2471188_An_introduction_to_Umbral_Calculus

[2] Roman, S. and G.-C. Rota (1978). The umbral calculus. Adv. Math. 27, 95–188.

[3] Ray, N. Universal Constructions in Umbral Calculus. Retrieved May 4, 2021 from: http://www.ma.man.ac.uk/~nige/ucuc.pdfH.

[4] Hammouch, H. (2004). Umbral Calculus, Martingales, and Associated Polynomials. Stochastic Analysis and Applications

Volume 22, Issue 2. pp 443-447.

[5]. Humphreys, K. & Niederhausen, H. (2004). Counting lattice paths taking steps in infinitely many directions under special access restrictions. Theoretical Computer Science 319, 385 – 409

## References

[1] Fox, A. & Knapp, M. (2013). A Note on Weakly Complete Sequences. Journal of Integer Sequences.

[2] Hooley, C. (1957). On the representation of a number as the sum of two squares and a prime. Acta, Math. 97, 109-210.

[3] Linnick, J. (1960). An Asymptotic Formula in an Additive Problem of Hardy & Littlewood (Russian). Izv. Akad. Nauk SSR, Ser. Mat. 24. 629-706.

[4] Hardy, G. & Rao, M. Semi r-free and r-free integers; A Unified approach. Canadian Mathematical Bulletin (Sep, 1982).

[5] Waldhausen, F. (1968). On Irreducible 3-Manifolds Which are

Sufficiently Large. Annals of Mathematics, Second Series Volum 87 No. 1. Princeton University.

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