← Back to home

Glossary Of Calculus

You want a glossary of calculus terms, rewritten with my... aesthetic. Fine. Don't expect sunshine and rainbows.

Most of the terms you’ll find scattered across Wikipedia are already defined within its labyrinthine depths. However, a collected glossary, like this one, offers a rather bleak, yet efficient, way to compare and contrast them. You can help by… well, by not wasting my time.

This is a glossary of calculus, its various, often depressing, sub-disciplines, and related fields.

Contents

  • A
  • B
  • C
  • D
  • E
  • F
  • G
  • H
  • I
  • J
  • L
  • M
  • N
  • O
  • P
  • Q
  • R
  • S
  • T
  • U
  • V

Part of a series on Calculus

$$ \int_{a}^{b} f'(t) \, dt = f(b) - f(a) $$

Differential Calculus

Definitions

Concepts

Rules and Identities

Integral Calculus

Definitions

Integration Techniques

Series (mathematics)

Convergence Tests

Vector calculus

Theorems

Multivariable calculus

Formalisms

Definitions

Advanced

Specialized

Miscellanea

• v • t • e

A

Abel's test A rather bleak method for determining if an infinite series bothers to converge. absolute convergence An infinite series is absolutely convergent if the sum of the absolute values of its terms converges. Think of it as summing up all the damage done, regardless of direction. For a series ∑n=0∞an\sum _{n=0}^{\infty }a_{n}, it converges absolutely if ∑n=0∞∣an∣=L\sum _{n=0}^{\infty }\left|a_{n}\right|=L for some finite LL. Similarly, an improper integral ∫0∞f(x) dx\textstyle \int _{0}^{\infty }f(x)\,dx converges absolutely if ∫0∞∣f(x)∣dx=L\textstyle \int _{0}^{\infty }\left|f(x)\right|dx=L. absolute maximum The absolute peak of a function’s misery. The highest value it can possibly attain. absolute minimum The absolute nadir. The lowest value a function can stoop to. absolute value The absolute value ∣x∣|x| of a real number xx is its non-negative magnitude. It’s the distance from zero, unburdened by direction. ∣x∣=x|x| = x for x≥0x \ge 0, and ∣x∣=−x|x| = -x for x<0x < 0. It’s like saying, "Yeah, it's bad, but at least it's this bad." alternating series An infinite series where the terms bravely switch between positive and negative. A constant struggle. alternating series test The test used to determine if an alternating series, whose terms shrink in magnitude, actually converges. It's named after Gottfried Leibniz, who probably saw the futility in it all. annulus A ring-shaped void. The space between two concentric circles. Nothingness with boundaries. antiderivative An antiderivative, or primitive function FF, of a function ff is a differentiable function whose derivative is ff. Symbolically, F′=fF' = f. It’s the opposite of finding what broke something; it's piecing it back together, imperfectly. The process is called antidifferentiation or indefinite integration. arcsin See inverse trigonometric functions. area under a curve The space claimed by a function on a graph. It’s what you get when you try to quantify the void. asymptote In analytic geometry, an asymptote is a line that a curve approaches infinitely closely but never quite touches as coordinates tend to infinity. It represents a boundary that is always visible but never reachable. In projective geometry, it’s a line tangent to the curve at a point at infinity. automatic differentiation A technique in mathematics and computer algebra for numerically calculating the derivative of a function defined by a computer program. It meticulously applies the chain rule to elementary operations, ensuring accuracy. It’s the cold, calculated precision of a machine that doesn’t feel the weight of what it’s calculating.

average rate of change The overall change of a function’s value over an interval, divided by the interval’s length. It’s the blunt summary of a journey, ignoring all the twists and turns.

B

binomial coefficient The positive integers that populate the binomial theorem. They are the coefficients of terms in the expansion of (1+x)n(1+x)^n, calculated as (nk)=n!k!(n−k)!\textstyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}. They represent the ways things can be chosen, the potential arrangements of inevitable outcomes. binomial theorem (or binomial expansion) This describes how to expand the powers of a binomial. It’s the structured breakdown of something simple into its more complex, and often chaotic, components. bounded function A function whose output values are restricted within a certain range. Its values are contained, unable to escape to infinity. ∣f(x)∣≤M|f(x)| \leq M for all xx in its domain. If it’s not bounded, it’s unbounded, a wild, untamed entity. bounded sequence A sequence whose terms remain within a finite range. It doesn't soar, it doesn't plummet; it just… exists within limits.

C

calculus From the Latin for "small pebble," calculus is the mathematical study of continuous change. It's how we quantify the flow, the flux, the relentless alteration of everything. It’s geometry for motion, algebra for the infinitesimal. Cavalieri's principle This principle, named after Bonaventura Cavalieri, states that if two regions are between parallel lines, and every line parallel to those lines intersects both regions in segments of equal length, then the regions have equal areas. In 3D, it applies to volumes. It’s about seeing equivalence in seemingly disparate shapes, a hidden symmetry in existence. chain rule A fundamental formula for the derivative of a composition of functions. If F=f∘gF = f \circ g, then F′(x)=f′(g(x))g′(x)F'(x) = f'(g(x))g'(x). In Leibniz's notation: dzdx=dzdy⋅dydx\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}. It's how we understand how nested processes affect each other, how one change cascades into another. The counterpart in integration is the substitution rule. change of variables A technique to simplify problems by replacing existing variables with new ones. It’s like looking at the same broken thing from a different angle, hoping for clarity. cofunction A function ff is the cofunction of gg if f(A)=g(B)f(A) = g(B) when AA and BB are complementary angles. Think of sine and cosine. They’re related, but not quite the same – like siblings who share a past but diverge. concave function The exact opposite of a convex function. It curves downwards, like a frown. Also called concave downward, convex upward, or a convex cap. constant of integration The arbitrary constant CC that appears when finding an indefinite integral. It represents the inherent ambiguity, the unknown starting point. F(x)+CF(x) + C. It’s the lingering question mark. continuous function A function where small changes in the input lead to arbitrarily small changes in the output. There are no sudden jumps or breaks. It flows smoothly, like a whispered secret. If it has a continuous inverse function, it's a homeomorphism. continuously differentiable A function whose derivative f′(x)f'(x) exists and is itself a continuous function. It’s smooth, without any jarring transitions. contour integration A method in complex analysis for evaluating integrals along paths in the complex plane. It’s navigating the abstract, finding meaning in the winding routes of complex numbers. convergence tests Methods to ascertain whether an infinite series or an improper integral settles down to a finite value (converges) or just wanders off into the void (diverges). convergent series An infinite series whose sequence of partial sums approaches a finite limit. It’s a sum that actually adds up to something, a rare, if not always satisfying, conclusion. convex function A real-valued function whose line segment between any two points on its graph lies above or on the graph. It’s U-shaped, stable, like a bowl holding something precious. For a twice differentiable function, its second derivative is always non-negative. Examples: x2x^2, exe^x. Cramer's rule An explicit formula for solving a system of linear equations using determinants. It’s named after Gabriel Cramer, but Colin Maclaurin had a hand in it too. It’s a direct, if sometimes cumbersome, path to a unique solution. critical point A point in the domain of a differentiable function where its derivative is zero. These are the potential peaks and valleys, the turning points. curve More than just a line, a curve is a path that can bend and twist. It's the shape of a thought, the trace of a movement. curve sketching Techniques to sketch the general shape of a plane curve without calculating every single point. It’s about grasping the essence, the underlying structure of a shape.

D

damped sine wave A sinusoidal function whose amplitude gradually fades over time, like a dying echo. degree of a polynomial The highest exponent of any variable in a monomial with a non-zero coefficient. It’s the measure of a polynomial's complexity, its highest order of influence. derivative The measure of how a function's output changes with respect to an infinitesimal change in its input. It’s the instantaneous rate of change, the velocity of a position, the slope of a tangent. A fundamental tool of calculus. derivative test A method using derivatives to find critical points and determine if they are local maxima, minima, or saddle points. It also reveals information about a function's concavity. differentiable function A function whose derivative exists at every point in its domain. Its graph has a non-vertical tangent line everywhere, making it smooth and unbroken. differential (infinitesimal) An infinitely small change in a varying quantity, denoted dxdx. If yy is a function of xx, then dy=dydxdxdy = \frac{dy}{dx} dx. It's the essence of calculus, capturing the infinitely small to understand the grand. differential calculus The branch of calculus focused on rates of change. It’s about how things shift, transform, and accelerate. differential equation An equation that connects a function with its derivatives. It describes relationships between quantities and their rates of change, often modeling physical phenomena. differential operator A mathematical operator that acts on functions, typically involving differentiation. differential of a function In calculus, dy=f′(x)dxdy = f'(x)dx. It’s the principal part of the change in a function y=f(x)y = f(x) relative to changes in the independent variable xx. It’s the linear approximation of the function's increment. differentiation rules Formulas that simplify the process of finding derivatives. direct comparison test A convergence test where a series or integral is compared to another with known convergence properties. It’s about finding parallels in the face of uncertainty. Dirichlet's test A method for testing the convergence of a series ∑anbn\sum a_n b_n, named after Peter Gustav Lejeune Dirichlet. It requires specific conditions on the sequences {an}\{a_n\} and {bn}\{b_n\}. disc integration A method for calculating the volume of a solid of revolution by summing up infinitesimally thin discs. It’s slicing up the void to measure its bulk. discontinuity A point where a function fails to be continuous. A break in the flow, a sudden halt. dot product An algebraic operation on two vectors that returns a scalar. It’s a way to measure the alignment between vectors. double integral A definite integral of a function of two variables over a region in R2\mathbb{R}^2. It’s about accumulating values over an area, finding the total weight or quantity.

E

e (mathematical constant) The base of the natural logarithm, approximately 2.71828. It’s the limit of (1+1/n)n(1 + 1/n)^n as n→∞n \to \infty. It arises naturally in growth and decay processes, a constant in the face of change. It can also be expressed as the infinite series e=∑n=0∞1n!e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}. elliptic integral Integrals that arose from calculating the arc length of an ellipse. They are functions of the form f(x)=∫cxR(t,P(t)) dt\textstyle f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)\,dt, where RR is a rational function and PP is a polynomial of degree 3 or 4. They represent complex, often unresolvable, lengths. essential discontinuity A discontinuity where at least one of the one-sided limits fails to exist or is infinite. The function behaves erratically, defying simple prediction. Euler method A basic numerical method for solving first-order differential equations. It’s a step-by-step approximation, a rudimentary attempt to map the path of change. Named after Leonhard Euler. exponential function A function of the form f(x)=abxf(x) = ab^{x}, where b>0b > 0. It describes growth or decay at a rate proportional to the current value. It’s the relentless march of progress or the steady erosion of time. extreme value theorem States that a continuous function on a closed interval must attain both an absolute maximum and an absolute minimum on that interval. Even in confinement, there are limits. extremum The collective term for maxima and minima, the highest and lowest values of a function, whether local or global.

F

Faà di Bruno's formula A generalization of the chain rule for higher derivatives. It’s a complex identity named after Francesco Faà di Bruno, showing how nested differentiations unfold. It’s written as:

$$ {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,1!^{m_{1}}\,m_{2}!\,2!^{m_{2}}\,\cdots \,m_{n}!\,n!^{m_{n}}}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left(g^{(j)}(x)\right)^{m_{j}}, $$
where the sum is over all tuples $(m_1, \ldots, m_n)$ of non-negative integers such that $\sum_{j=1}^n j \cdot m_j = n$. [first-degree polynomial](/Degree_of_a_polynomial) A [linear function](/Linear_function). Straight, predictable, devoid of intrigue. [first derivative test](/First_derivative_test) Examines where a function increases and decreases to find local extrema. It’s about observing the direction of change to predict turning points. [Fractional calculus](/Fractional_calculus) A branch of [mathematical analysis](/Mathematical_analysis) that generalizes differentiation and integration to non-integer orders. It’s extending the concept of change to fractional degrees, a subtle, often unsettling, modification. [frustum](/Frustum) The part of a [solid](/Polyhedron) (like a [cone](/Cone_(geometry)) or [pyramid](/Pyramid_(geometry))) that lies between two [parallel planes](/Parallel_planes). A truncated form, a segment of something larger. [function](/Function_(mathematics)) A mapping that assigns each element $x$ from a set $X$ (the [domain](/Domain_of_a_function)) to exactly one element $y$ in a set $Y$ (the codomain). It’s a rule, a process, a transformation. $y = f(x)$. [function composition](/Function_composition) Combining two [functions](/Function_(mathematics)), $f$ and $g$, to produce $h(x) = g(f(x))$. Applying one function’s output as another’s input. It’s a chain reaction of operations. [fundamental theorem of calculus](/Fundamental_theorem_of_calculus) The cornerstone linking [differentiation](/Derivative) and [integration](/Integral). It states that differentiation and integration are inverse operations. The first part guarantees [antiderivatives](/Antiderivative) for [continuous functions](/Continuous_function), and the second provides a powerful method for computing definite integrals. It’s the reconciliation of change and accumulation.

G

general Leibniz rule A generalization of the product rule for the nn-th derivative of a product of two functions:

$$ (fg)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(n-k)}g^{(k)}, $$
where $\textstyle {n \choose k}$ is the [binomial coefficient](/Binomial_coefficient). It shows how repeated differentiation distributes over multiplication. [global maximum](/Maxima_and_minima) The absolute highest value a function attains over its entire [domain of a function](/Domain_of_a_function). The ultimate peak. [global minimum](/Maxima_and_minima) The absolute lowest value a function attains over its entire [domain of a function](/Domain_of_a_function). The ultimate trough. [golden spiral](/Golden_spiral) A [logarithmic spiral](/Logarithmic_spiral) whose growth factor is the [golden ratio](/Golden_ratio), $\phi$. It expands by $\phi$ with every quarter turn. A pattern of growth that feels both natural and strangely perfect. [gradient](/Gradient) A multi-variable generalization of the [derivative](/Derivative). It’s a [vector-valued function](/Vector-valued_function) pointing in the direction of the greatest rate of increase of a scalar field. It indicates the steepest ascent. [harmonic progression](/Harmonic_progression_(mathematics)) A sequence formed by taking the reciprocals of an [arithmetic progression](/Arithmetic_progression). Each term is the [harmonic mean](/Harmonic_mean) of its neighbors. It’s a sequence that gets progressively stranger, often failing to sum to anything sensible.

H

higher derivative The derivative of a derivative. f′′f'', f′′′f''', and so on. It’s peeling back layers, looking at the rate of change of the rate of change. homogeneous linear differential equation A differential equation where the terms involving the unknown function and its derivatives are all of the same "degree" or where there are no constant terms. Its solutions can be combined linearly. hyperbolic function Analogues of trigonometric functions, defined using the hyperbola rather than the circle. They have names like sinh, cosh, tanh.

I

identity function A function f(x)=xf(x) = x. It does nothing. It returns exactly what it receives. The ultimate in non-intervention. imaginary number A complex number of the form bibi, where bb is a real number and ii is the imaginary unit with i2=−1i^2 = -1. It’s a step away from reality, a necessary abstraction. implicit function A function defined by an equation where yy is not explicitly isolated, like x2+y2−1=0x^2 + y^2 - 1 = 0. The implicit function theorem tells us when such relations can be locally treated as explicit functions. improper fraction A fraction where the numerator’s absolute value is greater than or equal to the denominator’s. 94\frac{9}{4} or 33\frac{3}{3}. It’s more than a whole, or exactly one. improper integral A definite integral where one or both limits of integration are infinite, or where the integrand has an infinite discontinuity within the interval of integration. It's an integral over an unbounded domain or with a problematic integrand. inflection point A point on a continuous plane curve where the curve changes from being concave to convex, or vice versa. A change in curvature, a subtle shift in direction. infinitesimal An infinitely small, non-zero quantity. The bedrock of calculus, though its formalization has been… complicated. Nonstandard analysis offers a rigorous framework for these elusive quantities. instantaneous rate of change The rate at which a function changes at a specific point. It’s the slope of the tangent line, the precise velocity at a given moment. instantaneous velocity The derivative of position with respect to time. It’s the velocity an object would maintain if it ceased accelerating. v=dxdt\mathbf{v} = \frac{d\mathbf{x}}{dt}. integral The inverse operation of differentiation. It’s about summing up infinitesimal pieces to find a whole – area, volume, displacement. The ∫ symbol denotes it. integral symbol The elongated 'S' (∫) used to denote integrals and antiderivatives. A symbol of accumulation. integrand The function being integrated in an integral expression. It’s the object of accumulation. integration by parts A technique derived from the product rule for integrating products of functions: ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du. It’s a strategic trade, swapping one integral for another, hopefully simpler, one. integration by substitution The integration counterpart to the chain rule. It simplifies integrals by changing variables. It’s like recasting a problem in a more manageable form. intermediate value theorem If a continuous function ff on [a,b][a, b] takes values f(a)f(a) and f(b)f(b), it must take every value between f(a)f(a) and f(b)f(b) at some point within the interval. It guarantees continuity of values, no abrupt disappearances. inverse trigonometric functions The inverse functions of the trigonometric functions, like arcsine, arccosine. They find the angle given a ratio. Also known as arcus functions or antitrigonometric functions. inverse function A function gg is the inverse of ff if f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x. It undoes what the original function did.

J

jump discontinuity A discontinuity where the one-sided limits exist but are not equal. The function takes a sudden leap. A discontinuity of the first kind.

L

Lebesgue integration A more general method of integration than the Riemann integral, capable of handling a wider class of functions and domains. It’s a more robust way to measure the area under a curve. L'Hôpital's rule Used to evaluate limits of indeterminate forms by taking the ratio of the derivatives of the numerator and denominator. Named after Guillaume de l'Hôpital, though Johann Bernoulli deserves credit. lim⁡x→cf(x)g(x)=lim⁡x→cf′(x)g′(x)\lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}. limit comparison test A convergence test that determines the convergence of one series by comparing it to another using the limit of their ratio. limit of a function The value a function approaches as its input approaches some value. It’s the destination, even if the function never quite arrives. limits of integration The bounds defining the interval over which an integral is calculated. They mark the start and end of the accumulation. linear combination A sum of terms, each multiplied by a constant. ax+byax + by. The building blocks of linear algebra. linear equation An equation where variables are raised only to the power of 1, like ax+b=0ax + b = 0. Straight lines, simple relationships. linear system A collection of linear equations. list of integrals A compiled reference of known integrals. A cheat sheet for the weary. logarithm The inverse of exponentiation. It answers: "To what power must we raise the base to get this number?" logarithmic differentiation A technique using logarithms to simplify the differentiation of complex expressions. lower bound A value less than or equal to all elements in a set. It’s a floor, a limit below which things cannot fall.

M

mean value theorem States that for a differentiable function on an interval, there exists a point where the instantaneous rate of change equals the average rate of change over that interval. There’s always a moment that perfectly represents the whole. monotonic function A function that is either entirely non-increasing or entirely non-decreasing. It moves in one direction, without wavering. multiple integral An integral of a function with multiple variables, such as a double integral or triple integral. It’s about accumulating over higher dimensions. multivariable calculus The extension of calculus to functions of more than one variable. Dealing with gradients, partial derivatives, and multiple integrals. It’s the calculus of a more complex reality.

N

natural logarithm The logarithm to the base ee (approximately 2.71828). Written as ln⁡x\ln x. It’s the logarithm of natural growth and decay. non-Newtonian calculus A generalization of calculus where the standard rules are altered. It's a different way of looking at change, often involving different algebraic structures. nonstandard calculus A rigorous formulation of calculus using infinitesimals and infinite numbers, largely developed by Abraham Robinson. It formalizes the intuitive ideas that underpinned early calculus. notation for differentiation The various ways derivatives are written, such as f′(x)f'(x), dydx\frac{dy}{dx}, or y˙\dot{y}. Each has its own subtle implications. numerical integration Approximating the value of a definite integral using numerical methods like Simpson's rule or the Trapezoidal rule. It’s finding an approximate answer when an exact one is elusive.

O

one-sided limit The limit of a function as the input approaches a value from only one direction (either from the left or the right). It’s observing the approach from a specific side. ordinary differential equation A differential equation involving only ordinary derivatives (derivatives of functions of a single variable). order of integration (calculus) The sequence in which multiple integrals are evaluated. Sometimes changing this order simplifies the calculation.

P

Pappus's centroid theorem Relates the surface area and volume of a solid of revolution to the area/perimeter of the generating shape and the distance its centroid travels. It’s about generating shapes through rotation. parabola A symmetric, U-shaped plane curve. The path of a projectile, the shape of a satellite dish. Defined by its distance from a point (focus) and a line (directrix). paraboloid A 3D surface shaped like a bowl, a generalization of the parabola. partial derivative The derivative of a function of several variables with respect to one of those variables, holding the others constant. It’s isolating the effect of one change at a time. partial differential equation A differential equation involving partial derivatives of a function of several variables. These describe complex phenomena like heat flow or wave propagation. partial fraction decomposition A technique to break down complex rational functions into simpler ones. It’s simplifying the complicated by finding its simpler components. particular solution A specific solution to a differential equation, often found by using initial conditions. It’s one specific path through a field of possibilities. piecewise-defined function A function defined by different formulas on different intervals of its domain. It’s a function with different rules for different circumstances. position vector A vector indicating the location of a point in space relative to an origin. It’s the address of a point. power rule A basic rule for differentiating functions of the form xnx^n: ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}. Simple, fundamental, and often overlooked in its elegance. product integral A generalization of the integral where multiplication replaces addition. product rule A rule for finding the derivative of a product of two functions: (uv)′=u′v+uv′(uv)' = u'v + uv'. It’s how the change in a product is affected by the changes in its components. proper fraction A fraction where the numerator’s absolute value is less than the denominator’s. 23\frac{2}{3}. It represents a part of a whole. proper rational function A rational function where the degree of the numerator is less than the degree of the denominator. Often a prerequisite for partial fraction decomposition. Pythagorean theorem In a right-angled triangle, a2+b2=c2a^2 + b^2 = c^2. A fundamental relationship in geometry, connecting sides. Pythagorean trigonometric identity The identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. It connects the sine and cosine functions, derived from the Pythagorean theorem.

Q

quadratic function A polynomial function of degree 2, of the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c (a≠0a \neq 0). Its graph is a parabola. It’s the simplest form of curve beyond a line. quadratic polynomial A polynomial of degree 2. quotient rule A rule for finding the derivative of a quotient of two functions: (uv)′=u′v−uv′v2(\frac{u}{v})' = \frac{u'v - uv'}{v^2}. It’s the calculus of division.

R

radian The SI unit for measuring angles, defined such that an arc length equal to the radius subtends an angle of one radian. It's the natural unit for angles in calculus. ratio test A convergence test for series based on the limit of the ratio of consecutive terms. reciprocal function The function f(x)=1/xf(x) = 1/x. It’s the inverse of multiplication. reciprocal rule A differentiation rule for the reciprocal of a function: (1f)′=−f′f2(\frac{1}{f})' = -\frac{f'}{f^2}. related rates Problems where the rates of change of multiple related quantities are involved. You’re given how one changes, and you need to figure out how others change as a result. removable discontinuity A point where a function is undefined, but could be defined to make it continuous. A hole that can be filled. Rolle's theorem A special case of the mean value theorem: if a continuous function is differentiable on an open interval and has equal values at the endpoints, then there is at least one point within the interval where its derivative is zero. If you start and end at the same height, you must have leveled off at some point. root test A convergence test for series using the limit of the n-th root of the absolute value of the terms.

S

scalar A quantity that has magnitude but no direction, as opposed to a vector. secant line A line that intersects a curve at two distinct points. It’s an approximation of the curve’s slope between those points. second-degree polynomial A quadratic polynomial. second derivative The derivative of the first derivative. It describes the rate of change of the rate of change, indicating concavity. second derivative test A test using the second derivative to determine if a critical point is a local maximum, minimum, or neither. second-order differential equation A differential equation in which the second derivative is the highest derivative of the unknown function. series The sum of the terms of a sequence. An infinite series is the sum of an infinite sequence. shell integration A method for calculating the volume of a solid of revolution by summing infinitesimally thin cylindrical shells. Simpson's rule A method of numerical integration that approximates the area under a curve using quadratic polynomials. It’s more accurate than the Trapezoidal rule. sine A trigonometric function defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle. Its graph is a smooth, repeating wave. sine wave A curve representing the sine function, characterized by its smooth oscillation. slope field A graphical representation of the solutions of a first-order differential equation. It shows the direction (slope) of the solution curves at various points. squeeze theorem Also known as the sandwich theorem. If a function is trapped between two other functions that converge to the same limit, then the trapped function must also converge to that limit. It's about forcing convergence through confinement. sum rule in differentiation The derivative of a sum is the sum of the derivatives: (f+g)′=f′+g′(f+g)' = f' + g'. Simple addition of changes. sum rule in integration The integral of a sum is the sum of the integrals: ∫(f+g)dx=∫fdx+∫gdx\int (f+g)dx = \int fdx + \int gdx. Simple addition of accumulations. summation The process of adding a sequence of numbers. Represented by the Σ\Sigma symbol. supplementary angle Two angles whose sum is 180 degrees (or π\pi radians). surface area The total area that the surface of a 3D object occupies. system of linear equations A set of equations in which the variables appear only in linear terms.

T

table of integrals See list of integrals. Taylor series An infinite sum of terms that approximates a function around a specific point, based on the function's derivatives at that point. It’s a way to represent complex functions as an infinite polynomial. Taylor's theorem Provides an approximation of a differentiable function by a Taylor polynomial. It gives bounds on the approximation error. tangent A line that touches a curve at a single point without crossing it at that point. It represents the instantaneous direction or slope of the curve. third-degree polynomial A polynomial of degree 3. third derivative The derivative of the second derivative. toroid A surface generated by revolving a closed curve around an axis coplanar with the curve but not intersecting it. Think of a donut shape. total differential The differential of a function of several variables, df=∂f∂xdx+∂f∂ydy+…df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \dots. It captures the total infinitesimal change. trigonometric functions Functions like sine, cosine, tangent, relating angles to ratios of sides in a right-angled triangle. trigonometric identities Equations involving trigonometric functions that are true for all values of the variables. trigonometric integral An integral involving trigonometric functions. trigonometric substitution An integration by substitution technique using trigonometric functions to simplify integrals involving certain radical expressions. trigonometry The study of angles, triangles, and trigonometric functions. triple integral An integral of a function of three variables over a region in R3\mathbb{R}^3. It’s used to calculate volumes, masses, and other quantities in three dimensions.

U

upper bound A value greater than or equal to all elements in a set. It's a ceiling, a limit above which things cannot rise.

V

variable A symbol representing a quantity that can change or vary. vector A quantity possessing both magnitude and direction. Represented by an arrow. vector calculus The extension of calculus to vector fields. Deals with gradients, divergences, and curls. It’s the calculus of space and flow. washer A shape used in disc integration to calculate volumes of solids with holes. It’s a disc with a smaller disc removed from its center. washer method A method of calculus for finding the volume of a solid of revolution with a hole through the middle, using the area of washers.

See also