calculus

Science of Fluxions

written in his book, this is Isaac Newton's term for what we call calculus

Lambda Calculus

a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. name binding is the association of entities (data and/or code) with identifiers. an identifier bound to an object is said to reference that object

Product Rule

a formula used to find the derivatives of products of two or more functions. It may be stated as (f∘g)' = f'∘g + f∘g'

Cavalieri's Principle

a modern implementation of the method of indivisibles, which is as follows: 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes.

Paraboloid

a solid generated by the rotation of a parabola around its axis of symmetry. a solid having two or more nonparallel parabolic cross sections

Velocity

the _____ of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. This is equivalent to a specification of its speed and direction of motion (e.g. 60 km/h to the north). applications of differential calculus involve this.

Engineering

the application of mathematics, empirical evidence and scientific, economic, social, and practical knowledge in order to invent, innovate, design, build, maintain, research, and improve structures, machines, tools, systems, components, materials, and processes

Cycloid

the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage. It is an example of a roulette, a curve generated by a curve rolling on another curve

Economics

the social science that describes the factors that determine the production, distribution and consumption of goods and services

Infinity (or, Infinite)

(symbol: ∞) is an abstract concept describing something without any bound and is relevant in a number of fields, predominantly mathematics and physics. In mathematics, it is often treated as if it were a number (i.e., it counts or measures things: "an _____ number of terms") but it is not the same sort of number as natural or real numbers

Cavalieri

17th century Italian mathematician who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections (not Archimedes) his work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first

Euclidean Space

In geometry, this encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. Every point in three-dimensional _________ is determined by three coordinates.

Work

In physics, a force is said to do _____ if, when acting on a body, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the _____ done on the ball as it falls is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). Applications of integral calculus include computations involving this.

Center of Mass

In physics, the _________ of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero or the point where if a force is applied causes it to move in direction of force without rotation. The distribution of mass is balanced around the ________ and the average of the weighted position coordinates of the distributed mass defines its coordinates. Applications of integral calculus include computations involving this.

Higher Derivatives

Let f be a differentiable function, and let f ′(x) be its derivative. The derivative of f ′(x) (if it has one) is written f ′′(x) and is called the second derivative of f. Similarly, the derivative of a second derivative, if it exists, is written f ′′′(x) and is called the third derivative of f. Continuing this process, one can define, if it exists, the nth derivative as the derivative of the (n-1)th derivative. The nth derivative is also called the derivative of order n

Hyperreal Numbers

The system of ______ is a way of treating infinite and infinitesimal quantities. These _______, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + . . . + 1., Such a number is infinite, and its reciprocal is infinitesimal. this term was introduced by Edwin Hewitt in 1948. these numbers can be used to give a Leibniz-like development of the usual rules of calculus

Weierstrass

a 19th century German mathematician often cited as the "father of modern analysis". In his work he formalized the concept of limit and eliminated infinitesimals. Following his work, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus"

Gottfried Leibniz

a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy. considered to be one of two people who developed modern calculus in the 17th century. he was the first to publish his results on the development of calculus. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. He paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. he developed much of the notation used in calculus today.

Method of Fluxions

a book by Isaac Newton. The book was completed in 1671, and published in 1736. in it he described differential calculus and part of the title is his term for differential calculus

The Analyst

a book published by Berkeley in 1734 where he criticizes, amongst other things, the use of infinitesimals in modern calculus (such as by Newton) as being unrigorous.

Mathematical Physics

a branch of applied mathematics that applies methods to solve problems in the natural science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force.

Real Analysis (or, traditionally, the Theory of Functions of a Real Variable)

a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions

Geometry

a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space

Mathematical Analysis

a branch of mathematics that studies continuous change and includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. these theories are usually studied in the context of real and complex numbers and functions

Motion

a change in position of an object with respect to time. This is typically described in terms of displacement, distance (scalar), velocity, acceleration, time and speed. ______ of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame

Measure Theory (or, Measure)

a collection of propositions to illustrate the principles of a subject invented by Lebesgue to define integrals of all but the most pathological (abnormal) functions. In mathematical analysis, a (this) on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, this is a generalization of the concepts of length, area, and volume.

Mathematical Proof

a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, this can be traced back to self-evident or assumed statements, known as axioms. These are examples of deductive reasoning and are distinguished from inductive or empirical arguments; this must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases.

Process Calculus

a diverse family of related approaches for formally modelling concurrent systems. these provide a tool for the high-level description of interactions, communications, and synchronizations between a collection of independent agents or processes

Calculus of Variations

a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. functionals are often expressed as definite integrals involving functions and their derivatives

(ε, δ)-definition of limit

a formalization of the notion of limit. Let f be a function. To say that the limit of f of x as x approaches c equals L means that f(x) can be made as close as desired to L by making the independent variable x close enough, but not equal, to the value c. How close is "close enough to c" depends on how close one wants to make f(x) to L. It also of course depends on which function f is and on which number c is. Therefore let the positive number ε (epsilon) be how close one wishes to make f(x) to L; strictly one wants the distance to be less than ε. Further, if the positive number δ is how close one will make x to c, and if the distance from x to c is less than δ (but not zero), then the distance from f(x) to L will be less than ε. Therefore δ depends on ε. The limit statement means that no matter how small ε is made, δ can be made small enough. The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work. In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point. This definition also works for functions with more than one argument. For such functions, δ can be understood as the radius of a circle or a sphere or some higher-dimensional analogy centered at the point where the existence of a limit is being proven, in the domain of the function and, for which, every point inside maps to a function value less than ε away from the value of the function at the limit point.

Chain Rule

a formula for computing the derivative of the composition of two or more functions. That is, if f and g are functions, then this expresses the derivative of their composition f ∘ g (the function which maps x to f(g(x)) in terms of the derivatives of f and g and the product of functions as follows: (f ∘ g)' = (f' ∘ g) ∘ g'. This can be written more explicitly in terms of the variable. Let F = f ∘ g, or equivalently, F(x) = f(g(x)) for all x. Then one can also write F'(x) = f'(g(x)) g'(x).

Complex Plane (or, z-plane)

a geometric representation of the numbers (of the form a+bi) established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a number (of the form a+bi) represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis

Finite Difference (or Calculus of Finite Differences)

a mathematical expression of the form f(x + b) − f(x + a). If this is divided by b − a, one gets a difference quotient. The approximation of derivatives by this method plays a central role for the numerical solution of differential equations, especially boundary value problems

Method of Exhaustion

a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "_____" by the lower bound areas successively established by the sequence members. the development of this by Eudoxus and Liu Hui, foreshadowed the concept of the limit, in order to calculate areas and volumes of shapes, for example

Sphere

a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (analogous to a circular object in two dimensions)

Function

a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output

Analytic Function

a relation between a set of inputs and outputs that is locally given by a convergent power series (A series is convergent if the sequence of its partial sums { S_1, S_2, S_3, . . .} converges; in other words, it approaches a given number) (a power series usually arises as the Taylor series of some known function). There exist both real ______ and complex _______, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex ________ exhibit properties that do not hold generally for real _________. This relation can only be so if and only if its Taylor series about x_0 converges to the function in some neighborhood for every x_0 in its domain

Taylor Series

a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point

Cours d'Analyse

a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821 in which we find a broad range of foundational approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation

Infinite Series

a sequence of numbers in which an infinite number of terms are added successively in a given pattern; the sequence of partial sums of a given sequence. can be used to resolve Zeno's paradoxes.

Zeno's Paradoxes

a set of philosophical problems generally thought to have been devised by ancient Greek philosopher (ca. 490-430 BC) to support Parmenides's doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. some of these look at division by zero or sums of infinitely many numbers in regards to the study of motion and area.

Mathematical Logic

a subfield of mathematics exploring the applications of formal reasoning conducted or assessed according to strict principles of validity to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in this include the study of the expressive power of formal systems and the deductive power of formal proof systems

Parabola

a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The path of a projectile under the influence of gravity ideally follows a curve of this shape.

Science

a systematic enterprise that creates, builds and organizes knowledge in the form of testable explanations and predictions about the universe

Fundamental Theorem of Calculus

a theorem that links the concept of the derivative of a function with the concept of the function's integral. The first part of this theorem is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions. The second part of the theorem is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals

Fourier Series

a way to represent a (wave-like) function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). more advanced Calculus applications include this.

Principia Mathematica (or Principia, or Philosophiæ Naturalis Principia Mathematica)

a work in three books by Sir Isaac Newton, in Latin, first published 5 July 1687. It states Newton's laws of motion, forming the foundation of classical mechanics, also Newton's law of universal gravitation, and a derivation of Kepler's laws of planetary motion (which Kepler first obtained empirically). It is "justly regarded as one of the most important works in the history of science"

Division by Zero

a/0 where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a (assuming a≠0), and so the result is undefined. such a paradox of this in regards to motion and area can be resolved by calculus tools, especially the limit and the infinite series.

Isaac Newton

an English physicist and mathematician (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and as a key figure in the scientific revolution. considered to be one of two people who developed modern calculus in the 17th century. he developed the use of calculus in his laws of motion and gravitation. The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by him in an idiosyncratic notation which he used to solve problems of mathematical physics. In his works, he rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable

Maria Gaetana Agnesi

an Italian mathematician, philosopher, theologian and humanitarian who was the first woman to write a mathematics handbook and the first woman appointed as a Mathematics Professor at a University. she wrote one of the first and most complete works on finite and infinitesimal analysis in 1748

Power Series

an infinite set of quantities constituting a progression or having the several values determined by a common relation of the form Σ a_n x^n (where n is a positive integer). also, a generalization of this for more than one variable. more advanced Calculus applications include this.

Sequence

an ordered collection of objects in which repetitions are allowed. it contains members (also called elements, or terms). the number of elements (possibly infinite) is called its length. order matters, and exactly the same elements can appear multiple times at different positions in this collection.

Heuristics (or Heuristic Technique)

any approach to problem solving, learning, or discovery that employs a practical method not guaranteed to be optimal or perfect, but sufficient for the immediate goals. Where finding an optimal solution is impossible or impractical, these methods can be used to speed up the process of finding a satisfactory solution. These methods can be mental shortcuts that ease the cognitive load of making a decision

Integral

assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data

Differential Calculus

concerns rates of change and slopes of curves. the primary objects of study in this subfield are the derivative of a function, related notions such as the differential, and their applications.

Integral Calculus

concerns the accumulation of quantities and the areas under and between curves. it assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data

The Method of Mechanical Theorems (or, The Method)

contains the first explicit use of indivisibles (sometimes referred to as infinitesimals). used by Archimedes in the 2nd century BC. resembles the use of integral calculus

Alhazen

derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid

Arc Length

determining this quantity of an irregular arc segment is also called rectification of a curve. When rectified, the curve gives a straight line segment with the same length as the curve's _______. Historically, many methods were used for specific curves. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. Applications of integral calculus include computations involving this.

Elementary Algebra

encompasses some of the basic concepts of the mathematical study of operations and their applications to solving equations, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, this introduces quantities without fixed values, known as variables. This use of variables entails a use of certain notation and an understanding of the general rules of the operators introduced in arithmetic. this branch is not concerned with certain structures outside the realm of real and complex numbers

Foundations

in calculus, this refers to the rigorous development of a subject from precise axioms and definitions

Acceleration

in physics, is the rate of change of velocity of an object. this is the net result of any and all forces acting on the object, as described by Newton's Second Law. applications of differential calculus involve this.

Distributions

introduced by Schwartz to take the derivative of any function whatsoever. these are objects that generalize the classical notion of functions in mathematical analysis. They make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a ________ (adj. form) derivative. These are widely used in the theory of partial differential equations, where it may be easier to establish the existence of ______ (adj. form) solutions than classical solutions, or appropriate classical solutions may not exist

Algebra

one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, it is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. it is the study of operations and their applications to solving equations

Mathematical Rigour

refers to a process of adhering absolutely to certain constraints, or the practice of maintaining strict consistency with certain predefined parameters (in this case, mathematically). These constraints may be environmentally imposed, socially, or logically imposed, such as in proofs which must maintain consistent answers

Non-Standard Analysis

reformulates the calculus using a logically rigorous notion of infinitesimal numbers. originated in 1960s by Robinson. it uses technical machinery from mathematical logic to augment the real number system with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The resulting numbers are called hyperreal numbers, and they can be used to give a Leibniz-like development of the usual rules of calculus

Second Fundamental Theorem of Calculus

states that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. this part has key practical applications because it markedly simplifies the computation of definite integrals

First Fundamental Theorem of Calculus

states that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions

Adequality

technique used by de Fermat to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in mathematical analysis. represents equality up to an infinitesimal error term.

Ghosts of Departed Quantities

term used by Berkeley to describe infinitesimals. in his book The Analyst (1734), he describes infinitesimals: "They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ________?

Slope

the ____, or gradient of a line is a number that describes both the direction and the steepness of the line. This is often denoted by the letter m. The direction of a line is either increasing, decreasing, horizontal or vertical. A line is increasing if it goes up from left to right. The _____ is positive, i.e. m>0. A line is decreasing if it goes down from left to right. The _____ is negative, i.e. m<0. If a line is horizontal the slope is zero. This is a constant function. If a line is vertical the _____ is undefined. The steepness, incline, or grade of a line is measured by the absolute value of the _____. A _____ with a greater absolute value indicates a steeper line. This is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. Differential calculus looks at the _____ of a curve.

Derivative

the _____ of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). these are a fundamental tool of calculus

Fourth Power

the _______ of a number n is the result of multiplying four instances of n together. So: n^4 = n × n × n × n these are also formed by multiplying a number by its cube. furthermore, they are squares of squares. Alhazen derived a formula to calculate the sums of these and used the results to carry out what we now call integration

Propositional Calculus

the branch of mathematical logic concerned with the study of formal statements of a theorem or problem, typically including the demonstration (whether they are true or false) that are formed by other formal statements (of this nature) with the use of logical connectives, and how their value depends on the truth value of their components

Nova Methodus pro Maximis et Minimis

the first published work on the subject of calculus. It was published in by Gottfried Leibniz in the Acta Eruditorum in 1684. It is considered to be the birth of infinitesimal calculus

Pressure

the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Applications of integral calculus include computations involving this.

Calculus

the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their applications to solving equations. both forms of this look at the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. associated with the study of functions and limits, for example. this term comes from Latin and refers to a small stone used for counting. more generally, this refers to any method or system of calculation guided by the symbolic manipulation of expressions. "The ______ was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking." —John von Neumann The development of this was built on earlier concepts of instantaneous motion and area underneath curves

Integration

the process of calculating an integral. This is the inverse of differentiation, since _____ of a given function results in a function whose derivative is the given function. This is used in the calculation of such things as the areas and volumes of irregular shapes and solids

Volume

the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains

Area

the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane

Mathematical Optimization

the selection of a best element (with regard to some criteria) from some set of available alternatives. In the simplest case, a ________ problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. applications of differential calculus involve this.

Mathematics

the study of topics such as quantity (numbers), structure, space, and change

Series

the sum of the terms of a sequence.

Limit (or, Convergence)

the value that a function or sequence "approaches" as the input or index approaches some value. these are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals. can be used to resolve Zeno's paradoxes

Infinitesimals

things so small that there is no way to measure them. calculus has historically been called "the calculus of ________" or "_______ calculus"

Mathematics Education

this is the practice of teaching and learning mathematics, along with the associated scholarly research. calculus is a part of this.

Continuity (or, Continuous Function)

where a function for which small changes in the input result in small changes in the output