Glossary of calculus
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This glossary of calculus is a list of definitions about calculus, its sub-disciplines, and related fields.
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A
- Abel's test
- A method of testing for the convergence of an infinite series.
- absolute convergence
- An infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number . Similarly, an improper integral of a function, , is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if
- absolute maximum
- The highest value a function attains.
- absolute minimum
- The lowest value a function attains.
- absolute value
- The absolute value or modulus |x| of a from zero.
- alternating series
- An infinite serieswhose terms alternate between positive and negative.
- alternating series test
- Is the method used to prove that an Gottfried Leibnizand is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.
- annulus
- A ring-shaped object, a region bounded by two concentric circles.
- antiderivative
- An antiderivative, primitive function, primitive integral or indefinite integral[Note 1] of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as .[1][2] The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative.
- arcsin
- area under a curve
- asymptote
- In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.[3] In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.[4][5]
- automatic differentiation
- In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation or computational differentiation,[6][7] is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.
- average rate of change
B
- binomial coefficient
- Any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula
binomial expansion
)there exists
a real number M such that
for all
x in X. A function that is not bounded is said to be unbounded.
Sometimes, if f(x) ≤ A for all x in X, then the function is said to be bounded above by A. On the other hand, if f(x) ≥ B for all x in X, then the function is said to be bounded below by B.C
- calculus
- (From arithmetic operations.
- Cavalieri's principle
- Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:[9]
- 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.
- chain rule
- The chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f and g are functions, then the chain rule 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:
dependent variables, then z, via the intermediate variable of y, depends on x as well. The chain rule then states,substitution rule.
- change of variables
- Is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.
- cofunction
- A complementary angles.[10] This definition typically applies to trigonometric functions.[11][12] The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620).[13][14]
- concave function
- Is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.
- constant of integration
- The This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function is defined on an interval and is an antiderivative of , then the set of all antiderivatives of is given by the functions , where C is an arbitrary constant (meaning that any value for C makes a valid antiderivative). The constant of integration is sometimes omitted in lists of integrals for simplicity.
- continuous function
- Is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.
- continuously differentiable
- A function f is said to be continuously differentiable if the derivative f′(x) exists and is itself a continuous function.
- contour integration
- In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.[17][18][19]
- convergence tests
- Are methods of testing for the interval of convergence or divergence of an infinite series.
- convergent series
- In infinite sequenceof numbers. Given an infinite sequence , the nthpartial sumis the sum of the first n terms of the sequence, that is,
- convex function
- In mathematics, a real-valued function defined on an n-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. For a twice differentiable function of a single variable, if the second derivative is always greater than or equal to zero for its entire domain then the function is convex.[20] Well-known examples of convex functions include the quadratic function and the exponential function .
- Cramer's rule
- In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750,[21][22] although Colin Maclaurin also published special cases of the rule in 1748[23] (and possibly knew of it as early as 1729).[24][25][26]
- critical point
- A critical point or stationary point of a
- curve
- A curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.
- curve sketching
- In geometry, curve sketching (or curve tracing) includes techniques that can be used to produce a rough idea of overall shape of a plane curve given its equation without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features. Here input is an equation. In digital geometry it is a method of drawing a curve pixel by pixel. Here input is an array (digital image).
D
- damped sine wave
- Is a sinusoidal function whose amplitude approaches zero as time increases.[29]
- degree of a polynomial
- Is the highest degree of its degree of a term is the sum of the exponents of the variablesthat appear in it, and thus is a non-negative integer.
- derivative
- The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
- derivative test
- A derivative test uses the local minimum, or a saddle point. Derivative tests can also give information about the concavityof a function.
- differentiable function
- A differentiable function of one tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.
- differential (infinitesimal)
- The term differential is used in deltax). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise. Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If y is a function of x, then the differential dy of y is related to dx by the formula
- differential calculus
- Is a subfield of calculusintegral calculus, the study of the area beneath a curve.[31]
- differential equation
- Is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.
- differential operator
- .
- differential of a function
- In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by
Leibniz notationdy/dx, and this is consistent with regarding the derivative as the quotient of the differentials. One also writesnon-standard analysis.
- differentiation rules
- .
- direct comparison test
- A convergence test in which an infinite series or an improper integral is compared to one with known convergence properties.
- Dirichlet's test
- Is a method of testing for the convergence of a series. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.[32] The test states that if is a sequence of real numbers and a sequence of complex numbers satisfying
- for every positive integer N
- disc integration
- Also known in axis of revolution.
- divergent series
- Is an partial sums of the series does not have a finite limit.
- discontinuity
- discrete set, a dense set, or even the entire domain of the function.
- dot product
- In Cartesian coordinates of two vectors is widely used and often called "the" inner product (or rarely projection product) of Euclidean space even though it is not the only inner product that can be defined on Euclidean space; see also inner product space.
- double integral
- The multiple integral is a triple integrals.[33]
E
- e (mathematical constant)
- The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828,[34] and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series[35]
integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Modern mathematics defines an "elliptic integral" as any function
f which can be expressed in the form
Institutionum calculi integralis (published 1768–1870).[36]
where b is a positive real number, and in which the argument x occurs as an exponent. For real numbers c and d, a function of the form is also an exponential function, as it can be rewritten as
minimum
, each at least once. That is, there exist numbers c and d in [a,b] such that:
extremum
greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers
, have no minimum or maximum.F
- Faà di Bruno's formula
- Is an identity in mathematics generalizing the chain rule to higher derivatives, named after Francesco Faà di Bruno (1855, 1857), though he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast stated the formula in a calculus textbook,[40] considered the first published reference on the subject.[41]
Perhaps the most well-known form of Faà di Bruno's formula says that
Bell polynomialsBn,k(x1,...,xn−k+1):
- ,
right pyramid or right cone.[42]
G
- general Leibniz rule
- The general Leibniz rule,[45] named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if and are -times differentiable functions, then the product is also -times differentiable and its th derivative is given by
- global maximum
- In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).[46][47][48] Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
As defined in greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.
- global minimum
- In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).[49][50][51] Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
As defined in greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.
- golden spiral
- In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio.[52] That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.
- gradient
- Is a multi-variable generalization of the scalar-valued.
H
- harmonic progression
- In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. It is a sequence of the form
- higher derivative
- Let f be a differentiable function, and let f ′ be its derivative. The derivative of f ′ (if it has one) is written f ′′ and is called the second derivative of f. Similarly, the derivative of the second derivative, if it exists, is written f ′′′ 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. These repeated derivatives are called higher-order derivatives. The nth derivative is also called the derivative of order n.
- homogeneous linear differential equation
- A differential equation can be homogeneous in either of two respects.
A first order differential equationis said to be homogeneous if it may be written
- hyperbolic function
- Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
I
- identity function
- Also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f(x) = x.
- imaginary number
- Is a complex number that can be written as a real number multiplied by the imaginary unit i,[note 2] which is defined by its property i2 = −1.[54] The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.[55]
- implicit function
- In mathematics, an implicit equation is a relation of the form , where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is .
An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments).[56]: 204–206 Thus, an implicit function for in the context of the unit circle is defined implicitly by . This implicit equation defines as a function of only if and one considers only non-negative (or non-positive) values for the values of the function.
The continuously differentiable multivariatefunction.
- improper fraction
- Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise.[57][58] In general, a common fraction is said to be a proper fraction if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1.[59][60] It is said to be an improper fraction, or sometimes top-heavy fraction,[61] if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, –3/4, and 4/9; examples of improper fractions are 9/4, –4/3, and 3/3.
- improper integral
- In definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, , , or in some instances as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, in some cases with infinity as a limit of integration. Specifically, an improper integral is a limit of the form:
- inflection point
- In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuous plane curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.
- instantaneous rate of change
- The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. .
- instantaneous velocity
- If we consider v as velocity and x as the displacement (change in position) vector, then we can express the (instantaneous) velocity of a particle or object, at any particular time t, as the derivative of the position with respect to time:
- integral
- An integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. .
- integral symbol
- The integral symbol:
is used to denote antiderivatives in mathematics. .
- integrand
- The function to be integrated in an integral.
- integration by parts
- In calculus, and more generally in integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be readily derived by integrating the product rule of differentiation. If u = u(x) and du = u′(x) dx, while v = v(x) and dv = v′(x) dx, then integration by parts states that:Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts. .
- integration by substitution
- Also known as u-substitution, is a method for solving integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the chain rule for differentiation. .
- intermediate value theorem
- In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. This has two important corollaries:
- inverse trigonometric functions
- (Also called arcus functions,cosecantfunctions, and are used to obtain an angle from any of the angle's trigonometric ratios.
J
- jump discontinuity
- Consider the function
L
- Lebesgue integration
- In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.
- L'Hôpital's rule
- L'Hôpital's rule or L'Hospital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be evaluated by substitution, allowing easier evaluation of the limit. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the contribution of the rule is often attributed to L'Hôpital, the theorem was first introduced to L'Hôpital in 1694 by the Swiss mathematician Johann Bernoulli.
L'Hôpital's rule states that for functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if
for all x in I with x ≠ c, and exists, then
- limit comparison test
- The limit comparison test allows one to determine the convergence of one series based on the convergence of another.
- limit of a function
- .
- limits of integration
- .
- linear combination
- In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).[74][75][76] The concept of linear combinations is central to linear algebra and related fields of mathematics.
- linear equation
- A linear equation is an equation relating two or more variables to each other in the form of with the highest power of each variable being 1.
- linear system
- .
- list of integrals
- .
- logarithm
- .
- logarithmic differentiation
- .
- lower bound
- .
M
- mean value theorem
- .
- monotonic function
- .
- multiple integral
- .
- Multiplicative calculus
- .
- multivariable calculus
- .
N
- natural logarithm
- The natural logarithm of a number is its Parenthesesare sometimes added for clarity, giving ln(x), loge(x) or log(x). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity.
- non-Newtonian calculus
- .
- nonstandard calculus
- .
- notation for differentiation
- .
- numerical integration
- .
O
P
- Pappus's centroid theorem
- (Also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.
- parabola
- Is a plane curve that is mirror-symmetrical and is approximately U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define exactly the same curves.
- paraboloid
- .
- partial derivative
- .
- partial differential equation
- .
- partial fraction decomposition
- .
- particular solution
- .
- piecewise-defined function
- A function defined by multiple sub-functions that apply to certain intervals of the function's domain.
- position vector
- .
- power rule
- .
- product integral
- .
- product rule
- .
- proper fraction
- .
- proper rational function
- .
- Pythagorean theorem
- .
- Pythagorean trigonometric identity
- .
Q
- quadratic function
- In polynomial functionwith one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:rootsof the univariate function. The bivariate case in terms of variables x and y has the formsurface is called a quadric, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.
- quadratic polynomial
- .
- quotient rule
- A formula for finding the derivative of a function that is the ratio of two functions.
R
- radian
- Is the SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.[79] Separately, the SI unit of solid angle measurement is the steradian.
- ratio test
- .
- reciprocal function
- .
- reciprocal rule
- .
- Riemann integral
- .
- .
- removable discontinuity
- .
- Rolle's theorem
- .
- root test
- .
S
- scalar
- .
- secant line
- .
- second-degree polynomial
- .
- second derivative
- .
- second derivative test
- .
- second-order differential equation
- .
- series
- .
- shell integration
- .
- Simpson's rule
- .
- sine
- .
- sine wave
- .
- slope field
- .
- squeeze theorem
- .
- sum rule in differentiation
- .
- sum rule in integration
- .
- summation
- .
- supplementary angle
- .
- surface area
- .
- system of linear equations
- .
T
- table of integrals
- .
- Taylor series
- .
- Taylor's theorem
- .
- tangent
- .
- third-degree polynomial
- .
- third derivative
- .
- toroid
- .
- total differential
- .
- trigonometric functions
- .
- trigonometric identities
- .
- trigonometric integral
- .
- trigonometric substitution
- .
- trigonometry
- .
- triple integral
- .
U
V
- variable
- .
- vector
- .
- vector calculus
- .
W
- washer
- .
- washer method
- .
See also
- Outline of calculus
- Glossary of areas of mathematics
- Glossary of astronomy
- Glossary of biology
- Glossary of botany
- Glossary of chemistry
- Glossary of ecology
- Glossary of engineering
- Glossary of physics
- Glossary of probability and statistics
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- ISBN 978-0-321-58876-0.
- ISBN 978-0-495-01166-8.
- ISBN 978-0-547-16702-2.
- ISBN 978-0-321-58876-0.
- The Wolfram Demonstrations Project.
- MR 3203600.
- ^
Uno Ingard, K. (1988). "Chapter 2". Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.
- ISBN 978-81-7133-912-9.
- ISBN 0-07-010813-7.
- ^ "World Wide Words: Vulgar fractions". World Wide Words. Retrieved 2014-10-30.
- ^ Weisstein, Eric W. "Improper Fraction". MathWorld.
- ^ Laurel (31 March 2004). "Math Forum – Ask Dr. Math:Can Negative Fractions Also Be Proper or Improper?". Retrieved 2014-10-30.
- ^ "New England Compact Math Resources". Archived from the original on 2012-04-15. Retrieved 2019-06-16.
- ISBN 978-0-85950-159-0. Retrieved 2014-07-29.
- ^ "Brook Taylor". History.MCS.St-Andrews.ac.uk. Retrieved May 25, 2018.
- ^ "Brook Taylor". Stetson.edu. Archived from the original on January 3, 2018. Retrieved May 25, 2018.
- ^ Weisstein, Eric W. "Bolzano's Theorem". MathWorld.
- ^ Taczanowski, Stefan (1978-10-01). "On the optimization of some geometric parameters in 14 MeV neutron activation analysis". Nuclear Instruments and Methods. ScienceDirect. 155(3): 543–546. doi:10.1016/0029-554X(78)90541-4.
- ISBN 978-155608010-4.
- ^ Ebner, Dieter (2005-07-25). Preparatory Course in Mathematics (PDF) (6 ed.). Department of Physics, University of Konstanz. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.
- ISBN 978-87-7681-702-2. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.
- ISBN 978-956141314-6.
- ^ Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [14] Inverse trigonometric functions". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 15. Retrieved 2017-08-12. […] α = arcsin m: It is frequently read "arc-sinem" or "anti-sine m," since two mutually inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1m, is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. […]
- ^ Klein, Christian Felix (1924) [1902]. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). 1 (3rd ed.). Berlin: J. Springer.
- ISBN 978-0-48643480-3. Retrieved 2017-08-13.
- ISBN 978-0-486-61348-2.
- ISBN 0-321-28713-4.
- ISBN 0-03-010567-6.
- ISBN 0-387-98258-2.
- ^ "Quadratic Equation -- from Wolfram MathWorld". Retrieved January 6, 2013.
- Bureau International des Poids et Mesures. Retrieved 2014-09-23.
Works cited
- Apostol, T (1967), Calculus, Vol. 1 (2nd ed.), Jon Wiley & Sons.
- Arbogast, L. F. A. (1800), Du calcul des derivations [On the calculus of derivatives] (in French), Strasbourg: Levrault, pp. xxiii+404.
- ISBN 978-0-471-96758-3.
- Craik, Alex D. D. (February 2005), "Prehistory of Faà di Bruno's Formula", Zbl 1088.01008.
- Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993). Solving ordinary differential equations I: Nonstiff problems. Berlin, New York: ISBN 978-3-540-56670-0.
- Johnson, Warren P. (March 2002), "The Curious History of Faà di Bruno's Formula" (PDF), Zbl 1024.01010.
Notes
- ^ The term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.[citation needed]
- ^ j is usually used in Engineering contexts where i has other meanings (such as electrical current)
- definite integrals. When the word integral is used without additional specification, the reader is supposed to deduce from the context whether it refers to a definite or indefinite integral. Some authors define the indefinite integral of a function as the set of its infinitely many possible antiderivatives. Others define it as an arbitrarily selected element of that set. Wikipedia adopts the latter approach.[citation needed]
- ^ The symbol J is commonly used instead of the intuitive I in order to avoid confusion with other concepts identified by similar I–like glyphs, e.g. identities.