List of definitions of terms and concepts commonly used in calculus
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This glossary of calculus is a list of definitions about calculus, its sub-disciplines, and related fields.
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
The absolute value or modulus|x| of a
real numberx is the
non-negative value of x without regard to its
sign. Namely, |x| = x for a
positivex, |x| = −x for a
negativex (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its
distance from zero.
Is the method used to prove that an
alternating series with terms that decrease in absolute value is a
convergent series. The test was used by
Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.
An antiderivative, primitive function, primitive integral or indefinite integral[Note 1] of a
functionf 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.
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]
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.
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
binomialpower(1 + x)n, and it is given by the formula
A
functionf defined on some
setX with
real or
complex values is called bounded, if the set of its values is
bounded. In other words,
there exists a real number M such that
for allx 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.
(From
Latincalculus, literally 'small pebble', used for counting and calculations, as on an
abacus)[8] is the
mathematical study of continuous change, in the same way that
geometry is the study of shape and
algebra is the study of generalizations of
arithmetic operations.
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.
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:
This may equivalently be expressed in terms of the variable. Let F = f∘g, or equivalently, F(x) = f(g(x)) for all x. Then one can also write
The chain rule may be written in
Leibniz's notation in the following way. If a variable z depends on the variable y, which itself depends on the variable x, so that y and z are therefore
dependent variables, then z, via the intermediate variable of y, depends on x as well. The chain rule then states,
The two versions of the chain rule are related; if and , then
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.
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.
The
indefinite integral of a given function (i.e., the
set of all
antiderivatives of the function) on a
connected domain is only defined
up to an additive constant, the constant of integration.[15][16] 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.
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.
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]
A series is convergent if the sequence of its partial sums tends to a
limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number such that for any arbitrarily small positive number , there is a (sufficiently large)
integer such that for all ,
If the series is convergent, the number (necessarily unique) is called the sum of the series.
Any series that is not convergent is said to be
divergent.
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.
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]
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).
Is the highest degree of its
monomials (individual terms) with non-zero coefficients. The
degree of a term is the sum of the exponents of the
variables that appear in it, and thus is a non-negative integer.
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.
A differentiable function of one
real variable is a function whose
derivative exists at each point in its
domain. As a result, the
graph of a differentiable function must have a (non-
vertical)
tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or
cusps.
The term differential is used in
calculus to refer to an
infinitesimal (infinitely small) change in some
varying quantity. For example, if x is a
variable, then a change in the value of x is often denoted Δx (pronounced delta x). 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
where dy/dx denotes the
derivative of y with respect to x. This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimal.
Is a subfield of calculus[30] concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being
integral calculus, the study of the area beneath a curve.[31]
Is a
mathematicalequation 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.
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
where is the
derivative of f with respect to x, and dx is an additional real
variable (so that dy is a function of x and dx). The notation is such that the equation
holds, where the derivative is represented in the
Leibniz notationdy/dx, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes
The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular
differential form, or analytical significance if the differential is regarded as a
linear approximation to the increment of a function. Traditionally, the variables dx and dy are considered to be very small (
infinitesimal), and this interpretation is made rigorous in
non-standard analysis.
Continuous functions are of utmost importance in
mathematics, functions and applications. However, not all
functions are continuous. If a function is not continuous at a point in its
domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a
discrete set, a
dense set, or even the entire domain of the function.
In
mathematics, the dot product or scalar product[note 1] is an
algebraic operation that takes two equal-length sequences of numbers (usually
coordinate vectors) and returns a single number. In
Euclidean geometry, the dot product of the
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.
The multiple integral is a
definite integral of a
function of more than one real
variable, for example, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in R2 are called
double integrals, and integrals of a function of three variables over a region of R3 are called
triple integrals.[33]
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]
In
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
functionf which can be expressed in the form
where R is a
rational function of its two arguments, P is a
polynomial of degree 3 or 4 with no repeated roots, and c is a constant..
For an essential discontinuity, only one of the two one-sided limits needs not exist or be infinite.
Consider the function
Then, the point is an essential discontinuity.
In this case, doesn't exist and is infinite – thus satisfying twice the conditions of essential discontinuity. So x0 is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from the term essential singularity which is often used when studying
functions of complex variables.
In
mathematics, an exponential function is a function of the form
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
States that if a real-valued
functionf is
continuous on the
closed interval [a,b], then f must attain a
maximum and a
minimum, each at least once. That is, there exist numbers c and d in [a,b] such that:
A related theorem is the boundedness theorem which states that a continuous function f in the closed interval [a,b] is
bounded on that interval. That is, there exist real numbers m and M such that:
The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its 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).[37][38][39]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
set theory, the maximum and minimum of a
set are the
greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of
real numbers, have no minimum or maximum.
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
where the sum is over all n-
tuples of nonnegative integers (m1, …, mn) satisfying the constraint
Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:
Combining the terms with the same value of m1 + m2 + ... + mn = k and noticing that mj has to be zero for j > n − k + 1 leads to a somewhat simpler formula expressed in terms of
Bell polynomialsBn,k(x1,...,xn−k+1):
The first derivative test examines a function's
monotonic properties (where the function is increasing or decreasing) focusing on a particular point in its domain. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then it will achieve a least value at that point. If the function fails to "switch", and remains increasing or remains decreasing, then no highest or least value is achieved.
and developing a
calculus for such operators generalizing the classical one.
In this context, the term powers refers to iterative application of a linear operator to a function, in some analogy to
function composition acting on a variable, i.e. f∘2(x) = f ∘ f (x) = f ( f (x) ).
Is a process or a relation that associates each element x of a
setX, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. If the function is called f, this relation is denoted y = f(x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f.[43] The symbol that is used for representing the input is the
variable of the function (one often says that f is a function of the variable x).
Is an operation that takes two
functionsf and g and produces a function h such that h(x) = g(f(x)). In this operation, the function g is
applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z.
The fundamental theorem of calculus is a
theorem that links the concept of
differentiating a
function with the concept of
integrating a function. The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the
antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This implies the existence of
antiderivatives for
continuous functions.[44] Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many
antiderivatives. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by
symbolic integration avoids
numerical integration to compute integrals. This provides generally a better numerical accuracy.
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
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
set theory, the maximum and minimum of a
set are the
greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of
real numbers, have no minimum or 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).[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
set theory, the maximum and minimum of a
set are the
greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of
real numbers, have no minimum or maximum.
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.
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
where −a/d is not a
natural number and kis a natural number.
Equivalently, a sequence is a harmonic progression when each term is the
harmonic mean of the neighboring terms.
It is not possible for a harmonic progression (other than the trivial case where a = 1 and k = 0) to sum to an
integer. The reason is that, necessarily, at least one denominator of the progression will be divisible by a
prime number that does not divide any other denominator.[53]
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.
where f and g are
homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form
which is easy to solve by
integration of the two members.
Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of
linear differential equations, this means that there are no constant terms. The solutions of any linear
ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.
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.
Is a
complex number that can be written as a
real number multiplied by the
imaginary uniti,[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]
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
implicit function theorem provides conditions under which some kinds of relations define an implicit function, namely relations defined as the
indicator function of the
zero set of some
continuously differentiablemultivariate function.
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.
In
mathematical analysis, an improper integral is the
limit of a
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:
or
in which one takes a limit in one or the other (or sometimes both) endpoints (
Apostol 1967, §10.23).
In
differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a
continuousplane curve at which the curve changes from being
concave (concave downward) to
convex (concave upward), or vice versa.
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. .
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:
From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (v vs. t graph) is the displacement, x. In calculus terms, the
integral of the velocity function v(t) is the displacement function x(t). In the figure, this corresponds to the yellow area under the curve labeled s (s being an alternative notation for displacement).
Since the derivative of the position with respect to time gives the change in position (in
metres) divided by the change in time (in
seconds), velocity is measured in
metres per second (m/s). Although the concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment. .
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. .
In calculus, and more generally in
mathematical analysis, integration by parts or partial integration is a process that finds the
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:
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:
If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).[64]
The
image of a continuous function over an interval is itself an interval. .
Then, the point x0 = 1 is a jump discontinuity.
In this case, a single limit does not exist because the one-sided limits, L− and L+, exist and are finite, but are not equal: since, L− ≠ L+, the limit L does not exist. Then, x0 is called a jump discontinuity, step discontinuity, or discontinuity of the first kind. For this type of discontinuity, the function f may have any value at x0.
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 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
FrenchmathematicianGuillaume 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
intervalI except possibly at a point c contained in I, if
for all x in I with x ≠ c, and exists, then
The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be evaluated directly.
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.
The natural logarithm of a number is its
logarithm to the
base of the
mathematical constante, where e is an
irrational and
transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x.[77]Parentheses are 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.
(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.
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.
In
algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a
polynomial function with 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:
with at least one of the
coefficientsa, b, c, d, e, or f of the second-degree terms being non-zero.
A univariate (single-variable) quadratic function has the form[78]
in the single variable x. The
graph of a univariate quadratic function is a
parabola whose axis of symmetry is parallel to the y-axis, as shown at right.
If the quadratic function is set equal to zero, then the result is a
quadratic equation. The solutions to the univariate equation are called the
roots of the univariate function.
The bivariate case in terms of variables x and y has the form
with at least one of a, b, c not equal to zero, and an equation setting this function equal to zero gives rise to a
conic section (a
circle or other
ellipse, a
parabola, or a
hyperbola).
In general there can be an arbitrarily large number of variables, in which case the resulting
surface is called a
quadric, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.
Is the
SI unit for measuring
angles, and is the standard unit of angular measure used in many areas of
mathematics. The length of an arc of a
unit circle is numerically equal to the measurement in radians of the
angle that it
subtends; one radian is just under 57.3
degrees (expansion at OEIS:
A072097). The unit was formerly an
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 .
^"Calculus". OxfordDictionaries. Archived from
the original on April 30, 2013. Retrieved 15 September 2017.
^Howard Eves, "Two Surprising Theorems on Cavalieri Congruence", The College Mathematics Journal, volume 22, number 2, March, 1991), pages 118–124
^Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [10] Functions of complementary angles". Written at Ann Arbor, Michigan, USA.
Trigonometry. Vol. Part I: Plane Trigonometry. New York, USA:
Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. pp. 11–12. Retrieved 2017-08-12.
^Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862),
p. 253-255Archived 2011-07-21 at the
Wayback Machine.
^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.
^Hazewinkel, Michiel (1994) [1987]. Encyclopaedia of Mathematics (unabridged reprint ed.). Kluwer Academic Publishers / Springer Science & Business Media.
ISBN978-155608010-4.
^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.
^Mejlbro, Leif (2010-11-11). Stability, Riemann Surfaces, Conformal Mappings - Complex Functions Theory (PDF) (1 ed.). Ventus Publishing ApS / Bookboon.
ISBN978-87-7681-702-2. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.
^Durán, Mario (2012). Mathematical methods for wave propagation in science and engineering. 1: Fundamentals (1 ed.). Ediciones UC. p. 88.
ISBN978-956141314-6.
^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.
^Klein, Christian Felix (2004) [1932]. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. (Translation of 3rd German ed.). Dover Publications, Inc. / The Macmillan Company.
ISBN978-0-48643480-3. Retrieved 2017-08-13.
^Dörrie, Heinrich (1965). Triumph der Mathematik. Translated by Antin, David. Dover Publications. p. 69.
ISBN978-0-486-61348-2.
^j is usually used in Engineering contexts where i has other meanings (such as electrical current)
^Antiderivatives are also called general integrals, and sometimes integrals. The latter term is generic, and refers not only to indefinite integrals (antiderivatives), but also to
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.