A volume element is the differential element whose volume integral over some range in a given coordinate system gives the volume of a solid,(1)In , the volume of the infinitesimal -hypercube bounded by , ..., has volume given by the wedge product(2)(Gray 1997).The use of the antisymmetric wedge product instead of the symmetric product is a technical refinement often omitted in informal usage. Dropping the wedges, the volume element for curvilinear coordinates in is given by(3)(4)(5)(6)(7)where the latter is the Jacobian and the are scale factors.
Vardi's integral is the beautiful definite integral(1)(2)(3)(OEIS A115252; Gradshteyn and Ryzhik 1980, p. 532; Bailey et al. 2007, p. 160; Moll 2006), where is the gamma function.Other examples similar to these include(4)(5)(Vardi 1988; Gradshteyn and Ryzhik 1980, pp. 571-572).
For a given bounded function over a partition of a given interval, the upper sum is the sum of box areas using the supremum of the function in each subinterval .
The definite integral(1)where , , and are real numbers and is the natural logarithm.
Laplace's integral is one of the following integral representations of the Legendre polynomial ,(1)(2)It can be evaluated in terms of the hypergeometricfunction.
A singular integral is an integral whose integrand reaches an infinite value at one or more points in the domain of integration. Even so, such integrals can converge, in which case, they are said to exist. (If they do not converge, they are said not to exist.) The most commonly encountered example of a singular integral is the Hilbert transform. (However, note that the logarithmic integral is not singular, since it converges in the classical Riemann sense.)In general, singular integrals can be defined by eliminating a small space including the singularity, and then taking the limit as this small space disappears.
Integration under the integral sign is the use of the identity(1)to compute an integral. For example, consider(2)for . Multiplying by and integrating between and gives(3)(4)But the left-hand side is equal to(5)so it follows that(6)(Woods 1926, pp. 145-146).
Serret's integral is given by(1)(2)(OEIS A102886; Serret 1844; Gradshteyn andRyzhik 2000, eqn. 4.291.8; Boros and Moll 2004, p. 243).
An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive. The Riemann integral is the simplest integral definition and the only one usually encountered in physics and elementary calculus. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."The Riemann integral of the function over from to is written(1)Note that if , the integral is written simply(2)as opposed to .Every definition of an integral is based on a particular measure. For instance, the Riemann integral is based on Jordan measure, and the Lebesgue integral is based..
An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. Improper integrals cannot be computed using a normal Riemann integral.For example, the integral(1)is an improper integral. Some such integrals can sometimes be computed by replacing infinite limits with finite values(2)and then taking the limit as ,(3)(4)(5)Improper integrals of the form(6)with one infinite limit and the other nonzero may also be expressed as finite integrals over transformed functions. If decreases at least as fast as , then let(7)(8)(9)(10)and(11)(12)If diverges as for , let(13)(14)(15)(16)(17)and(18)If diverges as for , let(19)(20)(21)and(22)(23)If the integral diverges exponentially, then let(24)(25)(26)and(27)..
The Riemann integral is the definite integral normally encountered in calculus texts and used by physicists and engineers. Other types of integrals exist (e.g., the Lebesgue integral), but are unlikely to be encountered outside the confines of advanced mathematics texts. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."The Riemann integral is based on the Jordan measure,and defined by taking a limit of a Riemann sum,(1)(2)(3)where and , , and are arbitrary points in the intervals , , and , respectively. The value is called the mesh size of a partition of the interval into subintervals .As an example of the application of the Riemann integral definition, find the area under the curve from 0 to . Divide into segments,..
A repeated integral is an integral taken multiple times over a single variable (as distinguished from a multiple integral, which consists of a number of integrals taken with respect to different variables). The first fundamental theorem of calculus states that if is the integral of , then(1)Now, if , then(2)It follows by induction that if , then the -fold integral of is given by(3)(4)Similarly, if , then(5)
A double integral over three coordinates giving the area within some region ,If a plane curve is given by , then the area between the curve and the x-axis from to is given by
Fubini's theorem, sometimes called Tonelli's theorem, establishes a connection between a multiple integral and a repeated one. If is continuous on the rectangular region , then the equalityholds (Thomas and Finney 1996, p. 919).
Let be a path given parametrically by . Let denote arc length from the initial point. Then(1)(2)where .
Ahmed's integral is the definite integral(OEIS A096615; Ahmed 2002; Borwein et al.2004, pp. 17-20).This is a special case of a general result that also yields(OEIS A102521 and A098459) as additional cases (Borwein et al. 2004, p. 20), where is Catalan's constant.
A double integral is a two-fold multiple integral.Examples of definite double integrals evaluating to simple constants include(1)(2)(3)(4)where is Catalan's constant (Borwein et al. 2004, pp. 48-49), and(5)where is the Euler-Mascheroni constant (Sondow 2003, 2005; Borwein et al. 2004, pp. 48-49).
Abel's integral is the definite integral(1)(2)(3)(4)(5)(6)(7)(8)(OEIS A102047), where is a cosine integral.
Multivariable calculus is the branch of calculus that studies functions of more than one variable. Partial derivatives and multiple integrals are the generalizations of derivative and integral that are used. An important theorem in multivariable calculus is Green's theorem, which is a generalization of the first fundamental theorem of calculus to two dimensions.
A counterexample is a form of counter proof.Given a hypothesis stating that is true for all , show that there exists a such that is false, contradicting the hypothesis.
A sequence such that either (1) for every , or (2) for every .
The logarithmic derivative of a function is defined as the derivative of the logarithm of a function. For example, the digamma function is defined as the logarithmic derivative of the gamma function,
The smallest value of a set, function, etc. The minimum value of a set of elements is denoted or , and is equal to the first element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the minimum is 1. The maximum and minimum are the simplest order statistics.The minimum value of a variable is commonly denoted (cf. Strang 1988, pp. 286-287 and 301-303) or (Golub and Van Loan 1996, p. 84). In this work, the convention is used.The minimum of a set of elements is implemented in the Wolfram Language as Min[list] and satisfies the identities(1)(2)A continuous function may assume a minimum at a single point or may have minima at a number of points. A global minimum of a function is the smallest value in the entire range of the function, while a local minimum is the smallest value in some local neighborhood.For a function which is continuous at a point , a necessary but not sufficient condition for to have a local..
The Weierstrass substitution is the trigonometric substitution which transforms an integral of the forminto one of the formAccording to Spivak (2006, pp. 382-383), this is undoubtably the world's sneakiest substitution.The Weierstrass substitution can also be useful in computing a Gröbner basis to eliminate trigonometric functions from a system of equations (Trott 2006, p. 39).
A box integral for dimension with parameters and is defined as the expectation of distance from a fixed point of a point chosen at random over the unit -cube,(1)(Bailey et al. 2006).Two special cases include(2)(3)which, with , correspond to hypercube point picking (to a fixed vertex) and hypercube line picking, respectively.Hypercube point picking to the center isgiven by(4)
Integrals over the unit square arising in geometricprobability are(1)which give the average distances in square point picking from a point picked at random in a unit square to a corner and to the center, respectively.Unit square integrals involving the absolute valueare given by(2)(3)for and , respectively.Another simple integral is given by(4)(Bailey et al. 2007, p. 67). Squaring the denominator gives(5)(6)(7)(8)(9)(OEIS A093754; M. Trott, pers. comm.), where is Catalan's constant and is a generalized hypergeometric function. A related integral is given by(10)which diverges in the Riemannian sense, as can quickly seen by transforming to polar coordinates. However, taking instead Hadamard integral to discard the divergent portion inside the unit circle gives(11)(12)(13)(14)(OEIS A093753), where is Catalan's constant.A collection of beautiful integrals over the unit squareare given by Guillera and Sondow..
Faà di Bruno's formula gives an explicit equation for the th derivative of the composition . If and are functions for which all necessary derivatives are defined, then(1)where and the sum is over all partitions of , i.e., values of , ..., such that(2)(Roman 1980).It can also be expressed in terms of Bell polynomial as(3)(M. Alekseyev, pers. comm., Nov. 3, 2006).Faà di Bruno's formula can be cast in a framework that is a special case of a Hopf algebra (Figueroa and Gracia-Bondía 2005).The first few derivatives for symbolic and are given by(4)(5)(6)
A smooth function is a function that has continuous derivatives up to some desired order over some domain. A function can therefore be said to be smooth over a restricted interval such as or . The number of continuous derivatives necessary for a function to be considered smooth depends on the problem at hand, and may vary from two to infinity. A function for which all orders of derivatives are continuous is called a C-infty-function.
The directional derivative is the rate at which the function changes at a point in the direction . It is a vector form of the usual derivative, and can be defined as(1)(2)where is called "nabla" or "del" and denotes a unit vector.The directional derivative is also often written in the notation(3)(4)where denotes a unit vector in any given direction and denotes a partial derivative.Let be a unit vector in Cartesian coordinates, so(5)then(6)
A vector derivative is a derivative taken with respect to a vector field. Vector derivatives are extremely important in physics, where they arise throughout fluid mechanics, electricity and magnetism, elasticity, and many other areas of theoretical and applied physics.The following table summarizes the names and notations for various vector derivatives.symbolvector derivativegradientLaplacian or vector Laplacian or directional derivativedivergencecurlconvective derivativeVector derivatives can be combined in different ways, producing sets of identities that are also very important in physics.Vector derivative identities involving the curl include(1)(2)(3)(4)(5)In Cartesian coordinates(6)(7)In spherical coordinates,(8)(9)(10)Vector derivative identities involving the divergenceinclude(11)(12)(13)(14)(15)In Cartesian coordinates,(16)(17)(18)(19)(20)(21)In spherical coordinates,(22)(23)(24)(25)(26)(27)By..
In general, "a" calculus is an abstract theory developed in a purely formal way."The" calculus, more properly called analysis (or real analysis or, in older literature, infinitesimal analysis), is the branch of mathematics studying the rate of change of quantities (which can be interpreted as slopes of curves) and the length, area, and volume of objects. The calculus is sometimes divided into differential and integral calculus, concerned with derivativesand integralsrespectively.While ideas related to calculus had been known for some time (Archimedes' method of exhaustion was a form of calculus), it was not until the independent work of Newton and Leibniz that the modern elegant tools and ideas of calculus were developed. Even so, many years elapsed until the subject was put on a mathematically rigorous footing by mathematicians such as Weierstrass...
The integral of over the unit disk is given by(1)(2)(3)(4)In general,(5)provided .Additional integrals include(6)(7)(8)
The Poisson integral with ,where is a Bessel function of the first kind.
In mathematics, a small positive infinitesimal quantity, usually denoted or , whose limit is usually taken as .The late mathematician P. Erdős also used the term "epsilons" to refer to children (Hoffman 1998, p. 4).
Graph the Riemann sum of as x goes from to using rectangles taking samples at the Maximum Minimum Left Right Midpoint Print estimated and actual areas? Rectangle Color Plot Color Light Gray Dark Gray Black White Red Orange Yellow Green Blue Purple Light Gray Dark Gray Black White Red Orange Yellow Green Blue Purple Replot Let a closed interval be partitioned by points , where the lengths of the resulting intervals between the points are denoted , , ..., . Let be an arbitrary point in the th subinterval. Then the quantityis called a Riemann sum for a given function and partition, and the value is called the mesh size of the partition.If the limit of the Riemann sums exists as , this limit is known as the Riemann integral of over the interval . The shaded areas in the above plots show the lower and upper sums for a constant mesh size...
For ,where is the gamma function.
Hadjicostas's formula is a generalization of the unitsquare double integral(1)(Sondow 2003, 2005; Borwein et al. 2004, p. 49), where is the Euler-Mascheroni constant. It states(2)for , where is the gamma function and is the Riemann zeta function (although care must be taken at because of the removable singularity present there). It was conjectured by Hadjicostas (2004) and almost immediately proved by Chapman (2004). The special case gives Beukers's integral for ,(3)(Beukers 1979). At , the formula is related to Beukers's integral for Apéry's constant , which is how interest in this class of integrals originally arose.There is an analogous formula(4)for , due to Sondow (2005), where is the Dirichlet eta function. This includes the special cases(5)(6)(7)(OEIS A094640; Sondow 2005) and(8)(9)(OEIS A103130), where is the Glaisher-Kinkelin constant (Sondow 2005)...
The term asymptotic means approaching a value or curve arbitrarily closely (i.e., as some sort of limit is taken). A line or curve that is asymptotic to given curve is called the asymptote of .Hardy and Wright (1979, p. 7) use the symbol to denote that one quantity is asymptotic to another. If , then Hardy and Wright say that and are of the same order of magnitude.
The operator representing the computation of a derivative,(1)sometimes also called the Newton-Leibniz operator. The second derivative is then denoted , the third , etc. The integral is denoted .The differential operator satisfies the identity(2)where is a Hermite polynomial (Arfken 1985, p. 718), where the first few cases are given explicitly by(3)(4)(5)(6)(7)(8)The symbol can be used to denote the operator(9)(Bailey 1935, p. 8). A fundamental identity for this operator is given by(10)where is a Stirling number of the second kind (Roman 1984, p. 144), giving(11)(12)(13)(14)and so on (OEIS A008277). Special cases include(15)(16)(17)A shifted version of the identity is given by(18)(Roman 1984, p. 146).
A fractional derivative of order 1/2. The semiderivative of is given byso the semiderivative of the constant function is given by
The second fundamental theorem of calculus holds for a continuous function on an open interval and any point in , and states that if is defined by the integral (antiderivative)thenat each point in , where is the derivative of .
The Leibniz integral rule gives a formula for differentiation of a definiteintegral whose limits are functions of the differential variable,(1)It is sometimes known as differentiation under the integral sign.This rule can be used to evaluate certain unusual definite integrals such as(2)(3)for (Woods 1926).Feynman (1997, pp. 69-72) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again."
A real function is said to be differentiable at a point if its derivative exists at that point. The notion of differentiability can also be extended to complex functions (leading to the Cauchy-Riemann equations and the theory of holomorphic functions), although a few additional subtleties arise in complex differentiability that are not present in the real case.Amazingly, there exist continuous functions which are nowhere differentiable. Two examples are the Blancmange function and Weierstrass function. Hermite (1893) is said to have opined, "I turn away with fright and horror from this lamentable evil of functions which do not have derivatives" (Kline 1990, p. 973).
Suppose is a function of that is twice differentiable at a stationary point . 1. If , then has a local minimum at . 2. If , then has a local maximum at . The extremum test gives slightly more general conditions under which a function with is a maximum or minimum.If is a two-dimensional function that has a local extremum at a point and has continuous partial derivatives at this point, then and . The second partial derivatives test classifies the point as a local maximum or local minimum.Define the second derivative testdiscriminant as(1)(2)Then 1. If and , the point is a local minimum. 2. If and , the point is a local maximum. 3. If , the point is a saddle point. 4. If , higher order tests must be used.
A generalization of the product rule for expressingarbitrary-order derivatives of products of functions,where is a binomial coefficient. This can also be written explicitly as(Roman 1980), where is the differential operator.
Implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable while treating the other variables as unspecified functions of .For example, the implicit equation(1)can be solved for(2)and differentiated directly to yield(3)Differentiating implicitly instead gives (4)(5)(6)(7)Plugging in verifies that this approach gives the same result as before.Implicit differentiation is especially useful when is needed, but it is difficult or inconvenient to solve for in terms of .
The first fundamental theorem of calculus states that, if is continuous on the closed interval and is the indefinite integral of on , then(1)This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.The second fundamental theorem of calculus holds for a continuous function on an open interval and any point in , and states that if is defined by(2)then(3)at each point in .The fundamental theorem of calculus along curves states that if has a continuous indefinite integral in a region containing a parameterized curve for , then(4)
If is differentiable at the point and is differentiable at the point , then is differentiable at . Furthermore, let and , then(1)There are a number of related results that also go under the name of "chain rules." For example, if , , and , then(2)The "general" chain rule applies to two sets of functions(3)(4)(5)and(6)(7)(8)Defining the Jacobi rotation matrix by(9)and similarly for and , then gives(10)In differential form, this becomes(11)(Kaplan 1984).
A related rates problem is the determination of the rate at which a function defined in terms of other functions changes.Related rates problems can be solved by computing derivatives for appropriate combinationsof functions using rules such as the chain rule(1)(for and ),product rule(2)quotient rule(3)sum rule(4)or power rule(5)
Let and be real numbers (usually taken as and ). The Dirichlet function is defined by(1)and is discontinuous everywhere. The Dirichletfunction can be written analytically as(2)Because the Dirichlet function cannot be plotted without producing a solid blend of lines, a modified version, sometimes itself known as the Dirichlet function (Bruckner et al. 2008), Thomae function (Beanland et al. 2009), or small Riemann function (Ballone 2010, p. 11), can be defined as(3)(Dixon 1991), illustrated above. This function is continuous at irrational and discontinuous at rational (although a small interval around an irrational point contains infinitely many rational points, these rationals will have very large denominators). When viewed from a corner along the line in normal perspective, a quadrant of Euclid's orchard turns into the modified Dirichlet function (Gosper)...
The pathological function(originally defined for ) that is continuous but differentiable only on a set of points of measure zero. The plots above show for (red), 3 (green), and 4 (blue).The function was published by Weierstrass but, according to lectures and writings by Kronecker and Weierstrass, Riemann seems to have claimed already in 1861 that the function is not differentiable on a set dense in the reals. However, Ullrich (1997) indicates that there is insufficient evidence to decide whether Riemann actually bothered to give a detailed proof for this claim. du Bois-Reymond (1875) stated without proof that every interval of contains points at which does not have a finite derivative, and Hardy (1916) proved that it does not have a finite derivative at any irrational and some of the rational points. Gerver (1970) and Smith (1972) subsequently proved that has a finite derivative (namely, 1/2) at the set of points where and are integers. Gerver..
A real-valued univariate function has a jump discontinuity at a point in its domain provided that(1)and(2)both exist and that .The notion of jump discontinuity shouldn't be confused with the rarely-utilized convention whereby the term jump is used to define any sort of functional discontinuity.The figure above shows an example of a function having a jump discontinuity at a point in its domain.Though less algebraically-trivial than removable discontinuities, jump discontinuities are far less ill-behaved than other types of singularities such as infinite discontinuities. This fact can be seen in a number of scenarios, e.g., in the fact that univariate monotone functions can have at most countably many discontinuities (Royden and Fitzpatrick 2010), the worst of which can be jump discontinuities (Zakon 2004).Unsurprisingly, the definition given above can be generalized to include jump discontinuities in multivariate real-valued..
A real-valued univariate function is said to have a removable discontinuity at a point in its domain provided that both and(1)exist while . Removable discontinuities are so named because one can "remove" this point of discontinuity by defining an almost everywhere identical function of the form(2)which necessarily is everywhere-continuous.The figure above shows the piecewise function(3)a function for which while . In particular, has a removable discontinuity at due to the fact that defining a function as discussed above and satisfying would yield an everywhere-continuous version of .Note that the given definition of removable discontinuity fails to apply to functions for which and for which fails to exist; in particular, the above definition allows one only to talk about a function being discontinuous at points for which it is defined. This definition isn't uniform, however, and as a result, some authors claim that, e.g.,..
A real-valued univariate function is said to have an infinite discontinuity at a point in its domain provided that either (or both) of the lower or upper limits of fails to exist as tends to .Infinite discontinuities are sometimes referred to as essential discontinuities, phraseology indicative of the fact that such points of discontinuity are considered to be "more severe" than either removable or jump discontinuities.The figure above shows the piecewise function(1)a function for which both and fail to exist. In particular, has an infinite discontinuity at .It is not uncommon for authors to say that univariate functions defined on a domain and admitting vertical asymptotes of the form have infinite discontinuities there though, strictly speaking, this terminology is incorrect unless such functions are defined piecewise so that . For example, the function has vertical asymptotes at , , though it has no discontinuities of any..
A function is said to be concave on an interval if, for any points and in , the function is convex on that interval (Gradshteyn and Ryzhik 2000).
A closed interval is an interval that includes all of its limit points. If the endpoints of the interval are finite numbers and , then the interval is denoted . If one of the endpoints is , then the interval still contains all of its limit points (although not all of its endpoints), so and are also closed intervals, as is the interval .
A function is said to be piecewise constant if it is locally constant in connected regions separated by a possibly infinite number of lower-dimensional boundaries. The Heaviside step function, rectangle function, and square wave are examples of one-dimensional piecewise constant functions. Examples in two dimensions include and (illustrated above) for a complex number, the real part, and the floor function. The nearest integer function is also piecewise constant.
A function is said to have bounded variation if, over the closed interval , there exists an such that(1)for all .The space of functions of bounded variation is denoted "BV," and has theseminorm(2)where ranges over all compactly supported functions bounded by and 1. The seminorm is equal to the supremum over all sums above, and is also equal to (when this expression makes sense).On the interval , the function (purple) is of bounded variation, but (red) is not. More generally, a function is locally of bounded variation in a domain if is locally integrable, , and for all open subsets , with compact closure in , and all smooth vector fields compactly supported in ,(3)div denotes divergence and is a constant which only depends on the choice of and .Such functions form the space . They may not be differentiable, but by the Riesz representation theorem, the derivative of a -function is a regular Borel measure . Functions of bounded variation also..
If a univariate real function has a single critical point and that point is a local maximum, then has its global maximum there (Wagon 1991, p. 87). The test breaks downs for bivariate functions, but does hold for bivariate polynomials of degree . Such exceptions include(1)(2)(3)(Rosenholtz and Smylie 1985, Wagon 1991). Note that equation (3) has discontinuous partial derivatives and , and and .
A function has the normal order if is approximately for almost all values of . More precisely, iffor every positive and almost all values of , then the normal order of is .
A discontinuity is point at which a mathematical object is discontinuous. The left figure above illustrates a discontinuity in a one-variable function while the right figure illustrates a discontinuity of a two-variable function plotted as a surface in . In the latter case, the discontinuity is a branch cut along the negative real axis of the natural logarithm for complex .Some authors refer to a discontinuity of a function as a jump,though this is rarely utilized in the literature.Though defined identically, discontinuities of univariate functions are considerably different than those of multivariate functions. One of the main differences between these cases exists with regards to classifying the discontinuities, a caveat discussed more at length below.In the case of a one-variable real-valued function , there are precisely three families of discontinuities that can occur. 1. The simplest type is the so-called removablediscontinuity...
Numerical differentiation is the process of finding the numerical value of a derivative of a given function at a given point. In general, numerical differentiation is more difficult than numerical integration. This is because while numerical integration requires only good continuity properties of the function being integrated, numerical differentiation requires more complicated properties such as Lipschitz classes. Numerical differentiation is implemented as ND[f, x, x0, Scale -> scale] in the Wolfram Language package NumericalCalculus` .There are many applications where derivatives need to be computed numerically. Thesimplest approach simply uses the definition of the derivativefor some small numerical value of .
A piecewise function is a function that is defined on a sequence of intervals. Acommon example is the absolute value,(1)Piecewise functions are implemented in the Wolfram Language as Piecewise[val1, cond1, val2, cond2, ...].Additional piecewise functions include the Heaviside step function, rectangle function, and triangle function.Semicolons and commas are sometimes used at the end of either the left or the right column, with particular usage apparently depending on the author. The words "if" and "for" are sometimes used in the right column, as is "otherwise" for the final (default) case.For example, Knuth (1996, pp. 175 and 180) uses the notations(2)(3)(4)both with and without the left-column commas. Similarly, Arfken (1985, pp. 488-489) uses(5)which lacks semicolons but only sometimes lacks right-column commas.In this work, commas and semicolons are not used...
The derivative of a function represents an infinitesimalchange in the function with respect to one of its variables.The "simple" derivative of a function with respect to a variable is denoted either or(1)often written in-line as . When derivatives are taken with respect to time, they are often denoted using Newton's overdot notation for fluxions,(2)The "d-ism" of Leibniz's eventually won the notation battle against the "dotage" of Newton's fluxion notation (P. Ion, pers. comm., Aug. 18, 2006).When a derivative is taken times, the notation or(3)is used, with(4)etc., the corresponding fluxion notation.When a function depends on more than one variable, a partial derivative(5)can be used to specify the derivative with respect to one or more variables.The derivative of a function with respect to the variable is defined as(6)but may also be calculated more symmetrically as(7)provided the..
A definite integral is an integral(1)with upper and lower limits. If is restricted to lie on the real line, the definite integral is known as a Riemann integral (which is the usual definition encountered in elementary textbooks). However, a general definite integral is taken in the complex plane, resulting in the contour integral(2)with , , and in general being complex numbers and the path of integration from to known as a contour.The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if is the indefinite integral for a continuous function , then(3)This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. Definite integrals may be evaluated in the Wolfram Language using Integrate[f, x, a, b].The question of which definite..
A monotonic function is a function which is either entirely nonincreasing or nondecreasing. A function is monotonic if its first derivative (which need not be continuous) does not change sign.The term monotonic may also be used to describe set functions which map subsets of the domain to non-decreasing values of the codomain. In particular, if is a set function from a collection of sets to an ordered set , then is said to be monotone if whenever as elements of , . This particular definition comes up frequently in measure theory where many of the families of functions defined (including outer measure, premeasure, and measure) begin by considering monotonic set functions.
A multiple integral is a set of integrals taken over variables, e.g.,An th-order integral corresponds, in general, to an -dimensional volume (i.e., a content), with corresponding to an area. In an indefinite multiple integral, the order in which the integrals are carried out can be varied at will; for definite multiple integrals, care must be taken to correctly transform the limits if the order is changed.In traditional mathematical notation, a multiple integral of a function that is first performed over a variable and then performed over a variable is writtenIn the Wolfram Language, this would be entered as Integrate[f[x, y], x, x1, x2, y, y1[x], y2[x]], where the order of the integration variables is specified in the order that the integral signs appear on the left, which is opposite to the actual order of integration...
Let be differentiable on the open interval and continuous on the closed interval . Then if , then there is at least one point where .Note that in elementary texts, the additional (but superfluous) condition is sometimes added (e.g., Anton 1999, p. 260).
If is continuous on a closed interval , and is any number between and inclusive, then there is at least one number in the closed interval such that .The theorem is proven by observing that is connected because the image of a connected set under a continuous function is connected, where denotes the image of the interval under the function . Since is between and , it must be in this connected set.The intermediate value theorem (or rather, the space case with , corresponding to Bolzano's theorem) was first proved by Bolzano (1817). While Bolzano's used techniques which were considered especially rigorous for his time, they are regarded as nonrigorous in modern times (Grabiner 1983).
Let be analytic on a domain , and assume that never vanishes. Then if there is a point such that for all , then is constant.Let be a bounded domain, let be a continuous function on the closed set that is analytic on , and assume that never vanishes on . Then the minimum value of on (which always exists) must occur on . In other words,
For what value of is a maximum? The maximum occurs at , where(1)which is zero at and gives a maximum of(2)(OEIS A073229).The function has inflection points at (OEIS A093157) and (OEIS A103476), which are the roots of(3)
A point at which the derivative of a function vanishes,A stationary point may be a minimum, maximum,or inflection point.
Let be a domain, and let be an analytic function on . Then if there is a point such that for all , then is constant. The following slightly sharper version can also be formulated. Let be a domain, and let be an analytic function on . Then if there is a point at which has a local maximum, then is constant.Furthermore, let be a bounded domain, and let be a continuous function on the closed set that is analytic on . Then the maximum value of on (which always exists) occurs on the boundary . In other words,The maximum modulus theorem is not always true on an unbounded domain.
(1)(2)where are partial derivatives.
Consider a function in one dimension. If has a relative extremum at , then either or is not differentiable at . Either the first or second derivative tests may be used to locate relative extrema of the first kind.A necessary condition for to have a minimum (maximum) at isandA sufficient condition is and (). Let , , ..., , but . Then has a local maximum at if is odd and , and has a local minimum at if is odd and . There is a saddle point at if is even.
An extremum is a maximum or minimum. An extremum may be local (a.k.a. a relative extremum; an extremum in a given region which is not the overall maximum or minimum) or global. Functions with many extrema can be very difficult to graph.Notorious examples include the functions and near , plotted above.Another pathological example is near 0 and 1, which hasextrema in the closed interval [0,1] (Mulcahy1996).
A point of a function or surface which is a stationary point but not an extremum. An example of a one-dimensional function with a saddle point is , which has(1)(2)(3)This function has a saddle point at by the extremum test since and .Surfaces can also have saddle points, which the second derivative test can sometimes be used to identify. Examples of surfaces with a saddle point include the handkerchief surface and monkey saddle.
If a function is continuous on a closed interval , then has both a maximum and a minimum on . If has an extremum on an open interval , then the extremum occurs at a critical point. This theorem is sometimes also called the Weierstrass extreme value theorem.The standard proof of the first proceeds by noting that is the continuous image of a compact set on the interval , so it must itself be compact. Since is compact, it follows that the image must also be compact.
A function has critical points at all points where or is not differentiable. A function has critical points where the gradient or or the partial derivative is not defined.
Let the least term of a sequence be a term which is smaller than all but a finite number of the terms which are equal to . Then is called the lower limit of the sequence.A lower limit of a seriesis said to exist if, for every , for infinitely many values of and if no number less than has this property.
Let be a nonempty set of real numbers that has a lower bound. A number is the called the greatest lower bound (or the infimum, denoted ) for iff it satisfies the following properties: 1. for all . 2. For all real numbers , if is a lower bound for , then .
Let the greatest term of a sequence be a term which is greater than all but a finite number of the terms which are equal to . Then is called the upper limit of the sequence.An upper limit of a seriesis said to exist if, for every , for infinitely many values of and if no number larger than has this property.
A number such that for all , there exists a member of the set different from such that .The topological definition of limit point of is that is a point such that every open set around it contains at least one point of different from .
The supremum is the least upper bound of a set , defined as a quantity such that no member of the set exceeds , but if is any positive quantity, however small, there is a member that exceeds (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., does not exist), it is denoted (or sometimes simply for short). The supremum is implemented in the Wolfram Language as MaxValue[f, constr, vars].More formally, the supremum for a (nonempty) subset of the affinely extended real numbers is the smallest value such that for all we have . Using this definition, always exists and, in particular, .Whenever a supremum exists, its value is unique. On the real line, the supremum of a set is the same as the supremum of its set closure.Consider the real numbers with their usual order. Then for any set , the supremum exists (in ) if and only if is bounded from above and nonempty...
Let be a nonempty set of real numbers that has an upper bound. Then a number is called the least upper bound (or the supremum, denoted ) for iff it satisfies the following properties: 1. for all . 2. For all real numbers , if is an upper bound for , then .
Let lim stand for the limit , , , , or , and suppose that lim and lim are both zero or are both . If(1)has a finite value or if the limit is , then(2)Historically, this result first appeared in l'Hospital's 1696 treatise, which was the first textbook on differential calculus. Within the book, l'Hospital thanks the Bernoulli brothers for their assistance and their discoveries. An earlier letter by John Bernoulli gives both the rule and its proof, so it seems likely that Bernoulli discovered the rule (Larson et al. 1999, p. 524).Note that l'Hospital's name is commonly seen spelled both "l'Hospital" (e.g., Maurer 1981, p. 426; Arfken 1985, p. 310) and "l'Hôpital" (e.g., Maurer 1981, p. 426; Gray 1997, p. 529), the two being equivalent in French spelling.L'Hospital's rule occasionally fails to yield useful results, as in the case of the function , illustrated above. Repeatedly applying..
Let for all in some open interval containing . Ifthen .
A theorem about (or providing an equivalent definition of) compact sets, originally due to Georg Cantor. Given a decreasing sequence of bounded nonempty closed setsin the real numbers, then Cantor's intersection theorem states that there must exist a point in their intersection, for all . For example, . It is also true in higher dimensions of Euclidean space.Note that the hypotheses stated above are crucial. The infinite intersection of open intervals may be empty, for instance . Also, the infinite intersection of unbounded closed sets may be empty, e.g., .Cantor's intersection theorem is closely related to the Heine-Borel theorem and Bolzano-Weierstrass theorem, each of which can be easily derived from either of the other two. It can be used to show that the Cantor set is nonempty.
The infimum is the greatest lower bound of a set , defined as a quantity such that no member of the set is less than , but if is any positive quantity, however small, there is always one member that is less than (Jeffreys and Jeffreys 1988). When it exists (which is not required by this definition, e.g., does not exist), the infimum is denoted or . The infimum is implemented in the Wolfram Language as MinValue[f, constr, vars].Consider the real numbers with their usual order. Then for any set , the infimum exists (in ) if and only if is bounded from below and nonempty.More formally, the infimum for a (nonempty) subset of the affinely extended real numbers is the largest value such that for all we have . Using this definition, always exists and, in particular, .Whenever an infimum exists, its value is unique...
A function is said to be nonincreasing on an interval if for all , where . Conversely, a function is said to be nondecreasing on an interval if for all with .
A function is said to be nondecreasing on an interval if for all , where . Conversely, a function is said to be nonincreasing on an interval if for all with .
Denote the th derivative and the -fold integral . Then(1)Now, if the equation(2)for the multiple integral is true for , then(3)(4)Interchanging the order of integration gives(5)But (3) is true for , so it is also true for all by induction. The fractional integral of of order can then be defined by(6)where is the gamma function.More generally, the Riemann-Liouville operatorof fractional integration is defined as(7)for with (Oldham and Spanier 1974, Miller and Ross 1993, Srivastava and Saxena 2001, Saxena 2002).The fractional integral of order 1/2 is called a semi-integral.Few functions have a fractional integral expressible in terms of elementaryfunctions. Exceptions include(8)(9)(10)(11)where is a lower incomplete gamma function and is the Et-function. From (10), the fractional integral of the constant function is given by(12)(13)A fractional derivative can also be similarly defined. The study of fractional derivatives and..
The solution to the differential equation(1)is(2)where(3)(4) is the Et-function, and is the gamma function.
The fractional derivative of of order (if it exists) can be defined in terms of the fractional integral as(1)where is an integer , where is the ceiling function. The semiderivative corresponds to .The fractional derivative of the function is given by(2)(3)(4)(5)(6)for . The fractional derivative of the constant function is then given by(7)(8)The fractional derivate of the Et-functionis given by(9)for .It is always true that, for ,(10)but not always true that(11)A fractional integral can also be similarly defined. The study of fractional derivatives and integrals is called fractional calculus.
The study of an extension of derivatives and integrals to noninteger orders. Fractional calculus is based on the definition of the fractional integral aswhere is the gamma function. From this equation, fractional derivatives can also be defined.
The derivative identity (1)(2)(3)(4)where denotes the derivative of . The Leibniz identity extends the product rule to higher-order derivatives.
The first fundamental theorem of calculus states that, if is continuous on the closed interval and is the indefinite integral of on , thenThis result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.
Suppose is continuous at a stationary point . 1. If on an open interval extending left from and on an open interval extending right from , then has a local maximum (possibly a global maximum) at . 2. If on an open interval extending left from and on an open interval extending right from , then has a local minimum (possibly a global minimum) at . 3. If has the same sign on an open interval extending left from and on an open interval extending right from , then has an inflection point at .
Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation.(1)The above partial derivative is sometimes denoted for brevity.Partial derivatives can also be taken with respect to multiple variables, as denoted for examples(2)(3)(4)Such partial derivatives involving more than one variable are called mixedpartial derivatives.For a "nice" two-dimensional function (i.e., one for which , , , , exist and are continuous in a neighborhood ), then(5)More generally, for "nice" functions, mixed partial derivatives must be equal regardless of the order in which the differentiation is performed, so it also is true that(6)If the continuity requirement for mixed partials is dropped, it is possible to construct functions for which mixed partials are not equal. An example is the function(7)which has and (Wagon 1991). This..
There are at least two meanings of the term "total derivative" in mathematics.The first is as an alternate term for the convectivederivative.The total derivative is the derivative with respect to of the function that depends on the variable not only directly but also via the intermediate variables . It can be calculated using the formulaThe total derivative of a function with respect to is implemented in the Wolfram Language as Dt[f[t, x, y, ...], t].
The Blancmange function, also called the Takagi fractal curve (Peitgen and Saupe 1988), is a pathological continuous function which is nowhere differentiable. Its name derives from the resemblance of its first iteration to the shape of the dessert commonly made with milk or cream and sugar thickened with gelatin.The iterations towards the continuous function are batrachions resembling the Hofstadter-Conway $10,000 sequence. The first six iterations are illustrated below. The th iteration contains points, where , and can be obtained by setting , lettingand looping over to 1 by steps of and to by steps of .
Let be the revenue for a production , the cost, and the profit. Thenand the marginal profit for the th unit is defined bywhere , , and are the derivatives of , , and , respectively.
The term limit comes about relative to a number of topics from several different branches of mathematics.A sequence of elements in a topological space is said to have limit provided that for each neighborhood of , there exists a natural number so that for all . This very general definition can be specialized in the event that is a metric space, whence one says that a sequence in has limit if for all , there exists a natural number so that(1)for all . In many commonly-encountered scenarios, limits are unique, whereby one says that is the limit of and writes(2)On the other hand, a sequence of elements from an metric space may have several - even infinitely many - different limits provided that is equipped with a topology which fails to be T2. One reads the expression in (1) as "the limit as approaches infinity of is ."The topological notion of convergence can be rewritten to accommodate a wider array of topological spaces by utilizing the language..
The term "pathological" is used in mathematics to refer to an example specifically cooked up to violate certain almost universally valid properties. Pathological problems often provide interesting examples of counterintuitive behavior, as well as serving as an excellent illustration of why very detailed conditions of applicability are required in order for many mathematical statements to be universally true.For example, the pathological Weierstrass and Blancmange functions are examples of a continuous function that is nowhere differentiable, a possibility that many students of calculus find quite surprising.In 1899, Poincaré remarked on the proliferation of pathological functions, "Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less..
The largest value of a set, function, etc. The maximum value of a set of elements is denoted or , and is equal to the last element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the maximum is 5. The maximum and minimum are the simplest order statistics.The maximum value of a variable is commonly denoted (Strang 1988, pp. 286-287 and 301-303) or (Golub and Van Loan 1996, p. 74). In this work, the convention is used.The maximum of a set of elements is implemented in the Wolfram Language as Max[list] and satisfies the identities(1)(2)Definite integrals include(3)(4)A continuous function may assume a maximum at a single point or may have maxima at a number of points. A global maximum of a function is the largest value in the entire range of the function, and a local maximum is the largest value in some local neighborhood.For a function which is continuous at a point , a necessary but not sufficient condition..
The Cauchy principal value of a finite integral of a function about a point with is given by(Henrici 1988, p. 261; Whittaker and Watson 1990, p. 117; Bronshtein and Semendyayev 1997, p. 283). Similarly, the Cauchy principal value of a doubly infinite integral of a function is defined byThe Cauchy principal value is also known as the principal value integral (Henrici 1988, p. 261), finite part (Vladimirov 1971), or partie finie (Vladimirov 1971).The Cauchy principal value of an integral having no nonsimple poles can be computed in the Wolfram Language using Integrate[f, x, a, b, PrincipalValue -> True]. Cauchy principal values of functions with possibly nonsimple poles can be computed numerically using the "CauchyPrincipalValue" method in NIntegrate.Cauchy principal values are important in the theory of generalized functions, where they allow extension of results to .Cauchy principal values..
There are several commonly used methods of defining the slippery, but extremely important, concept of a continuous function (which, depending on context, may also be called a continuous map). The space of continuous functions is denoted , and corresponds to the case of a C-k function.A continuous function can be formally defined as a function where the pre-image of every open set in is open in . More concretely, a function in a single variable is said to be continuous at point if 1. is defined, so that is in the domain of . 2. exists for in the domain of . 3. , where lim denotes a limit.Many mathematicians prefer to define the continuity of a function via a so-called epsilon-delta definition of a limit. In this formalism, a limit of function as approaches a point ,(1)is defined when, given any , a can be found such that for every in some domain and within the neighborhood of of radius (except possibly itself),(2)Then if is in and(3) is said to be continuous at..
The term Borel hierarchy is used to describe a collection of subsets of defined inductively as follows: Level one consists of all open and closed subsets of , and upon having defined levels , level is obtained by taking countable unions and intersections of the previous level. In particular, level two of the hierarchy consists of the collections of all Fsigma and Gdelta sets while subsequent levels are described by way of the rather confusingly-named collection of sets of the form , , , , , etc.The collection of sets across all levels of the Borel hierarchy is the Borel sigma-algebra. As such, the Borel hierarchy is fundamental to the study of measure theory.More general notions of the Borel hierarchy (and thus of Borel sets, etc.) are introduced and studied as part of various areas of set theory, topology, and mathematical logic...
An algorithm for finding the nearest local minimum of a function which presupposes that the gradient of the function can be computed. The method of steepest descent, also called the gradient descent method, starts at a point and, as many times as needed, moves from to by minimizing along the line extending from in the direction of , the local downhill gradient.When applied to a 1-dimensional function , the method takes the form of iteratingfrom a starting point for some small until a fixed point is reached. The results are illustrated above for the function with and starting points and 0.01, respectively.This method has the severe drawback of requiring a great many iterations for functions which have long, narrow valley structures. In such cases, a conjugate gradient method is preferable...
Lagrange multipliers, also called Lagrangian multipliers (e.g., Arfken 1985, p. 945), can be used to find the extrema of a multivariate function subject to the constraint , where and are functions with continuous first partial derivatives on the open set containing the curve , and at any point on the curve (where is the gradient).For an extremum of to exist on , the gradient of must line up with the gradient of . In the illustration above, is shown in red, in blue, and the intersection of and is indicated in light blue. The gradient is a horizontal vector (i.e., it has no -component) that shows the direction that the function increases; for it is perpendicular to the curve, which is a straight line in this case. If the two gradients are in the same direction, then one is a multiple () of the other, so(1)The two vectors are equal, so all of their components are as well, giving(2)for all , ..., , where the constant is called the Lagrange multiplier.The extremum..
At the age of 17, Bernard Mares proposed the definite integral (Borwein and Bailey2003, p. 26; Bailey et al. 2006)(1)(2)(OEIS A091473). Although this is within of ,(3)(OEIS A091494), it is not equal to it. Apparently, no closed-form solution is known for .Interestingly, the integral(4)(5)(Borwein et al. 2004, pp. 101-102) has a value fairly close to , but no other similar relationships seem to hold for other multipliers of the form or .The identity(6)can be expanded to yield(7)In fact,(8)where is a Borwein integral.