In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named in honour of English mathematician Brook Taylor. If the series uses the derivatives at zero, the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin.
The Taylor series of a real or complex function f(x) that is infinitely differentiable in a neighborhood of a real or complex number a, is the power series
which in a more compact form can be written as
where n! is the factorial of n and f (n)(a) denotes the nth derivative of f at the point a; the zeroth derivative of f is defined to be f itself and (x − a)^0 and 0! are both defined to be 1.
Often f(x) is equal to its Taylor series evaluated at x for all x sufficiently close to a. This is the main reason why Taylor series are important.
In the particular case where a = 0, the series is also called a Maclaurin series. Examples The Maclaurin series for any polynomial is the polynomial itself.
The Maclaurin series for (1 − x)^(− 1) is the geometric series
so the Taylor series for x^(− 1) at a = 1 is
By integrating the above Maclaurin series we find the Maclaurin series for -ln(1-x), where ln denotes the natural logarithm:
The Maclaurin series for the exponential function e^x at a = 0 is
The above expansion holds because the derivative of e^x is also e^x and e^0 equals 1. This leaves the terms (x − 0)^n in the numerator and n! in the denominator for each term in the infinite sum.
(As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sinx and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.)
A Monte Carlo method is a computational algorithm which relies on repeated random sampling to compute its results. Monte Carlo methods are often used when simulating physical and mathematical systems. Because of their reliance on repeated computation and random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer. Monte Carlo methods tend to be used when it is infeasible or impossible to compute an exact result with a deterministic algorithm.
몬테카를로 법(Monte Carlo method)은, 물리적, 수학적 시스템의 행동을 시뮬레이션하기 위한 계산 알고리즘이다. 다른 알고리즘과는 달리 통계학적이고, 일반적으로 무작위의 숫자를 사용한 비결정적인 방법이다. 스타니스와프 울람이 모나코의 유명한 도박의 도시 몬테카를로의 이름을 본따 명명하였다.
In mathematics, Monte Carlo integration is numerical quadrature using pseudorandom numbers. That is, Monte Carlo integration methods are algorithms for the approximate evaluation of definite integrals, usually multidimensional ones. The usual algorithms evaluate the integrand at a regular grid. Monte Carlo methods, however, randomly choose the points at which the integrand is evaluated.
Informally, to estimate the area of a domain D, first pick a simple domain d whose area is easily calculated and which contains D. Now pick a sequence of random points that fall within d. Some fraction of these points will also fall within D. The area of D is then estimated as this fraction multiplied by the area of d.
The traditional Monte Carlo algorithm distributes the evaluation points uniformly over the integration region. Adaptive algorithms such as VEGAS and MISER use importance sampling and stratified sampling techniques to get a better result.
In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle from the positive x-axis.
Definition The three coordinates (ρ, θ, φ) are defined as:
* ρ ≥ 0 is the distance from the origin to a given point P. * 0 ≤ θ ≤ 360° is the angle between the positive x-axis and the line from the origin to the P projected onto the xy-plane. * 0 ≤ φ ≤ 180° is the angle between the positive z-axis and the line formed between the origin and P.
θ is referred to as the azimuth, while φ is referred to as the zenith, colatitude or polar angle. θ and φ and lose significance when ρ = 0 and θ loses significance when sin(φ) = 0 (at φ = 0 and φ = 180°).
To plot a point from its spherical coordinates, go ρ units from the origin along the positive z-axis, rotate φ about the y-axis in the direction of the positive x-axis and rotate θ about the z-axis in the direction of the positive y-axis.
Coordinate system conversions As the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others.
Cartesian coordinate system The three spherical coordinates are obtained from Cartesian coordinates by: Note that the arctangent must be defined suitably so as to take account of the correct quadrant of y / x. The atan2 or equivalent function accomplishes this for computational purposes.
Conversely, Cartesian coordinates may be retrieved from spherical coordinates by:
Cylindrical coordinate system
The cylindrical coordinate system is a three-dimensional extrusion of the polar coordinate system, with an h coordinate to describe a point's height above or below the xy-plane. The full coordinate tuple is (r, θ, h).
Cylindrical coordinates may be converted into spherical coordinates by:
Spherical coordinates may be converted into cylindrical coordinates by:
The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted h) which measures the height of a point above the plane.
A point P is given as (r,θ,h). In terms of the Cartesian coordinate system:
* r is the distance from O to P', the orthogonal projection of the point P onto the XY plane. This is the same as the distance of P to the z-axis.
* θ is the angle between the positive x-axis and the line OP', measured counterclockwise.
* h is the same as z.
* Thus, the conversion function f from cylindrical coordinates to Cartesian coordinates is f(x,y,z) = (rcosθ,rsinθ,h). For use in physical sciences and technology, the recommended international standard notation is ρ, φ, z (ISO 31-11).
Some mathematicians indeed use (r,θ,z).
Cylindrical coordinates are useful in analyzing surfaces that are symmetrical about an axis, with the z-axis chosen as the axis of symmetry. For example, the infinitely long circular cylinder that has the Cartesian equation x2 + y2 = c2 has the very simple equation r = c in cylindrical coordinates. Hence the name "cylindrical" coordinates.
In mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point. To define the coordinates, two perpendicular directed lines (the x-axis or abscissa and the y-axis or ordinate), are specified, as well as the unit length, which is marked off on the two axes (see Figure 1). Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions.
Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by algebraic equations, namely equations satisfied by the coordinates of the points lying on the shape. Two-dimensional coordinate system
A Cartesian coordinate system in two dimensions is commonly defined by two axes, at right angles to each other, forming a plane (an xy-plane). The horizontal axis is normally labeled x, and the vertical axis is normally labeled y. In a three dimensional coordinate system, another axis, normally labeled z, is added, providing a third dimension of space measurement. The axes are commonly defined as mutually orthogonal to each other (each at a right angle to the other). (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles, and such systems are occasionally used today, although mostly as theoretical exercises.) All the points in a Cartesian coordinate system taken together form a so-called Cartesian plane. Equations that use the Cartesian coordinate system are called Cartesian equations.
The point of intersection, where the axes meet, is called the origin normally labeled O. The x and y axes define a plane that is referred to as the xy plane. Given each axis, choose a unit length, and mark off each unit along the axis, forming a grid. To specify a particular point on a two dimensional coordinate system, indicate the x unit first (abscissa), followed by the y unit (ordinate) in the form (x,y), an ordered pair.
The choice of letters comes from a convention, to use the latter part of the alphabet to indicate unknown values. In contrast, the first part of the alphabet was used to designate known values.
An example of a point P on the system is indicated in Figure 3, using the coordinate (3,5).
The intersection of the two axes creates four regions, called quadrants, indicated by the Roman numerals I (+,+), II (−,+), III (−,−), and IV (+,−). Conventionally, the quadrants are labeled counter-clockwise starting from the upper right ("northeast") quadrant. In the first quadrant, both coordinates are positive, in the second quadrant x-coordinates are negative and y-coordinates positive, in the third quadrant both coordinates are negative and in the fourth quadrant, x-coordinates are positive and y-coordinates negative (see table below.) Three-dimensional coordinate system
The three dimensional Cartesian coordinate system provides the three physical dimensions of space — length, width, and height. Figures 4 and 5, show two common ways of representing it. The three Cartesian axes defining the system are perpendicular to each other. The relevant coordinates are of the form (x,y,z). As an example, figure 4 shows two points plotted in a three-dimensional Cartesian coordinate system: P(3,0,5) and Q(−5,−5,7). The axes are depicted in a "world-coordinates" orientation with the z-axis pointing up. The x-, y-, and z-coordinates of a point can also be taken as the distances from the yz-plane, xz-plane, and xy-plane respectively. Figure 5 shows the distances of point P from the planes. The xy-, yz-, and xz-planes divide the three-dimensional space into eight subdivisions known as octants, similar to the quadrants of 2D space. While conventions have been established for the labelling of the four quadrants of the x-y plane, only the first octant of three dimensional space is labelled. It contains all of the points whose x, y, and z coordinates are positive. The z-coordinate is also called applicate.
수학에서 외적(外積)은 3차원 공간의 벡터들간의 이항연산의 일종이다. 연산의 결과가 스칼라인 내적(內積)과는 달리 연산의 결과가 벡터이기 때문에 벡터곱(vector product)이라고 불리기도 한다. 외적은 물리학의 각운동량, 로렌츠 힘등의 공식에 등장한다.
정의
두 벡터 a와 b의 외적은 a x b라 쓰고(수학자들은 a ^ b라고 쓰기도 한다.), 다음과 같이 정의된다.
식에서 θ는 a와 b가 이루는 각을 나타내며, n은 a와 b에 공통으로 수직인 단위벡터를 나타낸다.
위 정의에서의 문제점은 a와 b에 공통으로 수직인 방향이 두개라는 점이다. 즉, n이 수직이면, -n도 수직이다.
어느 것을 두 벡터의 외적으로 할 것인가는 벡터공간의 방향성(orientation)에 따라 달라진다. 오른손 좌표계에서는 a×b는, a, b, a×b가 오른손 좌표계 방향을 따르도록 정의되고, 왼손좌표계에선 마찬가지로 이 순서의 세 벡터가 왼손 좌표계 방향을 따르도록 정의된다.
In mathematics, the cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. By contrast, the dot product produces a scalar result. In many engineering and physics problems, it is handy to be able to construct a perpendicular vector from two existing vectors, and the cross product provides a means for doing so. The cross product is also known as the vector product, or Gibbs vector product.
The cross product is not defined except in three-dimensions (and the algebra defined by the cross product is not associative). Like the dot product, it depends on the metric of Euclidean space. Unlike the dot product, it also depends on the choice of orientation or "handedness." Certain features of the cross product can be generalized to other situations. For arbitrary choices of orientation, the cross product must be regarded not as a vector, but as a pseudovector. For arbitrary choices of metric, and in arbitrary dimensions, the cross product can be generalized by the exterior product of vectors, defining a two-form instead of a vector.
Definition
The cross product of two vectors a and b is denoted by a × b. In a three-dimensional Euclidean space, with a usual right-handed coordinate system, it is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
The cross product is given by the formula
where θ is the measure of the (non-obtuse) angle between a and b (0° ≤ θ ≤ 180°), a and b are the magnitudes of vectors a and b, and is a unit vector perpendicular to the plane containing a and b. If the vectors a and b are collinear (i.e., the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.
The direction of the vector is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of a and the middle finger in the direction of b. Then, the vector is coming out of the thumb (see the picture on the right).
Using the cross product requires the handedness of the coordinate system to be taken into account (as explicit in the definition above). If a left-handed coordinate system is used, the direction of the vector is given by the left-hand rule and points in the opposite direction.
This, however, creates a problem because transforming from one arbitrary reference system to another (e.g., a mirror image transformation from a right-handed to a left-handed coordinate system), should not change the direction of . The problem is clarified by realizing that the cross-product of two vectors is not a (true) vector, but rather a pseudovector. See cross product and handedness for more detail.
The dot product of two vectors (from an orthonormal vector space) a = [a1, a2, … , an] and b = [b1, b2, … , bn] is by definition:
where Σ denotes summation notation.
For example, the dot product of two three-dimensional vectors [1, 3, −5] and [4, −2, −1] is
Using matrix multiplication and treating the (column) vectors as n×1 matrices, the dot product can also be written as:
where aT denotes the transpose of the matrix a.
Using the example from above, this would result in a 1×3 matrix (i.e., vector) multiplied by a 3×1 vector (which, by virtue of the matrix multiplication, results in a 1×1 matrix, i.e., a scalar):
-------------------------------------------------------------------------------------------------- The dot product of two vectors a and b (sometimes called inner product, or, since its result is a scalar, the scalar product) is denoted by a ∙ b and is defined as:
where ||a|| and ||b|| denote the norm (or length) of a and b, and θ is the measure of the angle between a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point and then the length of a is multiplied with the length of that component of b that points in the same direction as a.
The dot product can also be defined as the sum of the products of the components of each vector:
where a and b are vectors of n dimensions; a1, a2, …, an are coordinates of a; and b1, b2, …, bn are coordinates of b.
This operation is often useful in physics; for instance, work is the dot product of force and displacement.
In calculus, l'Hôpital's rule (also spelled l'Hospital) uses derivatives to help compute limits with indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy computation of the limit. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital, who published the rule in his book Analyse des infiniment petits pour l'intelligence des lignes courbes (literal translation: Analysis of the infinitely small to understand curves) (1696), the first book to be written on differential calculus.
The rule is believed to be the work of Johann Bernoulli since l'Hôpital, a nobleman, paid Bernoulli a retainer of 300₣ per year to keep him updated on developments in calculus and to solve problems he had. (Moreover, the two signed a contract allowing l'Hôpital to use Bernoulli's discoveries in any way he wished.)[1] Among these problems was that of limits of indeterminate forms. When l'Hôpital published his book, he gave due credit to Bernoulli and, not wishing to take credit for any of the mathematics in the book, he published the work anonymously. Bernoulli, who was known for being extremely jealous, claimed to be the author of the entire work, and until recently, it was believed to be so. Nevertheless, the rule was named for l'Hôpital, who never claimed to have invented it in the first place[2].
The Stolz-Cesàro theorem is a similar result involving limits of sequences, and using finite difference operators rather than derivatives. Introduction
In simple cases, l'Hôpital's rule states that for functions f(x) and g(x), if:
or:
then:
where the prime (') denotes the derivative.
Among other requirements, for this rule to hold, the limit must exist. Other requirements are detailed below, in the formal statement. Proofs of l'Hôpital's rule Proof by Cauchy's mean value theorem
The most common proof of l'Hôpital's rule uses Cauchy's mean value theorem. With the indeterminate form 0 over 0
The case when
First, we expand continuously (or redefine) f(x) and g(x) by 0 for x = c. This doesn't change the limit since the limit doesn't depend on the value in the point (by definition).
According to Cauchy's mean value theorem there is a constant ξ in c < ξ < c + h such that:
Since f(c) = g(c) = 0,
If then and
With the indeterminate form infinity over infinity
In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. All of these approaches will be presented below.
In modern usage, there are six basic trigonometric functions, which are tabulated below along with equations relating them to one another. Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically or by other means and then derive these relations. A few other functions were common historically (and appeared in the earliest tables), but are now seldom used, such as the versine (1 − cos θ) and the exsecant (sec θ − 1). Many more relations between these functions are listed in the article about trigonometric identities.
.
.
.
is a unit vector perpendicular to the plane containing a and b. If the vectors a and b are collinear (i.e., the angle θ between them is either 0° or 180°), by the above formula, the cross product of a and b is the zero vector 0.
must exist.
then 
and
![\frac{f(x)}{g(x)} = \frac{f(y)}{g(x)} + \left [ 1 - \frac{g(y)}{g(x)} \right ] \frac{f'(\xi)}{g'(\xi)}](http://upload.wikimedia.org/math/4/b/2/4b2548d8cc3ac15edf419fb90d5b18cf.png)
and 
