Tuesday, January 1, 2013


 mathematics: the study of the measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are branches of mathematics.
 axiom or postulate: a proposition that is assumed without proof. 
 theorem: statement that has been proven on the basis of other theorems and/or axioms.
 arithmetic: the method or process of computation with figures: the most elementary branch of mathematics.
 algebra: the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations.
 geometry /dʒi'ɒmɪtri/ the branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles, and figures in space from their defining conditions by means of certain assumed properties of space.
 Calculus ('kælkjʊləs) focuses on limits, functions, derivatives, integrals, and infinite series. A branch of mathematics, developed independently by Newton and Leibniz. Both differential calculus and integral calculus are concerned with the effect on a function of an infinitesimal change in the independent variable as it tends to zero.
 trigonometry /trɪgə'nɒmɪtri/ the branch of mathematics that deals with the relations between the sides and angles of plane or spherical triangles, and the calculations based on them.
 subscript /'sʌbskrɪpt/ any character, number, or symbol written next to and slightly below another.
 superscript /'supərskrɪpt/ written or printed above the line; superior
 Addition addends sum
 Subtraction minuend subtrahend difference

 Multiplication factors or multipliers
 Division dividend divisor quotient remainder

 exponentiation /ɛkspoʊnɛnʃi'eɪʃən/ the raising of a number to any given power.
To raise a number to a whole (second, third, forth, fifth etc.) power
base of a power
index or an exponent of a power
value of a power
Extraction of a root
index (degree) of the root
value of the root
・prime ・composite

 prime numbers: have only 2 divisors, 1 and itself
 composite numbers: not prime numbers
 reciprocal for a number x, denoted by 1/x or x^(−1), is a number which when multiplied by x yields the multiplicative identity, 1.
 factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,
    5! = 5 x 4 x 3 x 2 x 1 = 120
The value of 0! is 1, according to the convention for an empty product.
 function /'fʌŋkʃən/ Also called correspondence, map, mapping, transformation. a relation between two sets in which one element of the second set is assigned to each element of the first set, as the expression y  = x²  ; operator.
・Even ・Odd

Even function: f(x) = f(-x) for all x
Odd function: f(x) = -f(-x) for all x
 inverse function g of function f:
f(x) = y
g(y) = x
g(f(x)) = x
Data is a collection of facts, such as values or measurements.
Qualitative data is descriptive information (it describes something)
Quantitative data, is numerical information (numbers).
Quantitative data can also be Discrete or Continuous:
    Discrete data can only take certain values (like whole numbers)
    Continuous data can take any value (within a range)
A Census is when you collect data for every member of the group (the whole "population").
A Sample is when you collect data just for selected members of the group.
 fraction /'frækʃən/ a number usually expressed in the form a/b.
 numerator /'numəreɪtər/ Arithmetic . the term of a fraction, usually above the line, that indicates the number of equal parts that are to be added together; the dividend placed over a divisor: The numerator of the fraction 2  / 3  is 2.
 denominator /dɪ'nɒməneɪtər/ Arithmetic . that term of a fraction, usually written under the line, that indicates the number of equal parts into which the unit is divided; divisor.
 Polynomial:  sum of  terms. Ex. -5x2y  The coefficient is –5, the variables are x and y, the degree of x is two, and the degree of y is one.The degree of the entire term is 2+1=3.
 Polynomial is an expression constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
 Fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed.
Every polynomial equation having complex coefficients and degree >=1 has at least one complex root.
A polynomial P(z) of degree n has  values n zi for which P(zi) = 0. Such values are called polynomial roots.
degree of a function

linear: f(x) = ax + b
quadratic:  f(x) = ax² + bx + c
cubic: f(x) = ax³ + bx² + cx + d
quartic: f(x) = ax⁴ + bx³ + cx² + dx + e,
Solution or a root is a number x that satisfies the equation f(x) = 0
 linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.  f(x) = ax + b
 linear equation can involve more than two variables. The general linear equation in n variables is:
a₁x₁ + a₂x₂ +... anxn = b
 quadratic /kwɒ'drætɪk/ Algebra. involving the square and no higher power of the unknown quantity; of the second degree.
The graph of a quadratic function is a parabola. The roots are the intersections with x-axis.
x = (-b ± √(b^2-4ac))/2a
The Peano axioms give a formal theory of the natural numbers. The axioms are:
    There is a natural number 0.
    Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a + 1.
    There is no natural number whose successor is 0.
    S is injective, i.e. distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
    If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.)
 integer is a number that can be written without a fractional or decimal component.
 rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
 irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
 real number is a value that represents a quantity along a continuous line.
 complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
 i: imaginary unit.
 i² = -1
 a + bi = 0 if and only if a = 0 and b = 0.
 (a+bi)+(c+di) = (a+c)+(b+d)i
 Complex conjugate of a+bi is a-bi.
 (a+bi)(a-bi) = a² + b²
To simplify a complex fraction, multiply the numerator and the denominator by the complex conjugate of the denominator.
 Modulus of a Complex Number |z| = √(a^2 + b^2) where z=a+bi
Pythagoras's theorem

(cosθ)^2 + (sinθ)^2  = 1
cos^2 + sin^2  = 1
Euler's number e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x = 0 is equal to 1.
e = limn→∞ (1 + 1/n)^n

Euler's formula
e^(iθ) = cosθ + i * sinθ

Euler's identity
e^(iπ) + 1 = 0

Exponential function
y = f(x) = e^x
d(e^x)/dx = e^x

Logarithm of a number x is the exponent y to which the base b, must be raised to produce that number x.
If b^y = x  then log b(x) = y.

Natural logarithm is the logarithm to the base e, where e is the Euler number.
log e (x) = ln(x) = log(x)

log(xy) = log(x) + log(y)
 hypotenuse /haɪ'pɒtnus, -yus/ the side of a right triangle opposite the right angle.

 sine /saɪn/ (in a right triangle) the ratio of the side opposite a given acute angle to the hypotenuse.
 cosine /'koʊsaɪn/(in a right triangle) the ratio of the side adjacent to a given angle to the hypotenuse.
 tangent /'tændʒənt/(in a right triangle) the ratio of the side opposite a given angle to the side adjacent to the angle.
 cosecant /koʊ'sikənt/  cscA = 1/sinA
 secant /'sikænt/ secA = 1/cosA
 cotangent /koʊ'tændʒənt/ cotA = 1/tanA

hyperbolic sine: sinh (/'sɪntʃ/ or /'ʃaɪn/)
hyperbolic cosine: cosh /'kɒʃ/
hyperbolic tangent: tanh (/'tæntʃ/ or /'θæn/)

hyperbolic cosecant: csch or cosech /'koʊʃæk/
hyperbolic secant: sech /'ʃæk/
hyperbolic cotangent: coth /'koʊθ/
A sequence is an ordered list.
Like a set, it contains members (also called elements, or terms).
The number of ordered elements (possibly infinite) is called the length of the sequence.
Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence.
Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.
For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last.
This sequence differs from (A, R, M, Y).
Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence.
Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...).
Finite sequences are sometimes known as strings or words and infinite sequences as streams.
The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0,1,1,2,3,5,8,13,21,34,...).
a₍n₎ = a₍n-1₎ + a₍n-2₎ with a₀ = 0 and a₁ = 1

(a₁,a₂,...,a₁₀), ak = k²
 Cauchy sequence
A sequence (x1, x2, x3, ...) of real numbers is called a Cauchy sequence, if for every positive real number ε, there is a positive integer N such that for all natural numbers m, n > N
    |xm - xn| < ε
 Series is the sum of the terms of a sequence.

example: ∑₍n=0→∞₎1/(2^n) = 1/2 + 1/4 + 1/8...
 power series (in one variable) is an infinite series of the form
f(x) = ∑₍n=0→∞₎ an( x-c)^n = a0 + a1 (x-c) + a2 (x-c)² + a3 (x-c)³ + ...
where an represents the coefficient of the nth term, c is a constant, and x varies around c.
 Fourier series: an infinite series that involves linear combinations of sines and cosines and approximates a given function on a specified domain.
 limit is the value that a function or sequence "approaches" as the input or index approaches some value.
 arc /ɑrk/ Geometry . any unbroken part of the circumference of a circle or other curved line.
 Radian is the ratio between the length of an arc and its radius. One radian is equal to 180/π degrees.
 radian /'reɪdiən/ the measure of a central angle subtending an arc equal in length to the radius: equal to 57.2958°. Abbreviation:  rad
 radius /'reɪdiəs/ a straight line extending from the center of a circle or sphere to the circumference or surface: The radius of a circle is half the diameter.
The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter, and is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes romanized as "pi" (/paɪ/).

π =  C/d where C is the circumference and d is the diameter.

d(x^n)/dx = nx^(n-1)
d(a+b)/dx = da/dx + db/dx

The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through domain of f exists and is equal to f(c).
lim₍x→c₎ f(x) = f(c)
 differential equation: an equation involving differentials or derivatives.
 differential /dɪfə'rɛnʃəl/ pertaining to or involving a derivative or derivatives.
 derivative /dɪ'rɪvətɪv/ Differential quotient. Differential coefficient.  The limit of the ratio of the increment of a function to the increment of a variable in it, as the latter tends to 0; the instantaneous change of one quantity with respect to another, as velocity, which is the instantaneous change of distance with respect to time.  The derivative is the slope of the line tangent to the graph of f(x).

 Differential equation: is an equation involving an unknown function and its derivatives.
 Order of the differential equation: is the order of the highest derivative of the unknown function.
 Linear differential equation of order n is a differential equation written in the following form:
f(x) = a0(x)y + a1(x)dy/dx + ... + an(x)dny/dxn
open interval I = (a,b)={x∈R | a < x < b}
closed interval I = [a,b]={x∈R | a ≤ x ≤ b}

Fundamental theorem of calculus
Let ƒ be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by
Then, F is continuous on [a,b], differentiable on the open interval (a, b), and F'(x) = f(x)
a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but the result of 3 − 8 is not.
 algebraic structure: consists of one or more sets closed under one or more operations, satisfying some axioms.
 identity element: is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them.
 monoid is an algebraic structure with a single associative binary operation and an identity element.
 semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element.
 ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition.
 abelian group (or commutative group) is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity).
 commutative ring is a ring in which the multiplication operation is commutative.
 group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element.
 field is a commutative ring whose nonzero elements form a group under multiplication.
magma is an algebraic structure that consists of a set M equipped with a single binary operation M X M → M. A binary operation is closed by definition, but no other axioms are imposed on the operation.
tuple is an ordered list of elements.
Euclidean Space (Cartesian space, n-space), is the space of all n-tuples of real numbers, (x1,x2, ...,xn). It is commonly denoted Rn.
Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two.
permutation of a set of objects is an arrangement of those objects into a particular order.
The number of permutations of n distinct objects is n! .
Fundamental theorem of arithmetic: any integer greater than 1 can be written as a unique product of prime numbers.
embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies).
partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.∂f/∂x
gradient of a scalar field y is a vector field ∇y that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
y = f(x1,x2,...xn)
∇y = (∂f/∂x1,∂f/∂x2,...∂f/∂xn)
random variable or stochastic variable is a variable whose value results from a measurement on some type of random process.
probability distribution is a function that describes the probability of a random variable taking certain values.
probability density function, or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the integral of this variable’s density over the region. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.
P[a ≤ X ≤ b] = ∫abf(x)dx
transcendental number is a number (possibly a complex number) that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients.

Gaussian function
f(x) = aek  
k = -(x-b)2/2c2
Normal distribution
f(x) = (1/√2πσ2)ek
k = -(x-μ)2/2σ2
Standard Deviation is square root of the Variance.
Variance is the average of the squared differences from the Mean.
Linear algebra is a branch of mathematics that studies vector spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often considered as a part of linear algebra.
matrix is a rectangular array of numbers, symbols, or expressions.
square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n.
eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, either remain proportional to the original vector (i.e., change only in magnitude, not in direction) or become zero. If A is a square matrix, a non-zero vector v is an eigenvector of A if there is a scalar λ (lambda) such that Av=λv
vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context.
normed vector space has 3 properties:
1.The zero vector, , has zero length; every other vector has a positive length.
∥∥> 0 if  ≠
Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
3. The triangle inequality holds. For any vectors x and y. (triangle inequality)
∥ + ∥ ≤ ∥∥ + ∥∥
linear transformation (or linear map, linear mapping, linear operator, linear function) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X
spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices.
null vector or zero vector or empty vector is the vector (0, 0, …, 0) in Euclidean space, all of whose components are zero.
homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces).
modular arithmetic (or clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus.Ex.:12-hour clock 15:00 ≡ 3:00 (mod 12)

 すうがく【数学】 mathematics. 数量および空間図形の性質について研究する学問。算術・代数学・幾何学・解析学・微分学・積分学などの総称。
 さんじゅつ【算術】1 計算の方法。算法。古くは数学全般をいった。2 旧制の小学校における教科名。現在の算数がほぼこれにあたる。
 さんすう【算数】 arithmetic. 小学校の教科の一。初歩の数学。数量や図形について基礎的知識を教えることを目標とする。
 だいすうがく【代数学】 algebra. 数の代わりに文字を用い、計算の法則・方程式の解法などを主に研究する数学の一分野。現在では、代数系の研究をいう。
 きかがく【幾何学】 geometry. 図形や空間の性質を研究する数学の一部門。紀元前300年ころ、ユークリッドによって集大成され、現在は、微分幾何学・代数幾何学・位相幾何学などに発展。幾何。
 かいせきがく【解析学】 Calculus. 微分積分学とそれから発展した数学の諸分科の総称。微分積分学・微分方程式論・積分方程式論・実関数論・複素関数論など。
 こうり【公理】axiom. 数学で、論証がなくても自明の真理として承認され、他の命題の前提となる根本命題。
 めいだい【命題】 proposition. 論理学で、判断を言語で表したもので、真または偽という性質を持つもの。3 数学で、真偽の判断の対象となる文章または式。定理または問題。
 ていり【定理】 theorem. ある理論体系において、その公理や定義をもとにして証明された命題で、それ以降の推論の前提となるもの。
 ぜんてい【前提】1 ある物事が成り立つための、前置きとなる条件。2 論理学で、推論において結論が導き出される根拠となる判断。
 ていぎ【定義】1 物事の意味・内容を他と区別できるように、言葉で明確に限定すること。2 論理学で、概念の内包を明瞭にし、その外延を確定すること。通常、その概念が属する最も近い類と種差を挙げることによってできる。
 たしざん【足し算】 addition. ある数に他の数を足して合計を求める計算。加法。寄せ算。加え算。
 ひきざん【引(き)算】 subtraction. ある数から他の数を引いて残りを求める計算。減法。
 かけざん【掛(け)算】 multiplication. ある数を他の数の表す回数だけ加えた合計を求める計算。乗法。
 わりざん【割(り)算】 division. ある数が他の数の何倍であるかを求める計算。除法。
 すうち【数値】1 計算や、計量・計測をして得られた数。2 文字式の中の文字に当てはまる具体的な数。
 けいさん【計算】1 物の数量をはかり数えること。勘定。2 加減乗除など、数式に従って処理し数値を引き出すこと。演算。3 結果や成り行きをある程度予測し、それを予定の一部に入れて考えること。
 せきぶん【積分】1 与えられた関数について、微分してこの関数になるすべての関数。また、それを求めること。不定積分。2 ある関数のグラフの区間を微小に分割し、各微小部分の幅とその関数値との積の和をつくり、微小部分の幅を限りなく小さくしていったときの和の極限値を求めること。また、その極限値。定積分。
 単項式・項 たんこうしきこう monomial. 数や文字だけの積
 多項式 たこうしき polynomial. 単項式の和
 整式 せいしき integral expression. 単項式と多項式を合わせたもの。
 係数 けいすう coefficient. 項の数の部分
 因数分解 いんすうぶんかい factorization. 整式Pを二つ以上の整式A,Bの積で表す。
 等式 とうしき equality. 二つの式P,Qを等号=で結びつけたP = Q。Pを左辺さへん、Qを右辺うへん、両方合わせて両辺りょうへん。
 不等式ふとうしき inequality.
 恒等式こうとうしき identity. 変数の値に関らず常に成り立つ等式。
 方程式 ほうていしき equation. 恒等式でない等式
 連立方程式 れんりつほうていしき simultaneous equations.
 公式 こうしき formula.
 自然数 しぜんすう natural
 定数 ていすう constant
 整数 せいすう integer
 実数 じっすう real number. 有理数と無理数の総称
 有理数 ゆうりすう rational number. 実数のうち、2個の整数の比によって表される数
 無理数 むりすう irrational number. 二つの整数の比で表すことができない数
 複素数 ふくそすう complex number
 虚数 きょすう imaginary
 分数 ぶんすう fraction: 二つの整数a・bの比として表される数
 分母 ぶんぼ denominator
 分子 ぶんし numerator
 小数 しょうすう decimal
 逆数 ぎゃくすう reciprocal of x is 1/x
 商 しょう quotient
 余り あまり remainder
 桁 けた digit
 平方 へいほう square
 立方 りっぽう cubic
 平方根 へいほうこん square root
 相加平均 そうかへいきん arithmetic mean = (a+b)/2
 相乗平均 そうじょうへいきん geometric mean = √(a*b)
 約数 やくすう divisor
 倍数 ばいすう multiple

 素数 そすう prime number
 偶数 ぐうすう even number par
 奇数 きすう odd number ímpar
 集合 しゅうごうset
 命題 めいだい proposition 
 pならばq    p ⇒ q
 p十分条件 q必要条件
 原点 げんてん origin
 線分 せんぶん segment
 直線 ちょくせん straight line
 半直線 はんちょくせん half line
 座標 ざひょう coordinates
 曲線 きょくせん curve
 三角形 さんかっけい triangle
 直角三角形 ちょっかくさんかくけい right triangle
 二等辺三角形 にとうへんさんかっけい isosceles triangle
 鋭角 えいかく acute angle
 直角 ちょっかく right angle
 鈍角 どんかく obtuse angle
 外角 がいかく exterior angle
 内角 ないかく interior angle
 三平方の定理 さんへいほうのていり Pythagoras theorem
 h^2  = o^2 + a^2

 斜辺 しゃへん hypotenuse h
 対辺 たいへん opposite side o
 隣辺 りんぺん adjacent side a

 正弦 せいげん サイン sine = o/h
 余弦 よげん コサイン cosine = a/h
 正接 せいせつ タンジェント tangent = o/a
 かく【角】 angle. 一点から出る二つの半直線がつくる図形。また、その開きの度合い。角度。
 ちょうてん【頂点】 vertex. 角をつくる二直線の交点。多面体の三つ以上の面の交わる点。円錐の母線が集まる点。放物線とその対称軸との交点。
 四角形 しかくけい quadrilateral
 四辺形 しへんけい quadrilateral
 長方形 ちょうほうけい rectangle
 六角形 ろっかくけい hexagon
 対角線 たいかくせん diagonal
 菱形 ひしがた lozenge
 台形 だいけい trapezoid
 多角形 たかくけい polygon
 正方形 せいほうけい regular square
 楕円 だえん ellipse
 放物線 ほうぶつせん parabola
 双曲線 そうきょくせん hyperbola
 円周 えんしゅう circumference
 半径 はんけい radius
 直径 ちょっけい diameter
 弧 こ arc
 弦 げん chord
 円周率 えんしゅうりつ π
 同心円 どうしんえん concentric circles
 球 きゅう sphere
 円筒 えんとう cylinder
 円錐 えんすい cone
 断面 だんめん cross-section
 立方体 りっぽうたい cube
 正八面体 octahedron
 正四面体 tetrahedron
 面積 めんせき area
 体積 たいせき volume
 平面 へいめん plane
 交点 こうてん intersection point
 接点 せってん contact point
 平行線 へいこうせん parallel
 接線 せっせん tangent
 割線 かっせん secant
 垂線 すいせん perpendicular
 垂直 すいちょく vertical
 水平 すいへい horizontal
 関数 かんすう function
 指数関数 しすうかんすう exponential function
 対数 たいすう logarithm
 p = loga(x)  ap = x
 等差数列 とうさすうれつ arithmetical progression. an+1 = an + d
 等比数列 とうひすうれつ geometric progression. an+1 = an * d
 階乗 かいじょう factorial
 n! = 1        if n = 0,
 n! = (n-1)!*n  if n > 0
 微分 びぶん derivative
 積分 せきぶん integral
 極限 きょくげん limit
 自転 rotation
 公転 revolution
 平行四辺形 parallelogram
 球面 きゅうめん spherical surface
 曲面 きょくめん curved surface
 螺旋 らせん spiral
 法線 ほうせん norm

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