高等数学英文板总结由刀豆文库小编整理，希望给你工作、学习、生活带来方便，猜你可能喜欢“高等数学英文版”。
函数
In mathematics, a function is a relation between a set of inputs and a set of permiible outputs with the property that each input is related to exactly one output.极限
In mathematics, a limit is the value that a function or sequence “approaches” as the input or index approaches some value.The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.In formulas, a limit is usually denoted “lim” as in limn → c(an)= L, and the fact of approaching a limit is represented by the right arrow(→)as in an → L.Suppose f is a real-valued function and c is a real number.The expreion limf(x)L
xcmeans that f(x)can be made to be as close to L as desired by making x sufficiently close to c.无穷小Infinitesimal In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, but not zero in size;or, so small that it cannot be distinguished from zero by any available means.无穷大 连续函数
In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output.介值定理
In mathematical analysis, the intermediate value theorem states that if a continuous function f with an interval [a, b] as its domain takes values f(a)and f(b)at each end of the interval, then it also takes any value between f(a)and f(b)at some point within the interval.This has two important specializations: If a continuous function has values of opposite sign inside an interval, then it has a root in that interval(Bolzano's theorem).[1] And, the image of a continuous function over an interval is itself an interval.导数
The derivative of a function of a real variable measures the sensitivity to change of a quantity(a function or dependent variable)which is determined by another quantity(the independent variable).Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, there is a corresponding value of y.This relationship can be written as y = f(x).If f(x)is the equation for a straight line, then there are two real numbers m and b such that y = m x + b.m is called the slope and can be determined from the formula:mchanginyy，where
changinxxthe symbol Δ(the uppercase form of the Greek letter Delta)is an abbreviation for “change in”.It follows that Δy = m Δx.A general function is not a line, so it does not have a slope.The derivative of f at the point x is the slope of the linear approximation to f at the point x.微分 罗尔定理
In calculus, Rolle's theorem eentially states that any real-valued differentiable function that attains equal values at two distinct points must have a stationary point somewhere between them;that is, a point where the first derivative(the slope of the tangent line to the graph of the function)is zero.If a real-valued function f is continuous on a closed interval [a, b], differentiable on the open interval(a, b), and f(a)= f(b), then there exists a c in the open interval(a, b)such that f/(c)0.This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case.It is also the basis for the proof of Taylor's theorem.拉格朗日中值定理Lagrange’s mean value theorem f(b)f(a)f'()
ba柯西中值定理Cauchy's mean value theorem
Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem.It states: If functions f and g are both continuous on the closed interval [a,b], and differentiable on the open interval(a, b), then there exists some c ∈(a,b), such that(f(b)f(a))g'(c)(g(b)g(a))f'(c)；Of course, if g(a)≠ g(b)and if
g′(c)≠ 0, this is equivalent to:
f'(c)f(b)f(a)。g'(c)g(b)g(a)洛必达法则L'Hôpital's rule In calculus, l'Hôpital's rule(pronounced: [lopiˈtal])uses derivatives to help evaluate limits involving indeterminate forms.Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit.In its simplest form, l'Hôpital's rule states that for functions f and g which are differentiable on an open interval I except poibly at a point c contained in I: If
limxcf(x)limg(x)0
xcor,andlimxcf'(x)exists, andg'(x)0for all x in I with x ≠ c, g'(x)thenlimxcf(x)f'(x)lim.g(x)g'(x)xcThe differentiation of the numerator and denominator often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be evaluated more easily.泰勒公式Taylor's theorem
Statement of the theorem The precise statement of the most basic version of Taylor's theorem is as follows: Taylor's theorem.Let k ≥ 1 be an integer and let the function f : R → R be k times differentiable at the point a ∈ R.Then there exists a function hk : R → R such that
This is called the Peano form of the remainder.不定积分Antiderivative In calculus, an antiderivative, primitive integral or indefinite integral[1] of a function f is a differentiable function F whose derivative is equal to f, i.e., F ′ = f.The proce of solving for antiderivatives is called antidifferentiation(or indefinite integration)and its opposite operation is called differentiation, which is the proce of finding a derivative.定积分Integration Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by
多元函数Functions with multiple inputs and outputs The concept of function can be extended to an object that takes a combination of two(or more)argument values to a single result.This intuitive concept is formalized by a function whose domain is the Cartesian product of two or more sets.重积分Multiple integral The multiple integral is a generalization of the definite integral to functions of more than one real variable, for example, f(x, y)or f(x, y, z).Integrals of a function of two variables over a region in R2 are called double integrals, and integrals of a function of three variables over a region of R3 are called triple integrals.曲线积分Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The terms path integral, curve integral, and curvilinear integral are also used;contour integral as well, although that is typically reserved for line integrals in the complex plane.对坐标的曲线积分Line integral of a scalar field For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂
U is defined as
where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a)and r(b)give the endpoints of C and.The function f is called the integrand, the curve C is the domain of integration, and the symbol ds may be intuitively interpreted as an elementary arc length.Line integrals of scalar fields over a curve C do not depend on the chosen parametrization r of C.Geometrically, when the scalar field f is defined over a plane(n=2), its graph is a surface z=f(x,y)in space, and the line integral gives the(signed)cro-sectional area bounded by the curve C and the graph of f.对弧长的曲线积分Line integral of a vector field For a vector field F : U ⊆ Rn → Rn, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as
where · is the dot product and r: [a, b] → C is a bijective parametrization of the curve C such that r(a)and r(b)give the endpoints of C.A line integral of a scalar field is thus a line integral of a vector field where the vectors are always tangential to the line.Line integrals of vector fields are independent of the parametrization r in absolute value, but they do depend on its orientation.Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.The line integral of a vector field along a curve is the integral of the corresponding 1-form under the musical isomorphism over the curve considered as an immersed 1-manifold.曲面积分Surface integral In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces.It can be thought of as the double integral analog of the line integral.Given a surface, one may integrate over its scalar fields(that is, functions which return scalars as values), and vector fields(that is, functions which return vectors as values).对坐标的曲面积分Surface integrals of scalar fields To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere.Let such a parameterization be x(s, t), where(s, t)varies in some region T in the plane.Then, the surface integral is given by
where the expreion between bars on the right-hand side is the magnitude of the cro product of the partial derivatives of x(s, t), and is known as the surface element.For example, if we want to find the surface area of some general
scalar
function,say,we
have
where
.So that , and.So,which is the familiar formula we get for the surface area of a general functional shape.One can recognize the vector in the second line above as the normal vector to the surface.Note that because of the presence of the cro product, the above formulas only work for surfaces embedded in three-dimensional space.This can be seen as integrating a Riemannian volume form on the parameterized surface, where the metric tensor is given by the first fundamental form of the surface.对面积的曲面积分Surface integrals of vector fields
Consider a vector field v on S, that is, for each x in S, v(x)is a vector.The surface integral can be defined component-wise according to the definition of the surface integral of a scalar field;the result is a vector.This applies for example in the expreion of the electric field at some fixed point due to an electrically charged surface, or the gravity at some fixed point due to a sheet of material.Alternatively, if we integrate the normal component of the vector field, the result is a scalar.Imagine that we have a fluid flowing through S, such that v(x)determines the velocity of the fluid at x.The flux is defined as the quantity of fluid flowing through S per unit time.This illustration implies that if the vector field is tangent to S at each point, then the flux is zero, because the fluid just flows in parallel to S, and neither in nor out.This also implies that if v does not just flow along S, that is, if v has both a tangential and a normal component, then only the normal component contributes to the flux.Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, and integrate the obtained field as above.We find the formula
The cro product on the right-hand side of this expreion is a surface normal determined by the parametrization.This formula defines the integral on the left(note the dot and the vector notation for the surface element).We may also interpret this as a special case of integrating 2-forms, where we identify the vector field with a 1-form, and then integrate its Hodge dual over the surface.This is equivalent to integrating
over the immersed surface, where
is the induced volume form on the surface, obtained by interior multiplication of the Riemannian metric of the ambient space with the outward normal of the surface.格林公式Green's theorem
Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C.If L and M are functions of(x, y)defined on an open region containing D and have continuous partial derivatives there, then
where the path of integration along C is counterclockwise.In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area.In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.高斯公式Divergence theorem
Suppose V is a subset of R(in the case of n = 3, V represents a volume in 3D space)which is compact and has a piecewise smooth boundary S(also indicated with ∂V = S).If F is a continuously differentiable vector field defined on a neighborhood of V, then we
nhave:
sThe left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V.The closed manifold ∂V is quite generally the boundary of V oriented by outward-pointing normals, and n is the outward pointing unit normal field of the boundary ∂V.(dS may be used as a shorthand for ndS.)The symbol within the two integrals strees once more that ∂V is a closed surface.In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume V, and the right-hand side represents the total flow acro the boundary ∂V.级数Series
A series is, informally speaking, the sum of the terms of a sequence.Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.[1] In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·.These can be written more compactly using the summation symbol ∑.幂级数Power series In mathematics, a power series(in one variable)is an infinite series of the form
where an represents the coefficient of the nth term, c is a constant, and x varies around c(for this reason one sometimes speaks of the series as being centered at c).This series usually arises as the Taylor series of some known function.In many situations c is equal to zero, for instance when considering a Maclaurin series.In such cases, the power series takes the simpler form
These power series arise primarily in analysis, but also occur in combinatorics(as generating functions, a kind of formal power series)and in electrical engineering(under the name of the Z-transform).The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at 1⁄10.In number theory, the concept of p-adic numbers is also closely related to that of a power series.微分方程Differential equation A differential equation is a mathematical equation that relates some function of one or more variables with its derivatives.Differential equations arise whenever a deterministic relation involving some continuously varying quantities(modeled by functions)and their rates of change in space and/or time(expreed as derivatives)are known or postulated.Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation.Only the simplest differential equations are solvable by explicit formulas;however, some properties of solutions of a given differential equation may be determined without finding their exact form.If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers.The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
FROM BODY TO SOUL高等数学第一讲 函数、极限和连续一、函数 1.函数的概念几种常见函数 绝对值函数： 符号函数： 取整函数： 分段函数：最大值最小值函数： 2.函数的特性有界性： 单调......
Dear Sir or Madam, I am writing to you to inquire if there would be any position available for me to work as a _______________________.please allow me to give m......
《工程应用数学A》课程总结无论我们做什么事都要不断地思考，不断地总结，学习也是这样，所以这次就借此机会对于这一学期所学内容进行一次总结，也算是对自我的一次思考。一、课程......
我的高等数学 学我所学，想我所想【摘要】《高等数学》教学中对于极限部分的要求很高，这主要是因为其特殊的地位决定的。然而极限部分绝大部分的运算令很多从中学进入高校的学......
高等数学难点总结函数（高等数学的主要研究对象）极限：数列的极限（特殊）——函数的极限（一般）极限的本质是通过已知某一个量（自变量）的变化趋势，去研究和探索另外一个量（因变量）的变化趋势......
