第一篇：高等数学英文板总结
函数
In mathematics, a function is a relation between a set of inputs and a set of permissible 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 expression 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 essentially 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 possibly 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 process of solving for antiderivatives is called antidifferentiation(or indefinite integration)and its opposite operation is called differentiation, which is the process of finding a derivative.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)cross-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 expression between bars on the right-hand side is the magnitude of the cross 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 cross 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 expression 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 cross product on the right-hand side of this expression 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 stresses 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 across 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(expressed 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.
第二篇：高等数学总结
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： 单调性： 奇偶性： 周期性：
3.反函数与复合函数
反函数：
复合函数：
第三篇：英文求职信摸板
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第四篇：高等数学上册总结
《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、课程主要知识
本课程主要以函数为起始，然后引出极限的定义以及极限的应用。然后以极限为基础介绍导数，微分。在微分中主要讲了一些求微分的定理，例如拉格朗日中值定理，柯西中值定理等等。其次讲了函数微积分，重点讲了一些求积分的方法，例如换元积分法，分部积分法。最后学习微分方程，这一块可以说是比较难的一章，什么一阶微分方程，二阶微分方程，二阶常系数齐次线性微分方程等等，计算量也比较大。所以总的来说全书的知识点都是相连起来的。后面知识总是以前面所学知识为基础，一层一层展开的。
二、个人学习心得体会
其实不瞒老师，我高中的时候数学不是太好，平时考试数学有就有点拖后腿，而且我高考数学只考了70多分。有一天老师说，高考没及格的同学数学一定要好好学，否则极有可能挂科。当时，我还不相信，至少认为这种事不会发生在我身上。自己平时在数学上多少也花了点功夫。可以说做的准备工作比高中还多。基本上在每次上课前
都能预习，课上也认真听，而且课也差不多都能听懂，作业也都是自己独立完成的。我想及格应该不是问题，但后来的第一次过程考核，我才发现差距在哪，题目基本上不怎么会写，而且后来成绩出来，刚好考了60分。当时心就碎了。感觉落差好大。于是感叹“高树”太高了！我想是不是我题目做少了，难道说大学学数学也要用题海战术吗？可是我看班里有些同学平时上课也不听，作业基本靠抄，有事没事就拿着手机看电子书，但是考试却比我高，我就很郁闷，难道是他们比我聪明还是他们另有技巧？
经过一段时间的学习之后，我发现课前预习很重要。课前预习能够让你上课更有效率，也不会那么累。老师上课在黑板上的板书很多都是书上的。如果你课前预习了，就会知道老师说的在哪，书上有没有，记笔记的时候就可以抓住重点。不用完整地抄下来。但是你不预习的话，因为不知道书上有没有或是哪里是重点就得全部抄下来，很浪费时间，这样一来一节课就全部用在记笔记上了，根本没什么时间去听课，上课也就不会有效率。所以课前预习很重要。其次必要的练习也不可缺少。比如说上课老师说的定理不太懂，这时候就需要用练习来加强对知识的理解。
三、本课程对个人的影响
高等数学在整个大学的学习过程中占有一定的重要地位，它不仅对以后将会学到的线性代数和概率统计有影响，而且还是考研必考的科目。对于我们网络工程专业准备考研的同学来说，这绝对是一个重
头戏。对于不准备考研的同学来说，也有一定的影响，它可以培养我们的逻辑思维能力、计算能力，使我们的思维更缜密。数学是科学之母，任何学科的发展都离不开它。所以高数一定要学好。
四、总结
学习如逆水行舟不进则退，对于高数这门课程尤其是这样。因为只要你一节课没跟上就会步步跟不上，所以高数的学习不能放松，必须抓紧。相信我能学好！一定可以的！
第五篇：高等数学极限总结
【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生感到困窘。本文立足教材的基本概念阐述，着重介绍极限运算过程中极具技巧的解决思路。希望以此文能对学习者有所帮助。
【关键词】高等数学 极限 技巧
《高等数学》极限运算技巧
《高等数学》的极限与连续是前几章的内容，对于刚入高校的学生而言是入门部分的重要环节。是“初等数学”向“高等数学”的起步阶段。
一，极限的概念
从概念上来讲的话，我们首先要掌握逼近的思想，所谓极限就是当函数的变量具有某种变化趋势（这种变化趋势是具有唯一性），那么函数的应变量同时具有一种趋势，而且这种趋势是与自变量的变化具有对应性。通俗的来讲，函数值因为函数变量的变化而无限逼近某一定值，我们就将这一定值称为该函数在变量产生这种变化时的极限！
从数学式子上来讲，逼近是指函数的变化，表示为。这个问题不再赘述，大家可以参考教科书上的介绍。
二，极限的运算技巧
我在上课时，为了让学生好好参照我的结论，我夸过这样一个海口，我说，只要你认真的记住这些内容，高数部分所要求的极限内容基本可以全部解决。现在想来这不是什么海口，数学再难也是基本的内容，基本的方法，关键是技巧性。我记得blog中我做过一道极限题，当时有网友惊呼说太讨巧了！其实不是讨巧，是有规律可循的！今天我写的内容希望可以对大家的学习有帮助!
我们看到一道数学题的时候，首先是审题，做极限题，首先是看它的基本形式，是属于什么形式采用什么方法。这基本上时可以直接套用的。
1，连续函数的极限
这个我不细说，两句话，首先看是不是连续函数，是连续函数的直接带入自变量。
2，不定型
我相信所有学习者都很清楚不定型的重要性，确实。那么下面详细说明一些注意点以及技巧。
第一，所有的含有无穷小的，首先要想到等价无穷小代换，因为这是最能简化运算的。等价代换的公式主要有六个：
需要注意的是等价物穷小代换是有适用条件的，即：在含有加减运算的式子中不能直接代换，在部分式子的乘除因子也不能直接代换，那么如果一般方法解决不了问题的话，必须要等价代换的时候，必须拆项运算，不过，需要说明，拆项的时候要小心，必须要保证拆开的每一项极限都存在。
此外等价无穷小代换的使用，可以变通一些其他形式，比如：
等等。特别强调在运算的之前，检验形式，是无穷小的形式才能等价代换。
当然在一些无穷大的式子中也可以去转化代换，即无穷大的倒数是无穷小。这需要变通的看问题。
在无穷小的运算中，洛必答法则也是一种很重要的方法，但是洛必答法则适用条件比较单一，就是无穷小比无穷小。比较常见的采用洛必答法则的是无穷小乘无穷大的情况。(特
别说明无穷小乘无穷大可以改写为无穷小比无穷小或者无穷大比无穷大的形式，这根据做题的需要来进行）。
第二，在含有∞的极限式中，一般可分为下面几种情况：
（1），“∞/∞ ”形式
如果是幂函数形式的（包含幂函数四则运算形式），可以找高次项，提出高次项，这样其他一切项就都是无穷小了，只有高次项是常数。比如：
，这道题中，可以看到提出最高次x（注意不是）其他项都是“0”，原来的x都是常数1了。当然如果分式形式中，只有分子中含有高次项，那么该极限式极限不存在（是无穷大），如果只有分母中含有高次项，那么该极限式极限为0，如果分子分母都含有高次项，我们可以直接去看高次项的系数，基本原理其实就是上面所说的提高次项。比如上面的例子，可以直接写1/2。
如果不是纯幂函数形式，无法用提高次项的方法（提高次项是优先使用的方法），使用洛必达也是一种很好的方法。需要强调的是洛必达是一种解决“∞/∞ ”或“0/0 ”的基本方法，它的严格限制形式只有这两种，所以比较好观察。但是多数时候我们优先采用其他的方法来解决，这主要是考虑运算量的问题。
（2），“∞-∞ ”形式
“ ∞-∞”形式不能直接运算，需要转换形式，即转换成“∞/∞ ”或“0/0 ”的形式，基本解法同上。比如：
这道题是转换形式之后是“∞/∞ ”的形式，提高次项解。
（3）“ ”形式
这也是需要转换的一种基本形式。因为无穷大与无穷小之间的倒数关系，所以这种转换时比较简单也是比较容易解决的。转换之后的形式也是“∞/∞ ”或“0/0 ”的形式。
第三，“ ”
这种形式的解决思路主要有两种。
第一种是极限公式，这种形式也是比较直观的。比如：
这道题的基本接替思路是，检验形式是“式。
”，然后选用公式，再凑出公式的形式，最后直接套用公第二种是取对数消指数。简单来说，“ ”形式指数的存在是我们解题的主要困难。那么我们直接消掉指数就可以采用其他方法来解决了。比如上面那道题用取对数消指数的方法来解，是这样的：
可以看出尽管思路切入点不一样，但是这两种方法有异曲同工之妙。
三，极限运算思维的培养
极限运算考察的是一种基本能力，所以在做题或者看书的时候依赖的是基本概念和基本方法。掌握一定的技巧可以使学习事半功倍。而极限思维的培养则是对做题起到指导性的意义。如何培养，一方面要立足概念，另一方面则需要在具体的运算中体会，多做题多总结。
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