高等数学英文板总结（精选合集）

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

xcmeans 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:mchanginyy，where

changinxxthe 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'()

ba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

limxcf(x)limg(x)0

xcor,andlimxcf'(x)exists, andg'(x)0for all x in I with x ≠ c, g'(x)thenlimxcf(x)f'(x)lim.g(x)g'(x)xcThe 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：高等数学难点总结

高等数学难点总结 上册：

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

由极限可以推得的一些性质：局部有界性、局部保号性……应当注意到，由极限所得到的性质通常都是只在局部范围内成立

在提出极限概念的时候并未涉及到函数在该点的具体情况，所以函数在某点的极限与函数在该点的取值并无必然联系

连续：函数在某点的极限 等于 函数在该点的取值 连续的本质：自变量无限接近，因变量无限接近

导数的概念

本质是函数增量与自变量增量的比值在自变量增量趋近于零时的极限，更简单的说法是变化率

微分的概念：函数增量的线性主要部分，这个说法有两层意思，一、微分是一个线性近似，二、这个线性近似带来的误差是足够小的，实际上任何函数的增量我们都可以线性关系去近似它，但是当误差不够小时，近似的程度就不够好，这时就不能说该函数可微分了

不定积分：导数的逆运算 什么样的函数有不定积分

定积分：由具体例子引出，本质是先分割、再综合，其中分割的作用是把不规则的整体划作规则的许多个小的部分，然后再综合，最后求极限，当极限存在时，近似成为精确 什么样的函数有定积分

求不定积分（定积分）的若干典型方法：换元、分部，分部积分中考虑放到积分号后面的部分，不同类型的函数有不同的优先级别，按反对幂三指的顺序来记忆

定积分的几何应用和物理应用

高等数学里最重要的数学思想方法：微元法

微分和导数的应用：判断函数的单调性和凹凸性

微分中值定理，可从几何意义去加深理解

泰勒定理：本质是用多项式来逼近连续函数。要学好这部分内容，需要考虑两个问题：

一、这些多项式的系数如何求？

二、即使求出了这些多项式的系数，如何去评估这个多项式逼近连续函数的精确程度，即还需要求出误差（余项），当余项随着项数的增多趋向于零时，这种近似的精确度就是足够好的。下册

（一）：

多元函数的微积分：将上册的一元函数微积分的概念拓展到多元函数

最典型的是二元函数

极限：二元函数与一元函数要注意的区别，二元函数中两点无限接近的方式有无限多种（一元函数只能沿直线接近），所以二元函数存在的要求更高，即自变量无论以任何方式接近于一定点，函数值都要有确定的变化趋势

连续：二元函数和一元函数一样，同样是考虑在某点的极限和在某点的函数值是否相等

导数：上册中已经说过，导数反映的是函数在某点处的变化率（变化情况），在二元函数中，一点处函数的变化情况与从该点出发所选择的方向有关，有可能沿不同方向会有不同的变化率，这样引出方向导数的概念

沿坐标轴方向的导数若存在，称之为偏导数

通过研究发现，方向导数与偏导数存在一定关系，可用偏导数和所选定的方向来表示，即二元函数的两个偏导数已经足够表示清楚该函数在一点沿任意方向的变化情况

高阶偏导数若连续，则求导次序可交换

微分：微分是函数增量的线性主要部分，这一本质对一元函数或多元函数来说都一样。只不过若是二元函数，所选取的线性近似部分应该是两个方向自变量增量的线性组合，然后再考虑误差是否是自变量增量的高阶无穷小，若是，则微分存在

仅仅有偏导数存在，不能推出用线性关系近似表示函数增量后带来的误差足够小，即偏导数存在不一定有微分存在

若偏导数存在，且连续，则微分一定存在

极限、连续、偏导数和可微的关系在多元函数情形里比一元函数更为复杂

极值：若函数在一点取极值，且在该点导数（偏导数）存在，则此导数（偏导数）必为零

所以，函数在某点的极值情况，即函数在该点附近的函数增量的符号，由二阶微分的符号判断。对一元函数来说，二阶微分的符号就是二阶导数的符号，对二元函数来说，二阶微分的符号可由相应的二次型的正定或负定性判断。

级数敛散性的判别思路：首先看通项是否趋于零，若不趋于零则发散。若通项趋于零，看是否正项级数。若是正项级数，首先看能否利用比较判别法，注意等比级数和调和级数是常用来作比较的级数，若通项是连乘形式，考虑用比值判别法，若通项是乘方形式，考虑用根值判别法。若不是正项级数，取绝对值，考虑其是否绝对收敛，绝对收敛则必收敛。若绝对值不收敛，考察一般项，看是否交错级数，用莱布尼兹准则判断。若不是交错级数，只能通过最根本的方法判断，即看其前n项和是否有极限，具体问题具体分析。

比较判别法是充分必要条件，比值和根值法只是充分条件，不是必要条件。

函数项级数情况复杂，一般只研究幂级数。阿贝尔定理揭示了幂级数的重要性质：收敛区域存在一个收敛半径。所以对幂级数，关键在于求出收敛半径，而这可利用根值判别法解决。

逐项求导和逐项积分不改变幂级数除端点外的区域的敛散性，端点情况复杂，需具体分析。

一个函数能展开成幂级数的条件是：存在任意阶导数。展开后的幂级数能收敛于原来函数的条件是：余项（误差）要随着项数的增加趋于零。这与泰勒展开中的结论一致。

微分方程：不同种类的方程有不同的常见解法，但理解上并无难处。

第五篇：高等数学难点总结

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

由极限可以推得的一些性质：局部有界性、局部保号性……应当注意到，由极限所得到的性质通常都是只在局部范围内成立

在提出极限概念的时候并未涉及到函数在该点的具体情况，所以函数在某点的极限与函数在该点的取值并无必然联系

连续：函数在某点的极限 等于 函数在该点的取值 连续的本质：自变量无限接近，因变量无限接近

导数的概念

本质是函数增量与自变量增量的比值在自变量增量趋近于零时的极限，更简单的说法是变化率

微分的概念：函数增量的线性主要部分，这个说法有两层意思，一、微分是一个线性近似，二、这个线性近似带来的误差是足够小的，实际上任何函数的增量我们都可以线性关系去近似它，但是当误差不够小时，近似的程度就不够好，这时就不能说该函数可微分了

不定积分：导数的逆运算 什么样的函数有不定积分

定积分：由具体例子引出，本质是先分割、再综合，其中分割的作用是把不规则的整体划作规则的许多个小的部分，然后再综合，最后求极限，当极限存在时，近似成为精确 什么样的函数有定积分

求不定积分（定积分）的若干典型方法：换元、分部，分部积分中考虑放到积分号后面的部分，不同类型的函数有不同的优先级别，按反对幂三指的顺序来记忆

定积分的几何应用和物理应用

高等数学里最重要的数学思想方法：微元法

微分和导数的应用：判断函数的单调性和凹凸性

微分中值定理，可从几何意义去加深理解

泰勒定理：本质是用多项式来逼近连续函数。要学好这部分内容，需要考虑两个问题：

一、这些多项式的系数如何求？

二、即使求出了这些多项式的系数，如何去评估这个多项式逼近连续函数的精确程度，即还需要求出误差（余项），当余项随着项数的增多趋向于零时，这种近似的精确度就是足够好的 下册

（一）：

多元函数的微积分：将上册的一元函数微积分的概念拓展到多元函数

最典型的是二元函数

极限：二元函数与一元函数要注意的区别，二元函数中两点无限接近的方式有无限多种（一元函数只能沿直线接近），所以二元函数存在的要求更高，即自变量无论以任何方式接近于一定点，函数值都要有确定的变化趋势

连续：二元函数和一元函数一样，同样是考虑在某点的极限和在某点的函数值是否相等

导数：上册中已经说过，导数反映的是函数在某点处的变化率（变化情况），在二元函数中，一点处函数的变化情况与从该点出发所选择的方向有关，有可能沿不同方向会有不同的变化率，这样引出方向导数的概念

沿坐标轴方向的导数若存在，称之为偏导数

通过研究发现，方向导数与偏导数存在一定关系，可用偏导数和所选定的方向来表示，即二元函数的两个偏导数已经足够表示清楚该函数在一点沿任意方向的变化情况

高阶偏导数若连续，则求导次序可交换

微分：微分是函数增量的线性主要部分，这一本质对一元函数或多元函数来说都一样。只不过若是二元函数，所选取的线性近似部分应该是两个方向自变量增量的线性组合，然后再考虑误差是否是自变量增量的高阶无穷小，若是，则微分存在

仅仅有偏导数存在，不能推出用线性关系近似表示函数增量后带来的误差足够小，即偏导数存在不一定有微分存在

若偏导数存在，且连续，则微分一定存在

极限、连续、偏导数和可微的关系在多元函数情形里比一元函数更为复杂

极值：若函数在一点取极值，且在该点导数（偏导数）存在，则此导数（偏导数）必为零

所以，函数在某点的极值情况，即函数在该点附近的函数增量的符号，由二阶微分的符号判断。对一元函数来说，二阶微分的符号就是二阶导数的符号，对二元函数来说，二阶微分的符号可由相应的二次型的正定或负定性判断。

级数敛散性的判别思路：首先看通项是否趋于零，若不趋于零则发散。若通项趋于零，看是否正项级数。若是正项级数，首先看能否利用比较判别法，注意等比级数和调和级数是常用来作比较的级数，若通项是连乘形式，考虑用比值判别法，若通项是乘方形式，考虑用根值判别法。若不是正项级数，取绝对值，考虑其是否绝对收敛，绝对收敛则必收敛。若绝对值不收敛，考察一般项，看是否交错级数，用莱布尼兹准则判断。若不是交错级数，只能通过最根本的方法判断，即看其前n项和是否有极限，具体问题具体分析。

比较判别法是充分必要条件，比值和根值法只是充分条件，不是必要条件。

函数项级数情况复杂，一般只研究幂级数。阿贝尔定理揭示了幂级数的重要性质：收敛区域存在一个收敛半径。所以对幂级数，关键在于求出收敛半径，而这可利用根值判别法解决。

逐项求导和逐项积分不改变幂级数除端点外的区域的敛散性，端点情况复杂，需具体分析。

一个函数能展开成幂级数的条件是：存在任意阶导数。展开后的幂级数能收敛于原来函数的条件是：余项（误差）要随着项数的增加趋于零。这与泰勒展开中的结论一致。

微分方程：不同种类的方程有不同的常见解法，但理解上并无难处。下册

（二）定积分、二重积分、三重积分、第一类曲线积分、第一类曲面积分都可以概率为一种类型的积分，从物理意义上来理解是某个空间区域（直线段、平面区域、立体区域、曲线段、曲面区域）的质量，其中被积元可看作区域的微小单元，被积函数则是该微小单元的密度

这些积分最终都是转化成定积分来计算

第二类曲线积分的物理意义是变力做功（或速度环量），第二类曲面积分的物理意义是流量

在研究上述七类积分的过程中，发现其实被积函数都是空间位置点的函数，于是把这种以空间位置作为自变量的函数称为场函数

场函数有标量场和向量场，一个向量场相当于三个标量场

场函数在一点的变化情况由方向导数给出，而方向导数最大的方向，称为梯度方向。梯度是一个向量，任何方向的方向导数，都是梯度在这个方向上的投影，所以梯度的模是方向导数的最大值

梯度方向是函数变化最快的方向，等位面方向是函数无变化的方向，这两者垂直

梯度实际上一个场函数不均匀性的量度

梯度运算把一个标量场变成向量场

一条空间曲线在某点的切向量，便是该点处的曲线微元向量，有三个分量，它建立了第一类曲线积分与第二类曲线积分的联系

一张空间曲面在某点的法向量，便是该点处的曲面微元向量，有三个分量，它建立了第一类曲面积分和第二类曲面积分的联系

物体在一点处的相对体积变化率由该点处的速度场决定，其值为速度场的散度 散度运算把向量场变成标量场

散度为零的场称为无源场

高斯定理的物理意义：对散度在空间区域进行体积分，结果应该是这个空间区域的体积变化率，同时这种体积变化也可看成是在边界上的流量造成的，故两者应该相等。即高斯定理把一个速度场在边界上的积分与速度场的散度在该边界所围的闭区域上的体积分联系起来

无源场的体积变化为零，这是容易理解的，相当于既无损失又无补充

物体在一点处的旋转情况由该点处的速度场决定，其值为速度场的旋度

旋度运算把向量场变成向量场

旋度为零的场称为无旋场

斯托克斯定理的物理意义：对旋度在空间曲面进行第二类曲面积分，结果应该表示的是这个曲面的旋转快慢程度，同时这种旋转也可看成是边界上的速度环量造成的，故两者应该相等。即斯托克斯定理把一个速度场在边界上形成的环量与该边界所围的曲面的第二类曲面积分联系起来。该解释是从速度环量的角度出发得到的，比高斯定理要难，不强求掌握。

无旋场的速度环量为零，这相当于一个区域没有旋转效应，这是容易理解的

格林定理是斯托克斯定理的平面情形

进一步考察无旋场的性质

旋度为零，相当于对旋度作的第二类曲面积分为零——即等号后边的第二类曲线积分为零，相当于该力场围绕一闭合空间曲线作做的功为零——即从该闭合曲线上任选一点出发，积分与路径无关——相当于所得到的曲线积分结果只于终点的选择有关，与路径无关，可看成终点的函数，这是一个场函数（空间位置的函数），称为势函数——所得的势函数的梯度正好就是原来的力场——因为力场函数是连续的，所以势函数有全微分

简单的概括起来就是：无旋场——积分与路径无关——梯度场——有势场——全微分

要注意以上这些说法之间的等价性

三定理（Gauss Stokes Green）的向量形式和分量形式都要熟悉