高等数学英文板总结.doc

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 means that f(x) can be made to be as close to L as desired by making x sufficiently close to c.无穷小InfinitesimalIn 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 (Bolzanos 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:， where the symbol (the uppercase form of the Greek letter Delta) is an abbreviation for change in. It follows that y = m x. 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, Rolles 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 . This version of Rolles theorem is used to prove the mean value theorem, of which Rolles theorem is indeed a special case. It is also the basis for the proof of Taylors theorem.拉格朗日中值定理Lagranges mean value theorem柯西中值定理Cauchys mean value theorem Cauchys 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 ；Of course, if g(a) g(b) and if g(c) 0, this is equivalent to: 。洛必达法则LHpitals ruleIn calculus, lHpitals rule (pronounced:lopital) 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, lHpitals 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 orand exists, andfor all x in I with x c, thenThe 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Taylors theorem Statement of the theoremThe precise statement of the most basic version of Taylors theorem is as follows: Taylors theorem.Let k1 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AntiderivativeIn calculus, an antiderivative, primitive integral or indefinite integral1 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IntegrationThrough 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 outputsThe 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 integralThe multiple integral is a generalization of the definite integral to functions of more than one real variable, for example, f(x,y) orf(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 fieldFor 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 fieldFor 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 integralIn 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 fieldsTo 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Greens theoremLet 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, Greens 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, Greens theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.高斯公式Divergence theoremSuppose V is a subset of (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 have:The 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 seriesIn 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 n