differential algebras

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本篇阅读材料“保持⼤脑年轻化的七种⽅法”选⾃《读者⽂摘》（原⽂标题：7 Anti-Aging Tips to Keep Your Brain Young）。

如果⼤家觉得⽐较简单，就当作泛读材料了解了解，认识⼏个新单词或新表达⽅式也不错。

如果⼤家觉得这些材料理解上有难度，不妨当做挑战⾃⼰的拔⾼训练，希望⼤家都有进步^^ 1. Move It Quick — what’s the No. 1 thing you can do for your brain’s health? Differential calculus, you say? Chess? Chaos theory? Nope, the best brain sharpener may be … sneakers? Yup. Once they’re on your feet, you can pump up your heart rate. “The best advice I can give to keep your brain healthy and young is aerobic exercise,” says Donald Stuss, PhD, a neuropsychologist and director of the Rotman Research Institute at Baycrest Centre for Geriatric Care in Toronto. differential calculus [数]微积分；微分学 chaos theory 混沌理论 pump up 给……打⽓；加速 aerobic exercise 有氧运动 Mark McDaniel, PhD, professor of psychology at Washington University in St. Louis, agrees, but adds, “I would suggest a combined program of aerobics and weight training. Studies show the best outcomes for those engaged in both types of exercise.” As we age, our brain cells, called neurons, lose the tree-branch-like connections between them. These connections, or synapses, are essential to thought. Quite literally, over time, our brains lose their heft. Perhaps the most striking brain research today is the strong evidence we now have that “exercise may forestall some kinds of mental decline,” notes McDaniel. It may even restore memory. Myriad animal studies have shown that, among other brain benefits, aerobic exercise increases capillary development in the brain, meaning more blood supply, more nutrients and — a big requirement for brain health — more oxygen. forestall v. 垄断；预先阻⽌ mental decline 智⼒下降 The preeminent exercise and brain-health researcher in humans is Arthur Kramer at the University of Illinois at Urbana-Champaign. In a dozen studies over the past few years, with titles such as “Aerobic Fitness Reduces Brain Tissue Loss in Aging Humans,” Kramer and his colleagues have proved two critical findings: Fit people have sharper brains, and people who are out of shape, but then get into shape, sharpen up their brains. This second finding is vital. There’s no question that working out makes you smarter, and it does so, Kramer notes, at all stages of life. Just as important, exercise staves off heart disease, obesity, diabetes and other maladies that increase the risk of brain problems as we age. preeminent adj. 卓越的 a dozen ⼀打；⼗⼆个 out of shape （⾝体）⾛形；变样 sharpen up 使……更敏锐；认真思考 stave off 避开；延缓 2. Feed It Another path to a better brain is through your stomach. We’ve all heard about antioxidants as cancer fighters. Eating foods that contain these molecules, which neutralize harmful free radicals, may be especially good for your brain too. Free radicals have nothing to do with Berkeley politics and everything to do with breaking down the neurons in our brains. Many colorful fruits and vegetables are packed with antioxidants, as are some beans, whole grains, nuts and spices. neutralize v. 抵消；中和；使……中⽴ have nothing to do with 与……⽆关 be packed with 挤满；塞满；充满 More important, though, is overall nutrition. In concert with a good workout routine, you should eat right to avoid the diseases that modern flesh is heir to. High blood pressure, diabetes, obesity and high cholesterol all make life tough on your brain, says Carol Greenwood, PhD, a geriatric research scientist at the University of Toronto. in concert with 和…相呼应；与…合作；和…⼀致 flesh is heir to ⼈所难免[共有]的 If your diet is heavy, then you’re probably also heavy. The same weight that burdens your legs on the stairs also burdens your brain for the witty reply or quick problem solving. The best things you can eat for your body, Greenwood notes, are also the best things you can eat for your brain. Your brain is in your body, after all. Greenwood’s recommendation is to follow the dietary guidelines from the American Diabetes Association (available at ). 3. Speed It Up Sorry to say, our brains naturally start slowing down at the cruelly young age of 30 (yes, 30). It used to be thought that this couldn’t be helped, but a barrage of new studies show that people of any age can train their brains to be faster and, in effect, younger. “Your brain is a learning machine,” says Michael Merzenich, PhD, a neuroscientist at the University of California, San Francisco. Given the right tools, we can train our brains to act like they did when we were younger. All that’s required is dedicated practice: exercises for the mind. a barrage of ⼤量的 in effect 实际上 Merzenich has developed a computer-based training regimen to speed up how the brain processes information (). Since much of the data we receive comes through speech, the Brain Fitness Program works with language and hearing to improve both speed and accuracy. Over the course of your training, the program starts asking you to distinguish sounds (between “dog” and “bog,” for instance) at an increasingly faster rate. It’s a bit like a tennis instructor, says Merzenich, shooting balls at you faster and faster over the course of the summer to keep you challenged. Though you may have started out slow, by Labor Day you’re pretty nimble. nimble adj. 敏捷的；聪明的 Similarly, Nintendo was inspired by the research of a Japanese doctor to develop a handheld game called Brain Age: Train Your Brain in Minutes a Day, which has sold more than two million copies in Japan. No software out there has yet been approved by the FDA as a treatment for cognitive impairment, but an increasing number of reputable scientific studies suggest that programs like Merzenich’s could help slow down typical brain aging, or even treat dementia. The biggest finding in brain research in the last ten years is that the brain at any age is highly adaptable, or “plastic,” as neurologists put it. If you ask your brain to learn, it will learn. And it may speed up in the process. Nintendo 任天堂（⽇本电⼦游戏公司及其开发的电脑游戏名称） FDA （美）⾷品及药物管理局（Food and Drug Administration） cognitive impairment 认知障碍；认知损害 brain aging 脑⽼化 To keep your brain young and supple, you can purchase software like Merzenich’s, or you can do one of a million new activities that challenge and excite you: playing Ping-Pong or contract bridge, doing jigsaw puzzles, learning a new language or the tango, taking accordion lessons, building a kit airplane, mastering bonsai technique, discovering the subtleties of beer-brewing and, sure, relearning differential calculus. supple adj. 灵活的；柔软的 jigsaw puzzles 拼图游戏；拼图玩具 “Anything that closely engages your focus and is strongly rewarding,” says Merzenich, will kick your brain into learning mode and necessarily notch it up. For his part, Merzenich, 64, has “4,000 hobbies,” including a wood shop and a vineyard. notch up 完成 4. Stay Calm So you may be saying to yourself, I have to sign up right now for Swahili and calculus and accordion lessons before my brain withers away! Stop! Breathe. Relax. Good. wither away 枯萎；幻灭 While challenging your brain is very important, remaining calm is equally so. In a paper on the brain and stress, Jeansok Kim of the University of Washington asserts, in no uncertain terms, that traumatic stress is bad for your brain cells. Stress can “disturb cognitive processes such as learning and memory, and consequently limit the quality of human life,” writes Kim. in no uncertain terms 明确地 One example is a part of the brain called the hippocampus, which is a primary locus of memory formation, but which canbe seriously debilitated by chronic stress. Of course, physical exercise is always a great destressor, as are calmer activities like yoga and meditation. And when you line up your mental calisthenics (your Swahili and swing lessons), make sure you can stay loose and have fun. 5. Give It a Rest Perhaps the most extreme example of the mental power of staying calm is the creative benefit of sleep. Next time you’re working on a complex problem, whether it be a calculus proof or choosing the right car for your family, it really pays to “sleep on it.” Researchers at Harvard Medical School have looked at the conditions under which people come up with creative solutions. In a study involving math problems, they found that a good night’s rest doubled participants’ chances of finding a creative solution to the problems the next day. The sleeping brain, they theorize, is vastly capable of synthesizing complex information. 6. Laugh a Little Humor stimulates the parts of our brain that use the “feel good” chemical messenger dopamine. That puts laughter in the category of activities you want to do over and over again, such as eating chocolate or having sex. Laughter is pleasurable, perhaps even “addictive,” to the brain. But can humor make us smarter? The jury is still out and more studies are needed, but the initial results are encouraging. Look for a feature on exciting new research about humor and intelligence in the September issue of Reader’s Digest. 7. Get Better With Age In our youth-obsessed culture, no one’s suggesting a revision to the Constitution allowing 20-year-olds to run for President. The age requirement remains at 35. You’ve heard about the wisdom and judgment of older people? Scientists are starting to understand how wisdom works on a neurological level.。

漫谈微分几何、多复变函数与代数几何（Differential geometry, functions of complex variable and algebraic geometry）Differential geometry and tensor analysis, developed with the development of differential geometry, are the basic tools for mastering general relativity. Because general relativity's success, to always obscure differential geometry has become one of the central discipline of mathematics.Since the invention of differential calculus, the birth of differential geometry was born. But the work of Euler, Clairaut and Monge really made differential geometry an independent discipline. In the work of geodesy, Euler has gradually obtained important research, and obtained the famous Euler formula for the calculation of normal curvature. The Clairaut curve of the curvature and torsion, Monge published "analysis is applied to the geometry of the loose leaf paper", the important properties of curves and surfaces are represented by differential equations, which makes the development of classical differential geometry to reach a peak. Gauss in the study of geodesic, through complicated calculation, in 1827 found two main curvature surfaces and its product in the periphery of the Euclidean shape of the space not only depends on its first fundamental form, the result is Gauss proudly called the wonderful theorem, created from the intrinsic geometry. The free surface of space from the periphery, the surface itself as a space to study. In 1854, Riemann made the hypothesis about geometric foundation, and extended the intrinsic geometry of Gauss in 2 dimensional curved surface, thus developing n-dimensional Riemann geometry, with the development of complex functions. A group of excellentmathematicians extended the research objects of differential geometry to complex manifolds and extended them to the complex analytic space theory including singularities. Each step of differential geometry faces not only the deepening of knowledge, but also the continuous expansion of the field of knowledge. Here, differential geometry and complex functions, Lie group theory, algebraic geometry, and PDE all interact profoundly with one another. Mathematics is constantly dividing and blending with each other.By shining the charming glory and the differential geometric function theory of several complex variables, unit circle and the upper half plane (the two conformal mapping establishment) defined on Poincare metric, complex function theory and the differential geometric relationships can be seen distinctly. Poincare metric is conformal invariant. The famous Schwarz theorem can be explained as follows: the Poincare metric on the unit circle does not increase under analytic mapping; if and only if the mapping is a fractional linear transformation, the Poincare metric does not change Poincare. Applying the hyperbolic geometry of Poincare metric, we can easily prove the famous Picard theorem. The proof of Picard theorem to modular function theory is hard to use, if using the differential geometric point of view, can also be in a very simple way to prove. Differential geometry permeates deep into the theory of complex functions. In the theory of multiple complex functions, the curvature of the real differential geometry and other series of calculations are followed by the analysis of the region definition metric of the complex affine space. In complex situations, all of the singular discrete distribution, and in more complex situations, because of the famous Hartogsdevelopment phenomenon, all isolated singularities are engulfed by a continuous region even in singularity formation is often destroyed, only the formation of real codimension 1 manifold can avoid the bad luck. But even this situation requires other restrictions to ensure safety". The singular properties of singularities in the theory of functions of complex functions make them destined to be manifolds. In 1922, Bergman introduced the famous Bergman kernel function, the more complex function or Weyl said its era, in addition to the famous Hartogs, Poincare, Levi of Cousin and several predecessors almost no substantive progress, injected a dynamic Bergman work will undoubtedly give this dead area. In many complex function domains in the Bergman metric metric in the one-dimensional case is the unit circle and Poincare on the upper half plane of the Poincare, which doomed the importance of the work of Bergman.The basic object of algebraic geometry is the properties of the common zeros (algebraic families) of any dimension, affine space, or algebraic equations of a projective space (defined equations),The definitions of algebraic clusters, the coefficients of equations, and the domains in which the points of an algebraic cluster are located are called base domains. An irreducible algebraic variety is a finite sub extension of its base domain. In our numerical domain, the linear space is the extension of the base field in the number field, and the dimension of the linear space is the number of the expansion. From this point of view, algebraic geometry can be viewed as a study of finite extension fields. The properties of algebraic clusters areclosely related to their base domains. The algebraic domain of complex affine space or complex projective space, the research process is not only a large number of concepts and differential geometry and complex function theory and applied to a large number of coincidence, the similar tools in the process of research. Every step of the complex manifold and the complex analytic space has the same influence on these subjects. Many masters in related fields, although they seem to study only one field, have consequences for other areas. For example: the Lerey study of algebraic topology that it has little effect on layer, in algebraic topology, but because of Serre, Weil and H? Cartan (E? Cartan, eldest son) introduction, has a profound impact on algebraic geometry and complex function theory. Chern studies the categories of Hermite spaces, but it also affects algebraic geometry, differential geometry and complex functions. Hironaka studies the singular point resolution in algebraic geometry, but the modification of complex manifold to complex analytic space and blow up affect the theory of complex analytic space. Yau proves that the Calabi conjecture not only affects algebraic geometry and differential geometry, but also affects classical general relativity. At the same time, we can see the important position of nonlinear ordinary differential equations and partial differential equations in differential geometry. Cartan study of symmetric Riemann space, the classification theorem is important, given 1, 2 and 3 dimensional space of a Homogeneous Bounded Domain complete classification, prove that they are all homogeneous symmetric domains at the same time, he guessed: This is also true in the n-dimensional equivalent relation. In 1959, Piatetski-Shapiro has two counterexample and find the domain theory of automorphic function study in symmetry, in the 4 and 5dimensional cases each find a homogeneous bounded domain, which is not a homogeneous symmetric domain, the domain he named Siegel domain, to commemorate the profound work on Siegel in 1943 of automorphic function. The results of Piatetski-Shapiro has profound impact on the theory of complex variable functions and automorphic function theory, and have a profound impact on the symmetry space theory and a series of topics. As we know, Cartan transforms the study of symmetric spaces into the study of Lie groups and Lie algebras, which is directly influenced by Klein and greatly develops the initial idea of Klein. Then it is Cartan developed the concept of Levi-Civita connection, the development of differential geometry in general contact theory, isomorphic mapping through tangent space at each point on the manifold, realize the dream of Klein and greatly promote the development of differential geometry. Cartan is the same, and concluded that the importance of the research in the holonomy manifold twists and turns, finally after his death in thirty years has proved to be correct. Here, we see the vast beauty of differential geometry.As we know, geodesic ties are associated with ODE (ordinary differential equations), minimal surfaces and high dimensional submanifolds are associated with PDE (partial differential equations). These equations are nonlinear equations, so they have high requirements for analysis. Complex PDE and complex analysis the relationship between Cauchy-Riemann equations coupling the famous function theory, in the complex case, the Cauchy- Riemann equations not only deepen the unprecedented contact and the qualitative super Cauchy-Riemann equations (the number of variables is greater than the number of equations) led to a strange phenomenon. This makes PDE and the theory ofmultiple complex functions closely integrated with differential geometry.Most of the scholars have been studying the differential geometry of the intrinsic geometry of the Gauss and Riemann extremely deep stun, by Cartan's method of moving frames is beautiful and concise dumping, by Chern's theory of characteristic classes of the broad and profound admiration, Yau deep exquisite geometric analysis skills to deter.When the young Chern faced the whole differentiation, he said he was like a mountain facing the shining golden light, but he couldn't reach the summit at one time. But then he was cast as a master in this field before Hopf and Weil.If the differential geometry Cartan development to gradually change the general relativistic geometric model, then the differential geometry of Chern et al not only affect the continuation of Cartan and to promote the development of fiber bundle in the form of gauge field theory. Differential geometry is still closely bound up with physics as in the age of Einstein and continues to acquire research topics from physicsWhy does the three-dimensional sphere not give flatness gauge, but can give conformal flatness gauge? Because 3D balls and other dimension as the ball to establish flat space isometric mapping, so it is impossible to establish a flatness gauge; and n-dimensional balls are usually single curvature space, thus can establish a conformal flat metric. In differential geometry, isometry means that the distance between the points on the manifold before and after the mapping remains the same. Whena manifold is equidistant from a flat space, the curvature of its Riemann cross section is always zero. Since the curvature of all spheres is positive constant, the n-dimensional sphere and other manifolds whose sectional curvature is nonzero can not be assigned to local flatness gauge.But there are locally conformally flat manifolds for this concept, two gauge G and G, if G=exp{is called G, P}? G between a and G transform is a conformal transformation. Weyl conformal curvature tensor remains unchanged under conformal transformation. It is a tensor field of (1,3) type on a manifold. When the Weyl conformal curvature tensor is zero, the curvature tensor of the manifold can be represented by the Ricci curvature tensor and the scalar curvature, so Penrose always emphasizes the curvature =Ricci+Weyl.The metric tensor g of an n-dimensional Riemann manifold is conformally equivalent to the flatness gauge locally, and is called conformally flat manifold. All Manifolds (constant curvature manifolds) whose curvature is constant are conformally flat, so they can be given conformal conformal metric. And all dimensions of the sphere (including thethree-dimensional sphere) are manifold of constant curvature, so they must be given conformal conformal metric. Conversely, conformally flat manifolds are not necessarily manifolds of constant curvature. But a wonderful result related to Einstein manifolds can make up for this regret: conformally conformally Einstein manifolds over 3 dimensions must be manifolds of constant curvature. That is to say, if we want conformally conformally flat manifolds to be manifolds of constant curvature, we must call Ric= lambda g, and this is thedefinition of Einstein manifolds. In the formula, Ric is the Ricci curvature tensor, G is the metric tensor, and lambda is constant. The scalar curvature S=m of Einstein manifolds is constant. Moreover, if S is nonzero, there is no nonzero parallel tangent vector field over it. Einstein introduction of the cosmological constant. So he missed the great achievements that the expansion of the universe, so Hubble is successful in the official career; but the vacuum gravitational field equation of cosmological term with had a Einstein manifold, which provides a new stage for mathematicians wit.For the 3 dimensional connected Einstein manifold, even if does not require the conformal flat, it is also the automatic constant curvature manifolds, other dimensions do not set up this wonderful nature, I only know that this is the tensor analysis summer learning, the feeling is a kind of enjoyment. The sectional curvature in the real manifold is different from the curvature of the Holomorphic cross section in the Kahler manifold, and thus produces different results. If the curvature of holomorphic section is constant, the Ricci curvature of the manifold must be constant, so it must be Einstein manifold, called Kahler- Einstein manifold, Kahler. Kahler manifolds are Kahler- Einstein manifolds, if and only if they are Riemann manifolds, Einstein manifolds. N dimensional complex vector space, complex projective space, complex torus and complex hyperbolic space are Kahler- and Einstein manifolds. The study of Kahler-Einstein manifolds becomes the intellectual enjoyment of geometer.Let's go back to an important result of isometric mapping.In this paper, we consider the isometric mapping between M and N and the mapping of the cut space between the two Riemann manifolds, take P at any point on M, and select two non tangent tangent vectors in its tangent space, and obtain its sectional curvature. In the mapping, the two tangent vectors on the P point and its tangent space are transformed into two other tangent vectors under the mapping, and the sectional curvature of the vector is also obtained. If the mapping is isometric mapping, then the curvature of the two cross sections is equal. Or, to be vague, isometric mapping does not change the curvature of the section.Conversely, if the arbitrary points are set, the curvature of the section does not change in nature, then the mapping is not isometric mapping The answer was No. Even in thethree-dimensional Euclidean space on the surface can not set up this property. In some cases, the limit of the geodesic line must be added, and the properties of the Jacobi field can be used to do so. This is the famous Cartan isometry theorem. This theorem is a wonderful application of the Jacobi field. Its wide range of promotion is made by Ambrose and Hicks, known as the Cartan-Ambrose-Hicks theorem.Differential geometry is full of infinite charm. We classify pseudo-Riemannian spaces by using Weyl conformal curvature tensor, which can be classified by Ricci curvature tensor, or classified into 9 types by Bianchi. And these things are all can be attributed to the study of differential geometry, this distant view Riemann and slightly closer to the Klein point of the perfect combination, it can be seen that the great wisdom Cartan, here you can see the profound influence of Einstein.From the Hermite symmetry space to the Kahler-Hodge manifold, differential geometry is not only closely linked with the Lie group, but also connected with algebra, geometry and topologyThink of the great 1895 Poicare wrote the great "position analysis" was founded combination topology unabashedly said differential geometry in high dimensional space is of little importance to this subject, he said: "the home has beautiful scenery, where Xuyuan for." (Chern) topology is the beauty of the home. Why do you have to work hard to compute the curvature of surfaces or even manifolds of high dimensions? But this versatile mathematician is wrong, but we can not say that the mathematical genius no major contribution to differential geometry? Can not. Let's see today's close relation between differential geometry and topology, we'll see. When is a closed form the proper form? The inverse of the Poicare lemma in the region of the homotopy point (the single connected region) tells us that it is automatically established. In the non simply connected region is de famous Rham theorem tells us how to set up, that is the integral differential form in all closed on zero.Even in the field of differential geometry ignored by Poicare, he is still in a casual way deeply affected by the subject, or rather is affecting the whole mathematics.The nature of any discipline that seeks to be generalized after its creation, as is differential geometry. From the curvature, Euclidean curvature of space straight to zero, geometry extended to normal curvature number (narrow Riemann space) andnegative constant space (Lobachevskii space), we know that the greatness of non Euclidean geometry is that it not only independent of the fifth postulate and other alternative to the new geometry. It can be the founder of triangle analysis on it. But the famous mathematician Milnor said that before differential geometry went into non Euclidean geometry, non Euclidean geometry was only the torso with no hands and no feet. The non Euclidean geometry is born only when the curvature is computed uniformly after the metric is defined. In his speech in 1854, Riemann wrote only one formula: that is, this formula unifies the positive curvature, negative curvature and zero curvature geometry. Most people think that the formula for "Riemann" is based on intuition. In fact, later people found the draft paper that he used to calculate the formula. Only then did he realize that talent should be diligent. Riemann has explored the curvature of manifolds of arbitrary curvature of any dimension, but the quantitative calculations go beyond the mathematical tools of that time, and he can only write the unified formula for manifolds of constant curvature. But we know,Even today, this result is still important, differential geometry "comparison theorem" a multitude of names are in constant curvature manifolds for comparison model.When Riemann had considered two differential forms the root of two, this is what we are familiar with the Riemann metric Riemannnian, derived from geometry, he specifically mentioned another case, is the root of four four differential forms (equivalent to four yuan product and four times square). This is the contact and the difference between the two. But he saidthat for this situation and the previous case, the study does not require substantially different methods. It also says that such studies are time consuming and that new insights cannot be added to space, and the results of calculations lack geometric meaning. So Riemann studied only what is now called Riemann metric. Why are future generations of Finsler interested in promoting the Riemann's not wanting to study? It may be that mathematicians are so good that they become a hobby. Cartan in Finsler geometry made efforts, but the effect was little, Chern on the geometric really high hopes also developed some achievements. But I still and general view on the international consensus, that is the Finsler geometry bleak. This is also the essential reason of Finsler geometry has been unable to enter the mainstream of differential geometry, it no beautiful properties really worth geometers to struggle, also do not have what big application value. Later K- exhibition space, Cartan space will not become mainstream, although they are the extension of Riemannnian geometry, but did not get what the big development.In fact, sometimes the promotion of things to get new content is not much, differential geometry is the same, not the object of study, the more ordinary the better, but should be appropriate to the special good. For example, in Riemann manifold, homogeneous Riemann manifold is more special, beautiful nature, homogeneous Riemann manifolds, symmetric Riemann manifold is more special, so nature more beautiful. This is from the analysis of manifold Lie group action angle.From the point of view of metric, the complex structure is given on the even dimensional Riemann manifold, and the complexmanifold is very elegant. Near complex manifolds are complex manifolds only when the near complex structure is integrable. The complex manifold must be orientable, because it is easy to find that its Jacobian must be nonnegative, whereas the real manifold does not have this property in general. To narrow the scope of the Kahler manifold has more good properties, all complex Submanifolds of Kahler manifolds are Kahler manifolds, and minimal submanifolds (Wirtinger theorem), the beautiful results captured the hearts of many differential geometry and algebraic geometry, because other more general manifolds do not set up this beautiful results. If the first Chern number of a three-dimensional Kahler manifold is zero, the Calabi-Yau manifold can be obtained, which is a very interesting manifold for theoretical physicists. The manifold of mirrors of Calabi-Yau manifolds is also a common subject of differential geometry in algebraic geometry. The popular Hodge structure is a subject of endless appeal.Differential geometry, an endless topic. Just as algebraic geometry requires double - rational equivalence as a luxury, differential geometry requires isometric transformations to be difficult. Taxonomy is an eternal subject of mathematics. In group theory, there are single group classification, multi complex function theory, regional classification, algebraic geometry in the classification of algebraic clusters, differential geometry is also classified.The hard question has led to a dash of young geometry and old scholars, and the prospect of differential geometry is very bright.。

美国数学本科生、研究生基础课程参考书目在网上找书的时候恰好看到这个，看着觉得的确是经典书目大全，贴在这里供学弟学妹们参考：）其中所谓第几学年云云，各校要求不同，像我所在的学校，一般学生第一年选三到四门基础课（代数、分析、几何三大类中至少各挑一门），学年末进行qualifying笔试。

第二年开始选自己喜爱方向的高级课程，并通过qualifying口试。

第三年开始做research，并通过第二语言考试（法语或德语或俄语，一般人都选法语，因为代数几何经典大作都是法语的）. 而Princeton 就没有基础课，只有seminar类型的课。

第一学年几何与拓扑：1、James R. Munkres, Topology：较新的拓扑学的教材适用于本科高年级或研究生一级；2、Basic Topology by Armstrong：本科生拓扑学教材；3、Kelley, General Topology：一般拓扑学的经典教材，不过观点较老；4、Willard, General Topology：一般拓扑学新的经典教材；5、Glen Bredon, Topology and geometry：研究生一年级的拓扑、几何教材；6、Introduction to Topological Manifolds by John M. Lee：研究生一年级的拓扑、几何教材，是一本新书；7、from calculus to cohomology by Madsen：很好的本科生代数拓扑、微分流形教材。

代数：1、Abstract Algebra Dummit：最好的本科代数学参考书，标准的研究生一年级代数材；2、Algebra Lang：标准的研究生一、二年级代数教材，难度很高，适合作参考书；3、Algebra Hungerford：标准的研究生一年级代数教材，适合作参考书；4、Algebra M,Artin：标准的本科生代数教材；5、Advanced Modern Algebra by Rotman：较新的研究生代数教材，很全面；6、Algebra：a graduate course by Isaacs：较新的研究生代数教材；7、Basic algebra Vol I&II by Jacobson：经典的代数学全面参考书，适合研究生参考。

数学专业英语词汇(C)(转载)c function c类函数c manifold c廖c mapping c类映射ca set 上解析集calculability 可计算性calculable mapping 可计算映射calculable relation 可计算关系calculate 计算calculating automaton 计算自动机calculating circuit 计算电路calculating element 计算单元calculating machine 计算机calculating punch 穿孔计算机calculating register 计算寄存器calculating unit 计算装置calculation 计算calculation of areas 面积计算calculator 计算机calculus 演算calculus of approximations 近似计算calculus of classes 类演算calculus of errors 误差论calculus of finite differences 差分法calculus of probability 概率calculus of residues 残数计算calculus of variations 变分法calibration 校准canal 管道canal surface 管道曲面cancel 消去cancellation 消去cancellation law 消去律cancellation property 消去性质cancelling of significant figures 有效数字消去canonical basis 典范基canonical coordinates 标准坐标canonical correlation coefficient 典型相关系数canonical distribution 典型分布canonical ensemble 正则总体canonical equation 典型方程canonical equation of motion 标准运动方程canonical expression 典范式canonical factorization 典范因子分解canonical flabby resolution 典型松弛分解canonical form 标准型canonical function 标准函数canonical fundamental system 标准基本系统canonical homomorphism 标准同态canonical hyperbolic system 典型双曲线系canonical image 标准象canonical mapping 标准映射canonical representation 典型表示canonical sequence 标准序列canonical solution 标准解canonical system of differential equations 标准微分方程组canonical variable 典型变量canonical variational equations 标准变分方程canonical variational problem 标准变分问题cantor curve 康托尔曲线cantor discontinum 康托尔密断统cantorian set theory 经典集论cap 交cap product 卡积capacity 容量card 卡片card punch 卡片穿孔机card reader 卡片读数器cardinal 知的cardinal number 基数cardinal product 基数积cardioid 心脏线carrier 支柱carry 进位carry signal 进位信号cartan formula 嘉当公式cartan subalgebra 嘉当子代数cartan subgroup 嘉当子群cartesian coordinate system 笛卡儿坐标系cartesian coordinates 笛卡尔座标cartesian equation 笛卡儿方程cartesian folium 笛卡儿叶形线cartesian product 笛卡儿积cartesian space 笛卡儿空间cartography 制图学cascaded carry 逐位进位casimir operator 卡巫尔算子cassini oval 卡吾卵形线casting out 舍去casting out nines 舍九法catastrophe theory 突变理论categorical judgment 范畴判断categorical proposition 范畴判断categorical syllogism 直言三段论categorical theory 范畴论categoricity 范畴性category 范畴category of groups 群范畴category of modules 模的范畴category of sets 集的范畴category of topological spaces 拓扑空间的范畴catenary 悬链线catenary curve 悬链线catenoid 悬链曲面cauchy condensation test 柯微项收敛检验法cauchy condition for convergence 柯握敛条件cauchy criterion 柯握敛判别准则cauchy distribution 柯沃布cauchy filter 柯嗡子cauchy inequality 柯位等式cauchy integral 柯锡分cauchy integral formula 柯锡分公式cauchy kernel 柯嗡cauchy kovalevskaya theorem 柯慰仆吡蟹蛩箍ǘɡ眵cauchy mean value formula 广义均值定理cauchy net 柯硒cauchy principal value 柯蔚cauchy problem 柯问题cauchy process 柯锡程cauchy residue theorem 残数定理cauchy sequence 柯悟列causal relation 因果关系causality 因果律cause 原因cavity 空腔cavity coefficient 空胴系数cayley number 凯莱数cayley sextic 凯莱六次线cayley transform 凯莱变换ccr algebra ccr代数celestial body 天体celestial coordinates 天体坐标celestial mechanics 天体力学cell 胞腔cell complex 多面复形cellular approximation 胞腔逼近cellular automaton 细胞自动机cellular cohomology 胞腔上同调cellular cohomology group 胞腔上同岛cellular decomposition 胞腔剖分cellular homotopy 胞腔式同伦cellular map 胞腔映射cellular subcomplex 胞腔子复形center 中心center of a circle 圆心center of curvature 曲率中心center of expansion 展开中心center of force 力心center of gravity 重心center of gyration 旋转中心center of inversion 反演中心center of mass 质心center of pressure 压力中心center of principal curvature 助率中心center of projection 射影中心center of symmetry 对称中心centered process 中心化过程centered system of sets 中心集系centi 厘centigram 厘克centimetre 厘米central angle 圆心角central confidence interval 中心置信区间central conic 有心圆锥曲线central derivative 中心导数central difference 中心差分central difference operator 中心差分算子central divided difference 中心均差central element 中心元central extension 中心扩张central extension field 中心扩张域central limit theorem 中心极限定理central line 中线central moment 中心矩central point 中心点central processing unit 中央处理器central projection 中心射影central quadric 有心二次曲面central series 中心群列central symmetric vector field 中心对称向量场central symmetry 中心对称centralizer 中心化子centre 中心centre of a circle 圆心centre of gyration 旋转中心centre of projection 射影中心centre of similarity 相似中心centre of similitude 相似中心centrifugal force 离心力centripetal acceleration 向心加速度centroid 形心certain event 必然事件certainty 必然cesaro mean 纬洛平均cesaro method of summation 纬洛总求法chain 链chain complex 链复形chain condition 链条件chain equivalence 链等价chain equivalent 链等价的chain group 链群chain homotopic 链同伦的chain homotopy 链同伦chain index 链指数chain map 链变换chain of prime ideals 素理想链chain of syzygies 合冲链chain rule 链式法则chain transformation 链变换chainette 悬链线chamber complex 箱盒复形chance 偶然性;偶然的chance event 随机事件chance move 随机步chance quantity 随机量chance variable 机会变量change 变化change of metrics 度量的变换change of the base 基的变换change of the variable 变量的更换channel 信道channel width 信道宽度character 符号character group 特贞群character space 特贞空间characteriatic system 特寨characteristic 特征characteristic boundary value problem 特者值问题characteristic class 示性类characteristic cone 特斩characteristic conoid 特沾体characteristic curve 特怔线characteristic derivation 特阵导characteristic determinant 特招列式characteristic differential equation 特闸分方程characteristic direction 特战向characteristic equation 特战程characteristic exponent 特崭数characteristic function 特寨数characteristic functional 特蘸函characteristic group 特蘸characteristic index 特崭标characteristic initial value problem 特挣值问题characteristic linear system 特者性系统characteristic manifold 特瘴characteristic matrix 特肇阵characteristic number 特正characteristic of a logarithm 对数的首数characteristic parameter 特瘴数characteristic polynomial 特锗项式characteristic pontrjagin number 庞德里雅金特正characteristic root 特争characteristic ruled surface 特毡纹曲面characteristic series 特招characteristic set 特寨characteristic state 特宅characteristic strip 特狰characteristic subgroup 特沼群characteristic surface 特怔面characteristic value 矩阵的特盏characteristic vector 特镇量charge 电荷chart 图chebyshev function 切比雪夫函数chebyshev inequality 切比雪夫不等式chebyshev polynomial 切比雪夫多项式check 校验check digit 检验位check routine 检验程序check sum 检查和chevalley group 歇互莱群chi square distribution 分布chi squared test 检验chi squared test of goodness of fit 拟合优度检验choice function 选择函数chord 弦chord line 弦chord of contact 切弦chord of curvature 曲率弦chordal distance 弦距离christoffel symbol 克里斯托弗尔符号chromatic number 色数chromatic polynomial 色多项式cipher 数字circle 圆circle diagram 圆图circle method 圆法circle of contact 切圆circle of convergence 收敛圆circle of curvature 曲率圆circle of inversion 反演圆circle problem 圆内格点问题circuit free graph 环道自由图circuit rank 圈数circulant 循环行列式circulant matrix 轮换矩阵circular 圆的circular arc 圆弧circular cone 圆锥circular correlation 循环相关circular cylinder 圆柱circular disk 圆盘circular domain 圆形域circular frequency 角频率circular functions 圆函数circular helix 圆柱螺旋线circular measure 弧度circular motion 圆运动circular neighborhood 圆邻域circular orbit 圆轨道circular pendulum 圆摆circular permutation 循环排列circular ring 圆环circular section 圆截面circular sector 圆扇形circular segment 圆弓形circular slit domain 圆形裂纹域circular symmetry 圆对称circular transformation 圆变换circulation 循环circulation index 环粮数circulation of vector field 向量场的循环circulatory integral 围道积分circumcenter 外心circumcentre 外心circumcircle 外接圆circumcone 外切圆锥circumference 圆周circumscribe 外接circumscribed circle 外接圆circumscribed figure 外切形circumscribed polygon 外切多边形circumscribed quadrilateral 外切四边形circumscribed triangle 外切三角形circumsphere 外接球cissoid 蔓叶类曲线cissoidal curve 蔓叶类曲线cissoidal function 蔓叶类函数clairaut equation 克莱罗方程class 类class bound 组界class field 类域class field tower 类域塔class frequency 组频率class function 类函数class interval 组距class mean 组平均class number 类数class of conjugate elements 共轭元素类classical groups 典型群classical lie algebras 典型李代数classical mechanics 经典力学classical sentential calculus 经典语句演算classical set theory 经典集论classical statistical mechanics 经典统计力学classical theory of probability 经典概率论classification 分类classification statistic 分类统计classification theorem 分类定理classify 分类classifying map 分类映射classifying space 分类空间clear 擦去clifford group 克里福特群clifford number 克里福特数clockwise 顺时针的clockwise direction 顺时针方向clockwise rotation 顺时针旋转clopen set 闭开集closable linear operator 可闭线性算子closable operator 可闭算子closed ball 闭球closed circuit 闭合电路closed complex 闭复形closed convex curve 卵形线closed convex hull 闭击包closed cover 闭覆盖closed curve 闭曲线closed disk 闭圆盘closed domain 闭域closed equivalence relation 闭等价关系closed extension 闭扩张closed filter 闭滤子closed form 闭型closed formula 闭公式closed geodesic 闭测地线closed graph 闭图closed graph theorem 闭图定理closed group 闭群closed half plane 闭半平面closed half space 闭半空间closed hull 闭包closed interval 闭区间closed kernel 闭核closed linear manifold 闭线性廖closed loop system 闭圈系closed manifold 闭廖closed map 闭映射closed neighborhood 闭邻域closed number plane 闭实数平面closed path 闭路closed range theorem 闭值域定理closed region 闭域closed riemann surface 闭黎曼面closed set 闭集closed shell 闭壳层closed simplex 闭单形closed solid sphere 闭实心球closed sphere 闭球closed star 闭星形closed subgroup 闭子群closed subroutine 闭型子程序closed surface 闭曲面closed symmetric extension 闭对称扩张closed system 闭系统closed term 闭项closeness 附近closure 闭包closure operation 闭包运算closure operator 闭包算子closure property 闭包性质clothoid 回旋曲线cluster point 聚点cluster sampling 分组抽样cluster set 聚值集coadjoint functor 余伴随函子coalgebra 上代数coalition 联合coanalytic set 上解析集coarser partition 较粗划分coaxial circles 共轴圆cobase 共基cobordant manifolds 配边廖cobordism 配边cobordism class 配边类cobordism group 配边群cobordism ring 配边环coboundary 上边缘coboundary homomorphism 上边缘同态coboundary operator 上边缘算子cocategory 上范畴cochain 上链cochain complex 上链复形cochain homotopy 上链同伦cochain map 上链映射cocircuit 上环道cocommutative 上交换的cocomplete category 上完全范畴cocycle 上闭键code 代吗coded decimal notation 二进制编的十进制记数法codenumerable set 余可数集coder 编器codiagonal morphism 余对角射codifferential 上微分codimension 余维数coding 编码coding theorem 编码定理coding theory 编码理论codomain 上域coefficient 系数coefficient domain 系数域coefficient function 系数函数coefficient functional 系数泛函coefficient group 系数群coefficient of alienation 不相关系数coefficient of association 相伴系数coefficient of covariation 共变系数coefficient of cubical expansion 体积膨胀系数coefficient of determination 可决系数coefficient of diffusion 扩散系数coefficient of excess 超出系数coefficient of friction 摩擦系数coefficient of nondetermination 不可决系数coefficient of rank correlation 等级相关系数coefficient of regression 回归系数coefficient of the expansion 展开系数coefficient of thermal expansion 热膨胀系数coefficient of variation 变差系数coefficient of viscosity 粘性系数coefficient problem 系数问题coefficient ring 系数环coercive operator 强制算子cofactor 代数余子式cofiber 上纤维cofibering 上纤维化cofibration 上纤维化cofilter 余滤子cofinal set 共尾集cofinal subset 共尾子集cofinality 共尾性cofinite subset 上有限子集cofunction 余函数cogenerator 上生成元cogredient automorphism 内自同构coherence 凝聚coherence condition 凝聚条件coherent module 凝聚摸coherent ring 凝聚环coherent set 凝聚集coherent sheaf 凝聚层coherent stack 凝聚层coherent topology 凝聚拓扑coherently oriented simplex 协同定向单形cohomological dimension 上同惮数cohomological invariant 上同祷变量cohomology 上同调cohomology algebra 上同碟数cohomology class 上同掂cohomology functor 上同弹子cohomology group 上同岛cohomology group with coefficients g 有系数g的上同岛cohomology module 上同担cohomology operation 上同邓算cohomology ring 上同捣cohomology sequence 上同凋列cohomology spectral sequence 上同底序列cohomology theory 上同帝cohomotopy 上同伦cohomotopy group 上同伦群coideal 上理想coimage 余象coincidence 一致coincidence number 叠合数coincidence point 叠合点coincident 重合的coinduced topology 余导出拓扑cokernel 上核collect 收集collectionwise normal space 成集体正规空间collective 集体collinear diagram 列线图collinear points 共线点collinear vectors 共线向量collinearity 共线性collineation 直射变换collineation group 直射群collineatory transformation 直射变换collocation method 配置法collocation of boundary 边界配置collocation point 配置点colocally small category 上局部小范畴cologarithm 余对数colorable 可着色的column 列column finite matrix 列有限矩阵column matrix 列阵column rank 列秩column space 列空间column vector 列向量combination 组合combination principle 结合原理combination with repetitions 有复组合combination without repetition 无复组合combinatorial analysis 组合分析combinatorial closure 组合闭包combinatorial dimension 组合维数combinatorial geometry 组合几何学combinatorial manifold 组合廖combinatorial method 组合方法combinatorial optimization problem 组合最优化问题combinatorial path 组合道路combinatorial problem 组合最优化问题combinatorial sphere 组合球面combinatorial sum 组合和combinatorial theory of probabilities 概率组合理论combinatorial topology 组合拓朴学combinatorially equivalent complex 组合等价复形combinatories 组合分析combinatory logic 组合逻辑combinatory topology 组合拓朴学combined matrix 组合矩阵comma 逗点command 命令commensurability 可通约性commensurable 可通约的commensurable quantities 可公度量common denominator 公分母common difference 公差common divisor 公约数common factor 公因子common factor theory 公因子论common fraction 普通分数common logarithm 常用对数common measure 公测度common multiple 公倍元common perpendicular 公有垂线common point 公共点common ratio 公比common tangent of two circles 二圆公切线communality 公因子方差communication channel 通讯通道commutant 换位commutation law 交换律commutation relation 交换关系commutative 可换的commutative diagram 交换图表commutative group 交换群commutative groupoid 阿贝耳广群commutative law 交换律commutative lie ring 交换李环commutative ordinal numbers 交换序数commutative ring 交换环commutativity 交换性commutator 换位子commutator group 换位子群commute 交换compact 紧的compact convergence 紧收敛compact group 紧群compact open topology 紧收敛拓扑compact operator 紧算子compact set 紧集compact space 紧空间compact subgroup 紧子群compact support 紧支柱compactification 紧化compactification theorem 紧化定理compactness 紧性compactness theorem 紧性定理compactum 紧统comparability of cardinals 基数的可比较性comparable curve 可比曲线comparable function 可比的函数comparable topology 可比拓扑comparable uniformity 可比一致性comparison function 比较函数comparison method 比较法comparison series 比较用级数comparison test 比较检验comparison theorem 比较定理compass 两脚规compatibile condition 相容性条件compatibility 一致性compatibility condition 相容性条件compatible system of algebraic equations 相容代数方程组compatible topology 相容拓扑学compensate 补偿compensating method 补偿法compensation 补偿compensation of error 误差的补偿compiler 编译程序compiling routine 编译程序complanar line 共面线complele induction 数学归纳法complement 补集complement of an angle 余角complementary 补的complementary angle 余角complementary degree 余次数complementary divisor 余因子complementary event 余事件complementary function 余函数complementary graph 余图complementary ideal 余理想complementary laws 补余律complementary module 补模complementary modulus 补模数complementary set 补集complementary space 补空间complementary submodules 补子模complementary subset 余子集complementary subspace 补子空间complemented lattice 有补格complete abelian variety 完备阿贝耳簇complete accumulation point 完全聚点complete axiom system 完备公理系统complete category 完全范畴complete class 完备类complete continuity 完全连续性complete disjunction 完全析取complete elliptic integral 完全椭圆积分complete field 完全域complete field of sets 集的完全域complete graph 完全图complete group 完全群complete group variety 完备群簇complete homomorphism 完全同态complete induction 数学归纳法complete integral 完全积分complete intersection 完全交叉complete lattice 完全格complete linear system 完备线性系统complete local ring 完全局部环complete measure 完全测度complete measure space 完备测度空间complete metric space 完备度量空间complete normality axiom 完全正规性公理complete ordered field 全序域complete orthogonal sequence 完全正交序列complete orthogonal set 完全正交系complete orthogonal system 完全正交系complete orthonormal sequence 完备标准正交序列complete orthonormal system 完备标准正交系complete probability space 完全概率空间complete quadrangle 完全四点形complete quadrilateral 完全四边形complete reducibility theorem 完全可约性定理complete regularity separation axiom 完全正则性分离公理complete reinhardt domain 完全赖因哈耳特域complete set 完全集complete solution 完全积分complete space 完备空间complete subcategory 完全子范畴complete system 完备系complete system of functions 函数完备系complete system of fundamental sequences 完全基本序列系complete system of invariants 完全的不变量系complete tensor product 完全张量积completed shell 闭壳层completely additive 完全加性的completely additive family of sets 完全加性集族completely additive measure 完全加性测度completely compact set 完全紧集completely continuous function 完全连续函数completely continuous linear operator 完全连续线性算子completely continuous mapping 全连续映射completely continuous operator 全连续映射completely distributive lattice 完全分配格completely homologous maps 完全同党射completely independent system of axioms 完全独立公理系统completely integrable 完全可积的completely integrable system 完全可积组completely integrally closed 完全整闭的completely mixed game 完全混合对策completely monotone 完全单的completely monotonic function 完全单弹数completely monotonic sequence 完全单凋列completely multiplicative 完全积性的completely multiplicative function 完全积性函数completely primary ring 完全准素环completely reducible 完全可约的completely reducible group 完全可约群completely regular filter 完全正则滤子completely regular space 完全正则空间completely regular topology 完全正则拓扑completely separated sets 完全可离集completely specified automaton 完全自动机completely splitted prime ideal 完全分裂素理想completely transitive group 全可迁群completeness 完全性completeness theorem 完全性定理completion 完备化complex 复形complex analytic fiber bundle 复解析纤维丛complex analytic manifold 复解析廖complex analytic structure 复解析结构complex cone 线丛的锥面complex conjugate 复共轭的complex conjugate matrix 复共轭阵complex curve 复曲线complex curvelinear integral 复曲线积分complex domain 复域complex experiment 析因实验complex field 复数域complex flnction 复值函数complex fraction 繁分数complex group 辛群complex line 复线complex line bundle 复线丛complex manifold 复廖complex multiplication 复数乘法complex number 复数complex number plane 复数平面complex plane with cut 有割的复平面complex quantity 复量complex root 复根complex series 复级数complex sphere 复球面complex surface 线丛的曲面complex unit 单位复数complex valued function 复值函数complex variable 复变量complex vector bundle 复向量丛complex velocity potential 复速度位势complexity 复杂性complication 复杂化component 分量component of variance 方差的分量componentwise convergence 分量方式收敛composable 组成的compose 组成composite 合成composite divisor 合成除数composite function 合成函数composite functor 合成函子composite group 合成群composite hypothesis 复合假设composite number 合成数composite probability 复合概率composition 合成composition algebra 合成代数composition factor 合成因子composition homomorphism 合成同态composition of vector subspaces 向量子空间的合成composition operator 合成算子composition series 合成列compound determinant 复合行列式compound event 复合事件compound function 合成函数compound number 合成数compound probability 合成概率compound proportion 复比例compound rule 复合规则computable function 可计算函数computation 计算computational error 计算误差computational formula 计算公式computational mistake 计算误差compute 计算computer 计算机computing center 计算中心computing element 计算单元computing machine 计算机computing time 计算时间comultiplication 上乘法concave 凹的concave angle 凹角concave convex game 凹击对策concave curve 凹曲线concave function 凹函数concave polygon 凹多边形concavity 凹性concavo convex 凹击的concentration 集中;浓度concentration ellipse 同心椭圆concentric circles 同心圆concept 概念conchoid 蚌线conchoidal 蚌线的conclusion 结论concomitant variable 相伴变量concrete number 名数concurrent form 共点形式concurrent planes 共点面concyclic points 共圆点condensation of singularities 奇点的凝聚condensation point 凝聚点condensation principle 凝聚原理condition equation 条件方程condition for continuity 连续性条件condition number 条件数condition of connectedness 连通性条件condition of positivity 正值性条件conditional convergence 条件收敛conditional definition 条件定义conditional density 条件性密度conditional distribution 条件分布conditional entropy 条件熵conditional equation 条件方程conditional event 条件性事件conditional gradient method 条件梯度法conditional inequality 条件不等式conditional instability 条件不稳定conditional instruction 条件指令conditional jump 条件转移conditional mathematical expectation 条件数学期望conditional probability 条件概率conditional probability measure 条件概率测度conditional proposition 条件命题conditional sentence 条件命题conditional stability 条件稳定性conditional transfer of control 条件转移conditionally compact set 条件紧集conditionally complete 条件完备的conditionally convergent 条件收敛的conditionally convergent series 条件收敛级数conditionally well posed problems 条件适定的问题conditioned observation 条件观测conditioning number 条件数conditions of similarity 相似条件conduction 传导conductivity 传导率conductor 导体;前导子conductor ramification theorem 前导子分歧定理cone 锥cone of a complex 复形锥面cone of a simplex 单形锥面confidence belt 置信带confidence coefficient 置信系数confidence ellipse 置信椭圆confidence ellipsoid 置信椭面confidence interval 置信区间confidence level 置信水平confidence limit 置信界限confidence region 置信区域configuration 布局configuration space 构形空间confinal 共尾的confinality 共尾性confirmation 证实confluent divided difference 合六差confluent hypergeometric equation 合镣超几何微分方程confluent hypergeometric function 合连几何函数confluent hypergeometric series 合连几何级数confluent interpolation polynomial 汇合内插多项式confocal conic sections 共焦二次曲线confocal conics 共焦二次曲线confocal quadrics 共焦二次曲面conformable matrices 可相乘阵conformal 保角的conformal curvature tensor 保形曲率张量conformal differential geometry 保形微分几何学conformal geometry 保形几何conformal mapping 保角素示conformal projection 保形射影conformal representation 保角素示conformal transformation 保角映射conformally connected manifold 保形连通廖conformally geodesic lines 保形测地线confounding 混杂confrontation 比较confusion 混乱congruence 同余式congruence group 同余群congruence method 同余法congruence of lines 线汇congruence relation 同余关系congruence subgroup 同余子群congruence zeta function 同余函数congruent 同余的congruent mapping 合同映射congruent number 同余数congruent transformation 合同映射conic 圆锥曲线conic function 圆锥函数conic section 圆锥曲线conical helix 圆锥螺旋线conical surface 锥面conics 圆锥曲线论conjugate 共轭的conjugate axis 共轭轴conjugate class 共轭类conjugate complex 共轭复形conjugate complex number 共轭复数conjugate convex function 共轭击函数conjugate curve 共轭曲线conjugate curve of the second order 共轭二次曲线conjugate diameter 共轭直径conjugate direction 共轭方向conjugate dyad 共轭并向量conjugate element 共轭元素conjugate exponent 共轭指数conjugate field 共轭域conjugate foci 共轭焦点conjugate function 共轭函数conjugate gradient method 共轭梯度法conjugate hyperbola 共轭双曲线conjugate latin square 共轭拉丁平conjugate line 共轭直线conjugate number 共轭数conjugate operator 共轭算子conjugate points 共轭点conjugate quaternion 共轭四元数conjugate root 共轭根conjugate ruled surface 共轭直纹曲面conjugate series 共轭级数conjugate space 共轭空间conjugate transformation 共轭变换conjugate vector 共轭向量conjugation map 共轭映射conjugation operator 共轭算子conjunction 合取conjunctive normal form 合取范式connected 连通的connected asymptotic paths 连通渐近路线connected automaton 连通自动机connected category 连通范畴connected chain 连通链connected complex 连通复形connected component 连通分支connected curve 连通曲线connected domain 连通域connected graph 连通图connected group 连通群connected sequence of functors 函子的连通序列connected set 连通集connected space 连通空间connected sum 连通和connectedness 连通性connecting homomorphism 连通同态connecting morphism 连通同态connecting path 连接道路connection 联络connection component 连通分量connectivity 连通性connex 连通conoid 劈锥曲面conormal 余法线conormal image 余法线象conrol chart technique 控制图法consequence 后承consequent 后项conservation law 守恒律conservation of angular momentum 角动量守恒conservation of energy 能量守恒conservation of mass 质量守恒conservation of momentum 动量守恒conservative extension 守恒扩张conservative field of force 保守力场conservative force 保守力conservative measurable transformation 守恒可测变换conservative vector field 守恒向量场consistency 相容性consistency conditions 相容条件consistency of equations 方程组的相容性consistency problem 相容性问题consistencyproof 相容性的证明consistent axiom system 相容性公理系consistent equations 相容方程组consistent estimator 相容估计consistent system of equations 相容方程组consistent test 相容检验constancy of sign 符号恒性constant 常数constant coefficient 常系数constant field 常数域constant function 常值函数constant mapping 常值映射constant of integration 积分常数constant of proportionality 比例系数constant of structure 构造常数constant pressure chart 等压面图constant pressure surface 等压面constant sheaf 常数层constant sum game 常和对策constant term 常数项constant value 定值constituent 组分constitutional diagram 组分图constrained game 约束对策constrained maximization 约束最大化constrained minimization 约束最小化constrained optimization 约束最优化constraint 约束construct 准constructibility 可构成性constructible 可构成的constructible map 可构成映射constructible set 可构成集construction 构成construction problem 准题constructive dilemma 构造二难推论constructive existence proof 可构造存在证明constructive mathematics 可构造数学constructive ordinal number 可构造序数consumer's risk 用户风险contact 接触contact angle 接触角contact point 接触点contact surface 接触面contact transformation 切变换content 含量context sensitive grammar 上下文有关文法contiguity 接触contiguous confluent hypergeometric function 连接合连几何函数contiguous hypergeometric function 连接超几何函数contiguous map 连接映射contingency 随机性contingency table 列contingent 偶然事故continuability 可延拓性continuation method 连续法continued equality 连等式continued fraction 连分数continued fraction expansion 连分式展开式continued proportion 连比例continuity 连续性continuity axiom 连续性公理continuity condition 连续性条件continuity equation 连续方程continuity in the mean 均方连续性continuity interval 连续区间continuity method 连续法continuity of function 函数的连续性continuity on both sides 双边连续性continuity on the left 左连续性continuity on the right 右连续性continuity principle 连续性原理continuity theorem 连续性定理continuous 连续的continuous analyzer 连续分析器continuous approximation 连续近似continuous curve 连续曲线continuous differentiability 连续可微性continuous distribution 连续分布continuous distribution function 连续分布函数continuous dynamical system 连续动力系统continuous function 连续函数continuous function in the mean 均方连续函数continuous game 连续对策continuous geometry 连续几何continuous group 拓扑群continuous homology 连续同调continuous homology group 连续同岛continuous image 连续象continuous in x 依x连续的continuous limit 连续极限continuous map 连续映射continuous on the left 左方连续的。

Maple 14 内置库函数包清单内置程序包描述内置程序包：algcurves研究由多元多项式定义的一维代数簇(代数曲线)的工具Algebraic完成代数数的计算命令ArrayTools用于矩阵、向量、和数组的底层操作工具AudioTools声音文件的输入、输出、和操作的命令集合Bits位操作命令Cache缓存表操作命令CAD连接CAD系统codegen将Maple程序转化为其它语言CodeGeneration将Maple代码转化为其它语言CodeTools提高Maple代码效率和质量的命令combinat组合数学combstruct组合结构ContextMenu创建和修改关联菜单CUDA使用CUDA(R)技术加速线性代数程序CurveFitting曲线拟合Database数据库连接DEtools对常微分方程系统操作、求解、绘图。

DifferentialAlgebra简化和解耦多项式微分方程系统，并计算它们的幂级数解。

DifferentialGeomet微分几何、李代数、张量rydifforms微分形式处理DiscreteTransforms离散数据变换计算DocumentTools在Maple文件或图元件中实现编程Domains创建计算域ExcelTools访问和操作Excel格式的数据ExternalCalling在Maple内调用外部函数FileTools文件操作和处理finance金融计算GaussInt高斯整数genfunc处理有理广义函数geom3d三维欧几里得几何geometry二维欧几里得几何gfun判定和处理生成函数的命令集GraphTheory图论Groebner Groebner基计算group群论hashmset多数据集处理heap堆数据处理ImageTools图像处理InstallerBuilder创建Maple工具箱安装程序IntegerRelations符号常数的整数线性组合的近似浮点数IntegrationTools定积分和不定积分inttrans积分变换及其逆变换LargeExpression管理序列创建的工具LibraryTools函数库处理liesymm偏微分方程紧对称系统表示函数包LinearAlgebra线性代数程序包，矩阵和向量计算LinearFunctionalSy线性泛函方程组stemsLinearOperators线性泛函方程组求解ListTools列表处理Logic处理布尔逻辑表达式LREtools线性递归方程求解Maplets创建图形用户界面MathematicalFunct提供数学函数的信息ionsMathML生成或输出MathMLMatlab连接MATLABMatrixPolynomialA多项式矩阵处理lgebraMmaTranslator Mathematica函数转换器MTM Maple Toolbox for MATLAB接口工具箱MultiSeries渐近或级数展开近numapprox计算在给定区间的多项式逼近函数numtheory数论Optimization优化Ore_algebra线性算子代数OreTools伪线性代数OrthogonalSeries正交多项式orthopoly生成各种类型的正交多项式padic实数的p-adic逼近PDEtools偏微分方程求解Physics物理包plots绘图包plottools图形对象处理PolynomialIdeals多项式理想PolynomialTools多项式对象处理powseries幂级数ProcessControl统计过程控制的计算和可视化QDifferenceEquati线性Q-微分方程求解onsqueue队列数据结构RandomTools随机对象处理RationalNormalFor将有理函数构造为多项式典范形式和有理范式的工具包msRealDomain实数域RegularChains符号求解代数方程组RootFinding求方程的数值根ScientificConstants物理常数和元素周期表属性ScientificErrorAnal处理带有数值和公差的物理量ysisSecurity设置Maple引擎安全simplex单纯形Slode求线性常微分方程组的幂级数解SNAP符号－数值混合算法Sockets网络通讯工具SoftwareMetrics分析Maple程序和模块代码的复杂性SolveTools代数方程系统求解Spread电子表格处理stack堆栈数据结构处理Statistics统计和数据分析StringTools字符串处理Student大学数学教学包Student[Calculus1]大学数学－单变量微积分Student大学数学－线性代数[LinearAlgebra]Student大学数学－多元微积分[MultivariateCalculus]Student大学数学－数值分析基础[NumericalAnalysis]Student大学数学－微积分预备知识[Precalculus]Student大学数学－向量微积分[VectorCalculus]SumTools求不定和定和的封闭形式sumtools求不定和定和tensor张量计算及张量在广义相对论中的应用Threads线程Tolerances公差计算Typesetting编程方式使用标准工作表排版和2-D方程解析选项TypeTools拓展已有的类型Units单位转换和计算环境VariationalCalculu变分法sVectorCalculus多元和矢量微积分Worksheet生成和处理Maple工作表XMLTools使用XML文件下列程序包已经被弃用：diffalg见 DifferentialAlgebralinalg见 LinearAlgebra and VectorCalculusnetworks见 GraphTheorypolytools见 PolynomialToolsprocess见 Threadsstats见 Statisticsstudent见 Student更多信息，参考帮助 ?package，这里 package 是上面程序包列表的名称。

P1U1A Electrical Networks 电路network n. 网络，电路resistor n. 电阻器inductor n. 电感器capacitor n. 电容器passive network 无源网络active network 有源网络characteristic adj. 特性(的)；n. 特性曲线Ohm n. 欧姆Faraday n. 法拉第electric charge 电荷integral n. 积分increment n. 增量armature n. 电枢，衔铁，加固aforementioned adj. 上述的，前面提到的represent v. 代表，表示，阐明amplify v. 放大symbolic adj. 符号的，记号的mesh n. 网孔Kirchhoff’s first law 基尔霍夫第一定律loop current 回路电流voltage drop 电压降in series 串联differential adj. 微分的；n. 微分variable n. 变量outline n. 轮廓；v. 提出……的要点eliminate v. 消除，对消[1] In the case of a resistor, the voltage-current relationship is given by Ohm’s law, which states that the voltage across the resistor is equal to the current through the resistor multiplied by the value of the resistance.就电阻来说，电压—电流的关系由欧姆定律决定。

欧姆定律指出：电阻两端的电压等于电阻上流过的电流乘以电阻值。

Which做关系代词，以引出非限制性定语从句。

[2]It may be that the inductor voltage rather than the current is the variable of interest in the circuit.或许在电路中，人们感兴趣的变量是电感电压而不是电感电流。

9月1日数学研究生基础课程参考书目*这个计划是按照美国的体系制订的，美国一年级的研究生课程大概相当于我国重点大学数学本科大三、大四的水平第一学年秋季学期春季学期几何与拓扑I 几何与拓扑II1、James R. Munkres, Topology较新的拓扑学的教材适用于本科高年级或研究生一年级2、Basic Topology by Armstrong本科生拓扑学教材3、Kelley, General Topology一般拓扑学的经典教材，不过观点较老4、Willard, General Topology一般拓扑学新的经典教材5、Glen Bredon, Topology and geometry研究生一年级的拓扑、几何教材6、Introduction to Topological Manifolds by John M. Lee研究生一年级的拓扑、几何教材，是一本新书7、From calculus to cohomology by Madsen很好的本科生代数拓扑、微分流形教材代数I 代数II1、Abstract Algebra Dummit最好的本科代数学参考书，标准的研究生一年级代数教材2、Algebra Lang标准的研究生一、二年级代数教材，难度很高，适合作参考书3、Algebra Hungerford标准的研究生一年级代数教材，适合作参考书4、Algebra M,Artin标准的本科生代数教材5、Advanced Modern Algebra by Rotman较新的研究生代数教材，很全面6、Algebra：a graduate course by Isaacs较新的研究生代数教材7、Basic algebra V ol I&II by Jacobson经典的代数学全面参考书，适合研究生参考分析基础复分析I实分析I1、Walter Rudin, Principles of mathematical analysis本科数学分析的标准参考书2、Walter Rudin, Real and complex analysis标准的研究生一年级分析教材3、Lars V. Ahlfors, Complex analysis本科高年级和研究生一年级经典的复分析教材4、Functions of One Complex Variable I，J.B.Conway研究生级别的单变量复分析经典5、Lang, Complex analysis研究生级别的单变量复分析参考书6、Complex Analysis by Elias M. Stein较新的研究生级别的单变量复分析教材7、Lang, Real and Functional analysis研究生级别的分析参考书8、Royden, Real analysis标准的研究生一年级实分析教材9、Folland, Real analysis标准的研究生一年级实分析教材第二学年秋季学期春季学期代数III 代数IV1、Commutative ring theory, by H. Matsumura较新的研究生交换代数标准教材2、Commutative Algebra I&II by Oscar Zariski , Pierre Samuel经典的交换代数参考书3、An introduction to Commutative Algebra by Atiyah标准的交换代数入门教材4、An introduction to homological algebra ,by weibel较新的研究生二年级同调代数教材5、A Course in Homological Algebra by P.J.Hilton,U.Stammbach经典全面的同调代数参考书6、Homological Algebra by Cartan经典的同调代数参考书7、Methods of Homological Algebra by Sergei I. Gelfand, Yuri I. Manin高级、经典的同调代数参考书8、Homology by Saunders Mac Lane经典的同调代数系统介绍9、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud 高级的代数几何、交换代数的参考书，最新的交换代数全面参考代数拓扑I 代数拓扑II1、Algebraic Topology, A. Hatcher最新的研究生代数拓扑标准教材2、Spaniers "Algebraic Topology"经典的代数拓扑参考书3、Differential forms in algebraic topology, by Raoul Bott and Loring W. Tu 研究生代数拓扑标准教材4、Massey, A basic course in Algebraic topology经典的研究生代数拓扑教材5、Fulton , Algebraic topology：a first course很好本科生高年级和研究生一年级的代数拓扑参考书6、Glen Bredon, Topology and geometry标准的研究生代数拓扑教材，有相当篇幅讲述光滑流形7、Algebraic Topology Homology and Homotopy高级、经典的代数拓扑参考书8、A Concise Course in Algebraic Topology by J.P.May研究生代数拓扑的入门教材，覆盖范围较广9、Elements of Homotopy Theory by G.W. Whitehead高级、经典的代数拓扑参考书实分析II 泛函分析1、Royden, Real analysis标准研究生分析教材2、Walter Rudin, Real and complex analysis标准研究生分析教材3、Halmos，"Measure Theory"经典的研究生实分析教材，适合作参考书4、Walter Rudin, Functional analysis标准的研究生泛函分析教材5、Conway,A course of Functional analysis标准的研究生泛函分析教材6、Folland, Real analysis标准研究生实分析教材7、Functional Analysis by Lax高级的研究生泛函分析教材8、Functional Analysis by Yoshida高级的研究生泛函分析参考书9、Measure Theory, Donald L. Cohn经典的测度论参考书微分拓扑李群、李代数1、Hirsch, Differential topology标准的研究生微分拓扑教材，有相当难度2、Lang, Differential and Riemannian manifolds研究生微分流形的参考书，难度较高3、Warner,Foundations of Differentiable manifolds and Lie groups标准的研究生微分流形教材，有相当的篇幅讲述李群4、Representation theory: a first course, by W. Fulton and J. Harris李群及其表示论的标准教材5、Lie groups and algebraic groups, by A. L. Onishchik, E. B. Vinberg李群的参考书6、Lectures on Lie Groups W.Y.Hsiang李群的参考书7、Introduction to Smooth Manifolds by John M. Lee较新的关于光滑流形的标准教材8、Lie Groups, Lie Algebras, and Their Representation by V.S. Varadarajan最重要的李群、李代数参考书9、Humphreys, Introduction to Lie Algebras and Representation Theory , Springer-Verlag, GTM-9标准的李代数入门教材第三学年秋季学期春季学期微分几何I 微分几何II1、Peter Petersen, Riemannian Geometry标准的黎曼几何教材2、Riemannian Manifolds: An Introduction to Curvature by John M. Lee最新的黎曼几何教材3、doCarmo, Riemannian Geometry.标准的黎曼几何教材4、M. Spivak, A Comprehensive Introduction to Differential Geometry I—V全面的微分几何经典，适合作参考书5、Helgason , Differential Geometry,Lie groups,and symmetric spaces标准的微分几何教材6、Lang, Fundamentals of Differential Geometry最新的微分几何教材，很适合作参考书7、kobayashi/nomizu, Foundations of Differential Geometry经典的微分几何参考书8、Boothby,Introduction to Differentiable manifolds and Riemannian Geometry标准的微分几何入门教材，主要讲述微分流形9、Riemannian Geometry I.Chavel经典的黎曼几何参考书10、Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”V ol 1—3经典的现代几何学参考书代数几何I 代数几何II1、Harris,Algebraic Geometry: a first course代数几何的入门教材2、Algebraic Geometry Robin Hartshorne经典的代数几何教材，难度很高3、Basic Algebraic Geometry 1&2 2nd ed. I.R.Shafarevich.非常好的代数几何入门教材4、Principles of Algebraic Geometry by giffiths/harris全面、经典的代数几何参考书，偏复代数几何5、Commutative Algebra with a view toward Algebraic Geometry by Eisenbud 高级的代数几何、交换代数的参考书，最新的交换代数全面参考6、The Geometry of Schemes by Eisenbud很好的研究生代数几何入门教材7、The Red Book of Varieties and Schemes by Mumford标准的研究生代数几何入门教材8、Algebraic Geometry I : Complex Projective Varieties by David Mumford复代数几何的经典调和分析偏微分方程1、An Introduction to Harmonic Analysis,Third Edition Yitzhak Katznelson调和分析的标准教材，很经典2、Evans, Partial differential equations偏微分方程的经典教材3、Aleksei.A.Dezin，Partial differential equations，Springer-Verlag偏微分方程的参考书4、L. Hormander "Linear Partial Differential Operators, " I&II偏微分方程的经典参考书5、A Course in Abstract Harmonic Analysis by Folland高级的研究生调和分析教材6、Abstract Harmonic Analysis by Ross Hewitt抽象调和分析的经典参考书7、Harmonic Analysis by Elias M. Stein标准的研究生调和分析教材8、Elliptic Partial Differential Equations of Second Order by David Gilbarg偏微分方程的经典参考书9、Partial Differential Equations ，by Jeffrey Rauch标准的研究生偏微分方程教材复分析II 多复分析导论1、Functions of One Complex Variable II，J.B.Conway单复变的经典教材，第二卷较深入2、Lectures on Riemann Surfaces O.Forster黎曼曲面的参考书3、Compact riemann surfaces Jost黎曼曲面的参考书4、Compact riemann surfaces Narasimhan黎曼曲面的参考书5、Hormander " An introduction to Complex Analysis in Several Variables" 多复变的标准入门教材6、Riemann surfaces , Lang黎曼曲面的参考书7、Riemann Surfaces by Hershel M. Farkas标准的研究生黎曼曲面教材8、Function Theory of Several Complex Variables by Steven G. Krantz高级的研究生多复变参考书9、Complex Analysis: The Geometric Viewpoint by Steven G. Krantz高级的研究生复分析参考书专业方向选修课：1、多复分析2、复几何3、几何分析4、抽象调和分析5、代数几何6、代数数论7、微分几何8、代数群、李代数与量子群9、泛函分析与算子代数10、数学物理11、概率理论12、动力系统与遍历理论13、泛代数*数学基础：1、halmos ,native set theory2、fraenkel ,abstract set theory3、ebbinghaus ,mathematical logic4、enderton ,a mathematical introduction to logic5、landau, foundations of analysis6、maclane ,categories for working mathematican应该在核心课程学习的过程中穿插选修假设本科应有的水平分析Walter Rudin, Principles of mathematical analysisApostol , mathematical analysisM.spivak , calculus on manifoldsMunknes ,analysis on manifoldsKolmogorov/fomin , introductory real analysisArnold ,ordinary differential equations代数：linear algebra by Stephen H. Friedberglinear algebra by hoffmanlinear algebra done right by Axleradvanced linear algebra by Romanalgebra ,artina first course in abstract algebra by rotman几何：do carmo, differential geometry of curves and surfacesDifferential topology by PollackHilbert ,foundations of geometryJames R. Munkres, Topology12:54 | 阅读评论(2) | 固定链接 | Mathematics数学分析的一些课本一、中文的：最难的5套书：1、《数学分析新讲》（1、2、3册），张筑生，北大版2、《数学分析》（1、2、3册），方企勤，高等教育版3、《数学分析教程》（上、下册）常庚哲等，高等教育版4、《数学分析》（上、下册）黄玉民等，科学出版社5、《简明数学分析》王昆扬，高等教育版最抽象的教材：《数学分析》（上、下册），邹应，武汉大学数学基地班教材（个人认为是目前国内观点最高，最抽象的书）二、国外的书好书太多，菲赫金哥茨的《数学分析原理》太老了，他的那套《微积分学教程》3卷（共8本）才是他的成名作，不过也太老了。