Atnaujinkite slapukų nuostatas

All the Mathematics You Missed: But Need to Know for Graduate School [Kietas viršelis]

4.13/5 (205 ratings by Goodreads)
(Williams College, Massachusetts), Illustrated by
  • Formatas: Hardback, 376 pages, aukštis x plotis x storis: 236x158x31 mm, weight: 642 g, Worked examples or Exercises; 2 Tables, unspecified; 177 Line drawings, unspecified
  • Išleidimo metai: 26-Nov-2001
  • Leidėjas: Cambridge University Press
  • ISBN-10: 0521792851
  • ISBN-13: 9780521792851
Kitos knygos pagal šią temą:
  • Formatas: Hardback, 376 pages, aukštis x plotis x storis: 236x158x31 mm, weight: 642 g, Worked examples or Exercises; 2 Tables, unspecified; 177 Line drawings, unspecified
  • Išleidimo metai: 26-Nov-2001
  • Leidėjas: Cambridge University Press
  • ISBN-10: 0521792851
  • ISBN-13: 9780521792851
Kitos knygos pagal šią temą:
Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge. But few have such a background. This book will help students to see the broad outline of mathematics and to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential geometry, real analysis, point-set topology, probability, complex analysis, abstract algebra, and more. An annotated bibliography then offers a guide to further reading and to more rigorous foundations. This book will be an essential resource for advanced undergraduate and beginning graduate students in mathematics, the physical sciences, engineering, computer science, statistics, and economics who need to quickly learn some serious mathematics.

Recenzijos

"This book will fill an interesting niche in a library collection...it should be used by browsing students interested in making sure that they are prepared for success in their graduate programs." Choice "All the Mathematics You Missed...is a help for students going on to graduate school..Since many students beginning graduate school do not have the mathematical knowledge needed, All the Mathematics You Missed aims to fill in the gaps." Berkshire Eagle, Pittsfield, MA "From the preface: 'The goal of this book is to give people at least a rough idea of many topics that beginning graduate students at the best graduate schools are assumed to know." Mathematical Reviews "The writing is lucid mathematical exposition, at a level quite appropriate to beginning graduate students." The American Statistician "Before classes began, I jump started my graduate career with the help of this book. Even though I didn't believe that I could have missed much math, it became clear that my belief was wrong during the first week of class. While proving a theorem, my professor asked if anyone remembered a previous result from calculus. While I did not remember it from my days as an undergraduate, I had read about the theorem and had even seen a sketch of the proof in Garrity's book...This will be one of the books that I keep with me as I continue as a graduate student. It has certainly helped me understand concepts that I have missed." Elizabeth D. Russell, Math Horizons "Point set topology, complex analysis, differential forms, the curvature of surfaces, the axiom of choice, Lebesgue integration, Fourier analysis, algorithms, and differential equations.... I found these sections to be the high points of the book. They were a sound introduction to material that some but not all graduate students will need."

Charles Ashbacher, School Science and Mathematics

Daugiau informacijos

An essential resource for advanced undergraduate and beginning graduate students in quantitative subjects who need to quickly learn some serious mathematics.
Preface xiii
On the Structure of Mathematics xix
Brief Summaries of Topics xxiii
Linear Algebra xxiii
Real Analysis xxiii
Differentiating Vector-Valued Functions xxiii
Point Set Topology xxiv
Classical Stokes' Theorems xxiv
Differential Forms and Stokes' Theorem xxiv
Curvature for Curves and Surfaces xxiv
Geometry xxv
Complex Analysis xxv
Countability and the Axiom of Choice xxvi
Algebra xxvi
Lebesgue Integration xxvi
Fourier Analysis xxvi
Differential Equations xxvii
Combinatorics and Probability Theory xxvii
Algorithms xxvii
Linear Algebra
1(22)
Introduction
1(1)
The Basic Vector Space Rn
2(2)
Vector Spaces and Linear Transformations
4(2)
Bases and Dimension
6(3)
The Determinant
9(3)
The Key Theorem of Linear Algebra
12(2)
Similar Matrices
14(1)
Eigenvalues and Eigenvectors
15(5)
Dual Vector Spaces
20(1)
Books
21(1)
Exercises
21(2)
&epsis; and δ Real Analysis
23(24)
Limits
23(2)
Continuity
25(1)
Differentiation
26(2)
Integration
28(3)
The Fundamental Theorem of Calculus
31(4)
Pointwise Convergence of Functions
35(1)
Uniform Convergence
36(2)
The Weierstrass M-Test
38(2)
Weierstrass' Example
40(3)
Books
43(1)
Exercises
44(3)
Calculus for Vector-Valued Functions
47(16)
Vector-Valued Functions
47(2)
Limits and Continuity
49(1)
Differentiation and Jacobians
50(3)
The Inverse Function Theorem
53(3)
Implicit Function Theorem
56(4)
Books
60(1)
Exercises
60(3)
Point Set Topology
63(18)
Basic Definitions
63(3)
The Standard Topology on Rn
66(6)
Metric Spaces
72(1)
Bases for Topologies
73(2)
Zariski Topology of Commutative Rings
75(2)
Books
77(1)
Exercises
78(3)
Classical Stokes' Theorems
81(30)
Preliminaries about Vector Calculus
82(13)
Vector Fields
82(2)
Manifolds and Boundaries
84(3)
Path Integrals
87(4)
Surface Integrals
91(2)
The Gradient
93(1)
The Divergence
93(1)
The Curl
94(1)
Orientability
94(1)
The Divergence Theorem and Stokes' Theorem
95(2)
Physical Interpretation of Divergence Thm.
97(1)
A Physical Interpretation of Stokes' Theorem
98(1)
Proof of the Divergence Theorem
99(5)
Sketch of a Proof for Stokes' Theorem
104(4)
Books
108(1)
Exercises
108(3)
Differential Forms and Stokes' Thm.
111(34)
Volumes of Parallelepipeds
112(3)
Diff. Forms and the Exterior Derivative
115(9)
Elementary k-forms
115(3)
The Vector Space of k-forms
118(1)
Rules for Manipulating k-forms
119(3)
Differential k-forms and the Exterior Derivative
122(2)
Differential Forms and Vector Fields
124(2)
Manifolds
126(6)
Tangent Spaces and Orientations
132(5)
Tangent Spaces for Implicit and Parametric Manifolds
132(1)
Tangent Spaces for Abstract Manifolds
133(2)
Orientation of a Vector Space
135(1)
Orientation of a Manifold and its Boundary
136(1)
Integration on Manifolds
137(2)
Stokes' Theorem
139(3)
Books
142(1)
Exercises
143(2)
Curvature for Curves and Surfaces
145(16)
Plane Curves
145(3)
Space Curves
148(4)
Surfaces
152(5)
The Gauss-Bonnet Theorem
157(1)
Books
158(1)
Exercises
158(3)
Geometry
161(10)
Euclidean Geometry
162(1)
Hyperbolic Geometry
163(3)
Elliptic Geometry
166(1)
Curvature
167(1)
Books
168(1)
Exercises
169(2)
Complex Analysis
171(30)
Analyticity as a Limit
172(2)
Cauchy-Riemann Equations
174(5)
Integral Representations of Functions
179(8)
Analytic Functions as Power Series
187(4)
Conformal Maps
191(3)
The Riemann Mapping Theorem
194(2)
Several Complex Variables: Hartog's Theorem
196(1)
Books
197(1)
Exercises
198(3)
Countability and the Axiom of Choice
201(12)
Countability
201(4)
Naive Set Theory and Paradoxes
205(2)
The Axiom of Choice
207(1)
Non-measureable Sets
208(2)
Godel and Independence Proofs
210(1)
Books
211(1)
Exercises
211(2)
Algebra
213(18)
Groups
213(6)
Representation Theory
219(2)
Rings
221(2)
Fields and Galois Theory
223(5)
Books
228(1)
Exercises
229(2)
Lebesgue Integration
231(12)
Lebesgue Measure
231(3)
The Cantor Set
234(2)
Lebesgue Integration
236(3)
Convergence Theorems
239(2)
Books
241(1)
Exercises
241(2)
Fourier Analysis
243(18)
Waves, Periodic Functions and Trigonometry
243(1)
Fourier Series
244(6)
Convergence Issues
250(2)
Fourier Integrals and Transforms
252(4)
Solving Differential Equations
256(2)
Books
258(1)
Exercises
258(3)
Differential Equations
261(24)
Basics
261(1)
Ordinary Differential Equations
262(4)
The Laplacian
266(4)
Mean Value Principle
266(1)
Separation of Variables
267(3)
Applications to Complex Analysis
270(1)
The Heat Equation
270(3)
The Wave Equation
273(6)
Derivation
273(4)
Change of Variables
277(2)
Integrability Conditions
279(2)
Lewy's Example
281(1)
Books
282(1)
Exercises
282(3)
Combinatorics and Probability
285(22)
Counting
285(2)
Basic Probability Theory
287(3)
Independence
290(1)
Expected Values and Variance
291(3)
Central Limit Theorem
294(6)
Stirling's Approximation for n!
300(5)
Books
305(1)
Exercises
305(2)
Algorithms
307(20)
Algorithms and Complexity
308(1)
Graphs: Euler and Hamiltonian Circuits
308(5)
Sorting and Trees
313(3)
P = NP?
316(1)
Numerical Analysis: Newton's Method
317(7)
Books
324(1)
Exercises
324(3)
A Equivalence Relations 327