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All the Math You Missed: (But Need to Know for Graduate School) 2nd Revised edition [Minkštas viršelis]

4.13/5 (217 ratings by Goodreads)
(Williams College, Massachusetts)
  • Formatas: Paperback / softback, 416 pages, aukštis x plotis x storis: 250x174x20 mm, weight: 880 g, Worked examples or Exercises
  • Išleidimo metai: 01-Jul-2021
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1009009192
  • ISBN-13: 9781009009195
Kitos knygos pagal šią temą:
All the Math You Missed: (But Need to Know for Graduate School) 2nd Revised editionDidesnis paveikslėlis
  • Formatas: Paperback / softback, 416 pages, aukštis x plotis x storis: 250x174x20 mm, weight: 880 g, Worked examples or Exercises
  • Išleidimo metai: 01-Jul-2021
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1009009192
  • ISBN-13: 9781009009195
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781009009195.html
Raktažodžiai:
The second edition of this bestselling book provides an overview of the key topics in undergraduate mathematics, allowing beginning graduate students to fill in any gaps in their knowledge. With numerous examples, exercises and suggestions for further reading, it is a must-have for anyone looking to learn some serious mathematics quickly.

Beginning graduate students in mathematical sciences and related areas in physical and computer sciences and engineering are expected to be familiar with a daunting breadth of mathematics, but few have such a background. This bestselling book helps students fill in the gaps in their knowledge. Thomas A. Garrity 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 explanations are accompanied by numerous examples, exercises and suggestions for further reading that allow the reader to test and develop their understanding of these core topics. Featuring four new chapters and many other improvements, this second edition of All the Math You Missed is an essential resource for advanced undergraduates and beginning graduate students who need to learn some serious mathematics quickly.

Recenzijos

'Reading Garrity is like talking with your favorite uncle - he tells you the essential stories, in a clear and colorful way, and you get just what you need to explore further. The topics are well chosen (and there are more in this new edition). His points of view enrich the reader - not only do you learn what to know, but how to know it. I wish I had had this book when I started graduate school.' John McCleary, Vassar College 'I admired one of the intentions behind the first edition of Garrity's All the Math You Missed: to give students the tools to appreciate the applications of mathematics without painting a simplistic picture of 'Applied Mathematics'. In this second edition, he takes this idea to the next level by introducing four additional chapters, dealing primarily with number theory and category theory.' Robert Kotiuga, Boston University 'I felt like I was terribly underprepared for graduate school, and Garrity's book helped me fill in some of those gaps. But far more importantly, the welcoming tone made me see that I wasn't alone in feeling anxious, and it made grad school feel less intimidating.' Daniel Erman, University of Wisconsin, Madison 'Incoming graduate students would find the book most useful this book is designed to provide some useful guidance The writing is clear and easy to read.' Bill Satzer, MAA Reviews

Daugiau informacijos

Fill in any gaps in your knowledge with this overview of key topics in undergraduate mathematics, now with four new chapters.
Preface xiii
On the Structure of Mathematics xix
Brief Summaries of Topics xxiii
0.1 Linear Algebra
xxiii
0.2 Real Analysis
xxiii
0.3 Differentiating Vector-Valued Functions
xxiii
0.4 Point Set Topology
xxiii
0.5 Classical Stokes' Theorems
xxiv
0.6 Differential Forms and Stokes' Theorem
xxiv
0.7 Curvature for Curves and Surfaces
xxiv
0.8 Geometry
xxiv
0.9 Countability and the Axiom of Choice
xxv
0.10 Elementary Number Theory
xxv
0.11 Algebra
xxv
0.12 Algebraic Number Theory
xxv
0.13 Complex Analysis
xxv
0.14 Analytic Number Theory
xxvi
0.15 Lebesgue Integration
xxvii
0.16 Fourier Analysis
xxvii
0.17 Differential Equations
xxvii
0.18 Combinatorics and Probability Theory
xxvii
0.19 Algorithms
xxviii
0.20 Category Theory
xxviii
1 Linear Algebra 1(22)
1.1 Introduction
1(1)
1.2 The Basic Vector Space Rn
1(3)
1.3 Vector Spaces and Linear Transformations
4(2)
1.4 Bases, Dimension, and Linear Transformations as Matrices
6(3)
1.5 The Determinant
9(3)
1.6 The Key Theorem of Linear Algebra
12(1)
1.7 Similar Matrices
13(2)
1.8 Eigenvalues and Eigenvectors
15(4)
1.9 Dual Vector Spaces
19(1)
1.10 Books
20(1)
Exercises
21(2)
2 and δ Real Analysis 23(23)
2.1 Limits
23(2)
2.2 Continuity
25(1)
2.3 Differentiation
26(2)
2.4 Integration
28(3)
2.5 The Fundamental Theorem of Calculus
31(4)
2.6 Pointwise Convergence of Functions
35(1)
2.7 Uniform Convergence
36(3)
2.8 The Weierstrass M-Test
39(1)
2.9 Weierstrass' Example
40(3)
2.10 Books
43(1)
Exercises
44(2)
3 Calculus for Vector-Valued Functions 46(15)
3.1 Vector-Valued Functions
46(1)
3.2 Limits and Continuity of Vector-Valued Functions
47(1)
3.3 Differentiation and Jacobians
48(4)
3.4 The Inverse Function Theorem
52(2)
3.5 The Implicit Function Theorem
54(4)
3.6 Books
58(1)
Exercises
59(2)
4 Point Set Topology 61(17)
4.1 Basic Definitions
61(2)
4.2 The Standard Topology on Rn
63(7)
4.3 Metric Spaces
70(1)
4.4 Bases for Topologies
71(1)
4.5 Zariski Topology of Commutative Rings
72(3)
4.6 Books
75(1)
Exercises
76(2)
5 Classical Stokes' Theorems 78(29)
5.1 Preliminaries about Vector Calculus
79(13)
5.1.1 Vector Fields
79(1)
5.1.2 Manifolds and Boundaries
80(4)
5.1.3 Path Integrals
84(4)
5.1.4 Surface Integrals
88(2)
5.1.5 The Gradient
90(1)
5.1.6 The Divergence
90(1)
5.1.7 The Curl
91(1)
5.1.8 Orientability
92(1)
5.2 The Divergence Theorem and Stokes' Theorem
92(2)
5.3 A Physical Interpretation of the Divergence Theorem
94(2)
5.4 A Physical Interpretation of Stokes' Theorem
96(1)
5.5 Sketch of a Proof of the Divergence Theorem
97(5)
5.6 Sketch of a Proof of Stokes' Theorem
102(3)
5.7 Books
105(1)
Exercises
105(2)
6 Differential Forms and Stokes' Theorem 107(34)
6.1 Volumes of Parallelepipeds
107(4)
6.2 Differential Forms and the Exterior Derivative
111(9)
6.2.1 Elementary k-Forms
111(3)
6.2.2 The Vector Space of k-Forms
114(1)
6.2.3 Rules for Manipulating k-Forms
115(3)
6.2.4 Differential k-Forms and the Exterior Derivative
118(2)
6.3 Differential Forms and Vector Fields
120(2)
6.4 Manifolds
122(6)
6.5 Tangent Spaces and Orientations
128(5)
6.5.1 Tangent Spaces for Implicit and Parametric Manifolds
128(1)
6.5.2 Tangent Spaces for Abstract Manifolds
129(1)
6.5.3 Orientation of a Vector Space
130(1)
6.5.4 Orientation of a Manifold and Its Boundary
131(2)
6.6 Integration on Manifolds
133(2)
6.7 Stokes' Theorem
135(3)
6.8 Books
138(1)
Exercises
138(3)
7 Curvature for Curves and Surfaces 141(15)
7.1 Plane Curves
141(3)
7.2 Space Curves
144(4)
7.3 Surfaces
148(5)
7.4 The Gauss-Bonnet Theorem
153(1)
7.5 Books
154(1)
Exercises
154(2)
8 Geometry 156(9)
8.1 Euclidean Geometry
156(2)
8.2 Hyperbolic Geometry
158(3)
8.3 Elliptic Geometry
161(1)
8.4 Curvature
162(1)
8.5 Books
163(1)
Exercises
163(2)
9 Countability and the Axiom of Choice 165(11)
9.1 Countability
165(4)
9.2 Naive Set Theory and Paradoxes
169(2)
9.3 The Axiom of Choice
171(1)
9.4 Non-measurable Sets
172(1)
9.5 Godel and Independence Proofs
173(1)
9.6 Books
174(1)
Exercises
175(1)
10 Elementary Number Theory 176(16)
10.1 Types of Numbers
177(2)
10.2 Prime Numbers
179(2)
10.3 The Division Algorithm and the Euclidean Algorithm
181(2)
10.4 Modular Arithmetic
183(1)
10.5 Diophantine Equations
183(2)
10.6 Pythagorean Triples
185(2)
10.7 Continued Fractions
187(3)
10.8 Books
190(1)
Exercises
191(1)
11 Algebra 192(18)
11.1 Groups
192(6)
11.2 Representation Theory
198(2)
11.3 Rings
200(2)
11.4 Fields and Galois Theory
202(5)
11.5 Books
207(1)
Exercises
208(2)
12 Algebraic Number Theory 210(6)
12.1 Algebraic Number Fields
210(1)
12.2 Algebraic Integers
211(2)
12.3 Units
213(1)
12.4 Primes and Problems with Unique Factorization
214(1)
12.5 Books
215(1)
Exercises
215(1)
13 Complex Analysis 216(29)
13.1 Analyticity as a Limit
217(2)
13.2 Cauchy-Riemann Equations
219(5)
13.3 Integral Representations of Functions
224(8)
13.4 Analytic Functions as Power Series
232(3)
13.5 Conformal Maps
235(3)
13.6 The Riemann Mapping Theorem
238(2)
13.7 Several Complex Variables: Hartog's Theorem
240(1)
13.8 Books
241(1)
Exercises
242(3)
14 Analytic Number Theory 245(10)
14.1 The Riemann Zeta Function
245(2)
14.2 Riemann's Insight
247(1)
14.3 The Gamma Function
248(1)
14.4 The Functional Equation: A Hidden Symmetry
249(1)
14.5 Linking π(x) with the Zeros of ζ(s)
250(3)
14.6 Books
253(1)
Exercises
254(1)
15 Lebesgue Integration 255(11)
15.1 Lebesgue Measure
255(3)
15.2 The Cantor Set
258(2)
15.3 Lebesgue Integration
260(3)
15.4 Convergence Theorems
263(1)
15.5 Books
264(1)
Exercises
265(1)
16 Fourier Analysis 266(16)
16.1 Waves, Periodic Functions and Trigonometry
266(1)
16.2 Fourier Series
267(6)
16.3 Convergence Issues
273(2)
16.4 Fourier Integrals and Transforms
275(3)
16.5 Solving Differential Equations
278(3)
16.6 Books
281(1)
Exercises
281(1)
17 Differential Equations 282(22)
17.1 Basics
282(1)
17.2 Ordinary Differential Equations
283(4)
17.3 The Laplacian
287(4)
17.3.1 Mean Value Principle
287(1)
17.3.2 Separation of Variables
288(3)
17.3.3 Applications to Complex Analysis
291(1)
17.4 The Heat Equation
291(3)
17.5 The Wave Equation
294(6)
17.5.1 Derivation
294(4)
17.5.2 Change of Variables
298(2)
17.6 The Failure of Solutions: Integrability Conditions
300(2)
17.7 Lewy's Example
302(1)
17.8 Books
303(1)
Exercises
303(1)
18 Combinatorics and Probability Theory 304(23)
18.1 Counting
304(2)
18.2 Basic Probability Theory
306(2)
18.3 Independence
308(1)
18.4 Expected Values and Variance
309(3)
18.5 Central Limit Theorem
312(7)
18.6 Stirling's Approximation for n!
319(5)
18.7 Books
324(1)
Exercises
325(2)
19 Algorithms 327(18)
19.1 Algorithms and Complexity
327(1)
19.2 Graphs: Euler and Hamiltonian Circuits
328(4)
19.3 Sorting and Trees
332(4)
19.4 P=NP?
336(1)
19.5 Numerical Analysis: Newton's Method
337(6)
19.6 Books
343(1)
Exercises
343(2)
20 Category Theory 345(20)
20.1 The Basic Definitions
345(2)
20.2 Examples
347(1)
20.3 Functors
347(3)
20.3.1 Link with Equivalence Problems
347(1)
20.3.2 Definition of Functor
348(1)
20.3.3 Examples of Functors
349(1)
20.4 Natural Transformations
350(2)
20.5 Adjoints
352(4)
20.6 "There Exists" and "For All" as Adjoints
356(1)
20.7 Yoneda Lemma
357(5)
20.8 Arrow, Arrows, Arrows Everywhere
362(1)
20.9 Books
363(1)
Exercises
364(1)
Appendix Equivalence Relations 365(2)
Bibliography 367(8)
Index 375
Thomas A. Garrity is the Webster Atwell Class of 1921 Professor of Mathematics at Williams College, Massachusetts, where he was the director of the Williams College Project for Effective Teaching for many years. Among his awards are Rice University's Nicolas Salgo Outstanding Teaching award and the Haimo award of the MAA. His other books include Algebraic Geometry: A Problem Solving Approach (2013, co-authored) and Electricity and Magnetism for Mathematicians (2015).