The mathematical mechanic: using physical reasoning to solve problems
In this delightful book, Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can.
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton [u.a.]
Princeton Univ. Press
2009
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | In this delightful book, Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. |
| Beschreibung: | VIII, 186 S. Ill., graph. Darst. |
| ISBN: | 9780691140209 |
Internformat
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| 100 | 1 | |a Levi, Mark |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The mathematical mechanic |b using physical reasoning to solve problems |c Mark Levi |
| 264 | 1 | |a Princeton [u.a.] |b Princeton Univ. Press |c 2009 | |
| 300 | |a VIII, 186 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 520 | 3 | |a In this delightful book, Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Problem solving | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 0 | 7 | |a Mathematische Physik |0 (DE-588)4037952-8 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804140024511332352 |
|---|---|
| adam_text | Contents
1
Introduction
1
1.1
Math versus
Physics
1
1.2
What This Book Is About
2
1.3
A Physical versus a Mathematical Solution: An
Example
6
1.4
Acknowledgments
8
2
The Pythagorean Theorem
9
2.1
Introduction
9
2.2
The Fish Tank Proof of the Pythagorean Theorem
9
2.3
Converting a Physical Argument into a Rigorous
Proof
12
2.4
The Fundamental Theorem of Calculus
14
2.5
The Determinant by Sweeping
15
2.6
The Pythagorean Theorem by Rotation
16
2.7
Still Water Runs Deep
17
2.8
A Three-Dimensional Pythagorean Theorem
19
2.9
A Surprising Equilibrium
21
2.10
Pythagorean Theorem by Springs
22
2.11
More Geometry with Springs
23
2.12
A Kinetic Energy Proof: Pythagoras on Ice
24
2.13
Pythagoras and Einstein?
25
3
Minima and Maxima
27
3.1
The Optical Property of Ellipses
28
3.2
More about the Optical Property
31
3.3
Linear Regression (The Best Fit) via Springs
31
3.4
The Polygon of Least Area
34
3.5
The Pyramid of Least Volume
36
3.6
A Theorem on Centroids
39
3.7
An Isoperimetric Problem
40
3.8
The Cheapest Can
44
3.9
The Cheapest Pot
47
VI CONTENTS
3.10
The Best Spot in a Drive-in Theater
48
3.11
The Inscribed Angle
51
3.12
Fermat s Principle and Snell s Law
52
3.13
Saving a Drowning Victim by Fermat s Principle
57
3.14
The Least Sum of Squares to a Point
59
3.15
Why Does a Triangle Balance on the Point of
Intersection of the Medians?
60
3.16
The Least Sum of Distances to Four Points in Space
61
3.17
Shortest Distance to the Sides of an Angle
63
3.18
The Shortest Segment through a Point
64
3.19
Maneuvering a Ladder
65
3.20
The Most Capacious Paper Cup
67
3.21
Minimal-Perimeter Triangles
69
3.22
An Ellipse in the Corner
72
3.23
Problems
74
4
Inequalities by Electric Shorting
76
4.1
Introduction
76
4.2
The Arithmetic Mean Is Greater than the Geometric
Mean by Throwing a Switch
78
4.3
Arithmetic Mean
>
Harmonic Mean for
η
Numbers
80
4.4
Does Any Short Decrease Resistance?
81
4.5
Problems
83
5
Center of Mass: Proofs and Solutions
84
5.1
Introduction
84
5.2
Center of Mass of a Semicircle by Conservation of
Energy
85
5.3
Center of Mass of a Half-Disk (Half-Pizza)
87
5.4
Center of Mass of a Hanging Chain
88
5.5
Pappus s Centroid Theorems
89
5.6
Ceva s Theorem
92
5.7
Three Applications of Ceva s Theorem
94
5.8
Problems
96
6
Geometry and Motion
99
6.1
Area between the Tracks of a Bike
99
6.2
An Equal-Volumes Theorem
101
6.3
How Much Gold Is in a Wedding Ring?
102
6.4
The Fastest Descent
104
CONTENTS
VII
6.5
Finding Yt sin
t
and j- cos
t
by Rotation
106
6.6
Problems
108
7
Computing Integrals Using Mechanics
109
7.1
Computing
ƒ„
-^¿Ły
by Lifting a Weight
109
7.2
Computing f* sin tdt with a Pendulum
111
7.3
A Fluid Proof of Green s Theorem
J
12
8
The Euler-Lagrange Equation via Stretched Springs
115
8.1
Some Background on the Euler-Lagrange Equation
115
8.2
A Mechanical Interpretation of the Euler-Lagrange
Equation
117
8.3
A Derivation of the Euler-Lagrange Equation
118
8.4
Energy Conservation by Sliding a Spring
119
9
Lenses, Telescopes, and Hamiltonian Mechanics
120
9.1
Area-Preserving Mappings of the Plane: Examples
121
9.2
Mechanics and Maps
121
9.3
A (Literally
! )
Hand-Waving Proof of Area
Preservation
123
9.4
The Generating Function
124
9.5
A Table of Analogies between Mechanics and
Analysis
125
9.6
The Uncertainty Principle
126
9.7
Area Preservation in Optics
126
9.8
Telescopes and Area Preservation
129
9.9
Problems
131
10
A Bicycle Wheel and the Gauss-Bonnet Theorem
133
10.1
Introduction
133
10.2
The Dual-Cones Theorem
135
10.3
The Gauss-Bonnet Formula Formulation and
Background
138
10.4
The Gauss-Bonnet Formula by Mechanics
142
10.5
A Bicycle Wheel and the Dual Cones
143
10.6
The Area of a Country
146
11
Complex Variables Made Simple(r)
148
11.1
Introduction
148
11.2
How a Complex Number Could Have Been Invented
149
VIII
CONTENTS
11.3
Functions as Ideal Fluid Flows
150
11.4
A Physical Meaning of the Complex Integral
153
11.5
The Cauchy Integral Formula via Fluid Flow
154
11.6
Heat Flow and Analytic Functions
156
11.7
Riemann Mapping by Heat Flow
157
11.8
Euler s Sum via Fluid Flow
159
Appendix. Physical Background
161
A.I Springs
161
A.2 Soap Films
162
A.3 Compressed Gas
164
A.4 Vacuum
165
A.5 Torque
165
A.6 The Equilibrium of a Rigid Body
166
A.7 Angular Momentum
167
A.8 The Center of Mass
169
A.9 The Moment of Inertia
170
A.
10
Current
172
A.
11
Voltage
172
A.
12 Kirchhoff s
Laws
173
A.
13
Resistance and Ohm s Law
174
A.
14
Resistors in Parallel
174
A.
15
Resistors in Series
175
A.
16
Power Dissipated in a Resistor
176
A.
17
Capacitors and Capacitance
176
A.
18
The Inductance: Inertia of the Current
177
A.
19
An Electrical-Plumbing Analogy
179
A.20 Problems
181
Bibliography
183
Index
185
|
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| dewey-ones | 510 - Mathematics |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035739502 |
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| indexdate | 2024-07-09T21:53:24Z |
| institution | BVB |
| isbn | 9780691140209 |
| language | English |
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| physical | VIII, 186 S. Ill., graph. Darst. |
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| spelling | Levi, Mark Verfasser aut The mathematical mechanic using physical reasoning to solve problems Mark Levi Princeton [u.a.] Princeton Univ. Press 2009 VIII, 186 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier In this delightful book, Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. Mathematische Physik Problem solving Mathematical physics Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Problemlösen (DE-588)4076358-4 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 s Problemlösen (DE-588)4076358-4 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018015944&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Levi, Mark The mathematical mechanic using physical reasoning to solve problems Mathematische Physik Problem solving Mathematical physics Mathematische Physik (DE-588)4037952-8 gnd Problemlösen (DE-588)4076358-4 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4076358-4 |
| title | The mathematical mechanic using physical reasoning to solve problems |
| title_auth | The mathematical mechanic using physical reasoning to solve problems |
| title_exact_search | The mathematical mechanic using physical reasoning to solve problems |
| title_full | The mathematical mechanic using physical reasoning to solve problems Mark Levi |
| title_fullStr | The mathematical mechanic using physical reasoning to solve problems Mark Levi |
| title_full_unstemmed | The mathematical mechanic using physical reasoning to solve problems Mark Levi |
| title_short | The mathematical mechanic |
| title_sort | the mathematical mechanic using physical reasoning to solve problems |
| title_sub | using physical reasoning to solve problems |
| topic | Mathematische Physik Problem solving Mathematical physics Mathematische Physik (DE-588)4037952-8 gnd Problemlösen (DE-588)4076358-4 gnd |
| topic_facet | Mathematische Physik Problem solving Mathematical physics Problemlösen |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018015944&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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