Appendix B

NotationThis appendix contains a summary of the notation used in this book.

Math notationExpression Read as Used to

a, b, x, y denote variables= is equal to indicate two expressions are equal in value≡ is defined as define a variable in term an expression

a + b a plus b combine lengthsa− b a minus b find the difference in length

a× b ≡ ab a times b find the area of a rectanglea/b ≡ a

b a divided by b find the area of a rectangle

an a exponent n denote a multiplied by itself n timesa2 ≡ aa a squared find the area of a square

a3 ≡ aaa a cubed find the volume of a cube√2 ≡ a

12 square root of a find the side of a square given the area

3√

a ≡ a13 cube root of a find the side of a cube given the volume

a−1 ≡ 1a one over a denotes division by a

f(x) f of x denote the output of the function f ap-plied to the input x

f−1 f inverse denote the inverse function of f(x)if f(x) = y, then f−1(y) = x

ex e to the x denote the exponential function base eln(x) natural log of x the logarithm base e

ax a to the x denote the exponential function base aloga(x) log base a of x the logarithm base a

θ, φ theta, phi denote anglessin, cos, tan sin, cos, tan obtain trigonometric ratios

% percent denote proportions of a total a% ≡ a100

364

Set notationYou don’t need a lot of fancy notation to understand mathematics.It really helps, though, if you know a little bit of set notation.

Symbol Read as Denotes

{ . . . } the set . . . define a sets| such that describe or restrict the elements of a set

N the naturals the set N ≡ {0, 1, 2, 3, . . .}Z the integers the set Z ≡ {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .}Q the rationals the set of fractions of integersA the set of algebraic numbersR the set of real numbersC the set of complex numbers

⊂ subset one set contained in another⊆ subset or equal containment or equality∪ union the combined the elements from two sets∩ intersection the elements two sets have in common

S \ T S set minus T the elements of S that are not in Ta ∈ S a in S a is an element of the set Sa /∈ S a not in S a is not an element of the set S∀x for all x a statement that holds for all x∃x there exists x an existence statement�x doesn’t exist x a non-existence statement

Vectors notationExpression Denotes

�v a vector(vx, vy) vector in component notation

vx ı̂ + vy ̂ vector in unit vector notation��v�∠θ vector in length-and-direction notation��v� length of the vector �v

θ angle the vector �v makes with the x-axisv̂ ≡ �v

��v� unit length vector in the same direction as �v

�u · �v dot product of the vectors �u and �v�u× �v cross product of the vectors �u and �v

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Mechanics notationExpression Denotes

�F a forcem mass of an object

a(t) acceleration of an object as a function of timev(t) velocity of an object as a function of timex(t) position of an object as a function of time

�N normal force�Ffs static force of friction�Ffk kinetic force of friction

�Fg ≡ �W gravitational force; the weight of an objectUg gravitational potential energy�Fs force of a springUs spring potential energy�p momentum of a moving object

K kinetic energyT torque

Iobj moment of inertia of an objectα(t) angular acceleration of an object as a function of timeω(t) angular velocity of an object as a function of timeθ(t) angular position of an object as a function of time

L angular momentum of a spinning objectKr rotational kinetic energy of a spinning object

Calculus notationExpression Denotes

∞ infinity�, δ the Greek letters epsilon and delta

f(x) a functionlim

x→∞f(x) the limit of f(x) as x goes to infinity

limx→a

f(x) the limit of f(x) as x goes to a

f �(x) the derivative of f(x)

f ��(x) the second derivative of f(x)d

dx the derivative operatorF (x) the antiderivative function of f(x)R

f(x) dx the indefinite integral of f(x)R b

af(x) dx the definite integral of f(x) between x = a and x = b

F (x)˛̨β

αthe change in F (x): F (x)

˛̨β

α= F (β)− F (α)

an the sequence an : N→ RP∞n=0 an the infinite series of the sequence an

366

No BULLSHIT guide to

MATH & PHYSICS

Ivan Savov

September 1, 2013

Copyright c� Ivan Savov, 2012, 2013.All rights reserved.Minireference Publishing. Fourth edition (v4.0), first printing.ISBN: 978-0-9920010-0-1The author can be reached at ivan.savov@gmail.com.hg changeset: 230:fd00faf6115f.

Contents

Preface vii

Introduction 1

1 Math fundamentals 31.1 Solving equations . . . . . . . . . . . . . . . . . . . . . 41.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Variables . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Functions and their inverses . . . . . . . . . . . . . . . 121.5 Basic rules of algebra . . . . . . . . . . . . . . . . . . . 151.6 Solving quadratic equations . . . . . . . . . . . . . . . 191.7 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . 231.8 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . 271.9 Fractions . . . . . . . . . . . . . . . . . . . . . . . . . 301.10 The number line . . . . . . . . . . . . . . . . . . . . . 331.11 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 351.12 The Cartesian plane . . . . . . . . . . . . . . . . . . . 371.13 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 401.14 Function reference . . . . . . . . . . . . . . . . . . . . 46

Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Square . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Square root . . . . . . . . . . . . . . . . . . . . . . . . 49Absolute value . . . . . . . . . . . . . . . . . . . . . . 50Polynomial functions . . . . . . . . . . . . . . . . . . . 51Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . 55Exponential . . . . . . . . . . . . . . . . . . . . . . . . 56Natural logarithm . . . . . . . . . . . . . . . . . . . . 57Function transformations . . . . . . . . . . . . . . . . 57General quadratic function . . . . . . . . . . . . . . . 60General sine function . . . . . . . . . . . . . . . . . . . 62

1.15 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 62

i

CONTENTS

1.16 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . 661.17 Trigonometric identities . . . . . . . . . . . . . . . . . 711.18 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 731.19 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . 741.20 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . 771.21 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . 811.22 Solving systems of linear equations . . . . . . . . . . . 851.23 Compound interest . . . . . . . . . . . . . . . . . . . . 871.24 Set notation . . . . . . . . . . . . . . . . . . . . . . . . 901.25 Math exercises . . . . . . . . . . . . . . . . . . . . . . 96

2 Introduction to physics 972.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 972.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 1002.3 Introduction to calculus . . . . . . . . . . . . . . . . . 1062.4 Kinematics with calculus . . . . . . . . . . . . . . . . . 110

3 Vectors 1153.1 Great outdoors . . . . . . . . . . . . . . . . . . . . . . 1163.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.3 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.4 Vector products . . . . . . . . . . . . . . . . . . . . . . 1273.5 Complex numbers . . . . . . . . . . . . . . . . . . . . 129

4 Mechanics 1354.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1354.2 Projectile motion . . . . . . . . . . . . . . . . . . . . . 1394.3 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1484.4 Force diagrams . . . . . . . . . . . . . . . . . . . . . . 1504.5 Momentum . . . . . . . . . . . . . . . . . . . . . . . . 1634.6 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.7 Uniform circular motion . . . . . . . . . . . . . . . . . 1774.8 Angular motion . . . . . . . . . . . . . . . . . . . . . . 1854.9 Simple harmonic motion . . . . . . . . . . . . . . . . . 1954.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 2074.11 Mechanics exercises . . . . . . . . . . . . . . . . . . . . 208

5 Calculus 2135.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2135.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 2155.3 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 2265.4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.5 Limit formulas . . . . . . . . . . . . . . . . . . . . . . 2375.6 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 2415.7 Derivative formulas . . . . . . . . . . . . . . . . . . . . 244

ii

CONTENTS

5.8 Derivative rules . . . . . . . . . . . . . . . . . . . . . . 2455.9 Higher derivatives . . . . . . . . . . . . . . . . . . . . 2505.10 Optimization algorithm . . . . . . . . . . . . . . . . . 2555.11 Implicit differentiation . . . . . . . . . . . . . . . . . . 2605.12 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 2645.13 Riemann sums . . . . . . . . . . . . . . . . . . . . . . 2745.14 The fundamental theorem of calculus . . . . . . . . . . 2805.15 Techniques of integration . . . . . . . . . . . . . . . . 2865.16 Applications of integration . . . . . . . . . . . . . . . . 3055.17 Improper integrals . . . . . . . . . . . . . . . . . . . . 3135.18 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 3145.19 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 3175.20 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 3285.21 Calculus exercises . . . . . . . . . . . . . . . . . . . . . 329

End matter 345Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 345Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . 345Further reading . . . . . . . . . . . . . . . . . . . . . . . . . 346

A Constants, units, and conversion ratios 361Fundamental constants of Nature . . . . . . . . . . . . . . . 361Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362Derived units . . . . . . . . . . . . . . . . . . . . . . . . . . 362Other units and conversions . . . . . . . . . . . . . . . . . . 363

B Notation 364Math notation . . . . . . . . . . . . . . . . . . . . . . . . . 364Set notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 365Vectors notation . . . . . . . . . . . . . . . . . . . . . . . . 365Mechanics notation . . . . . . . . . . . . . . . . . . . . . . . 366Calculus notation . . . . . . . . . . . . . . . . . . . . . . . . 366

C Formulas 367Calculus formulas . . . . . . . . . . . . . . . . . . . . . . . . 367Mechanics formulas . . . . . . . . . . . . . . . . . . . . . . . 370

iii

Placement exam

The answers1 to this placement exam will tell you where to start reading.

1. What is the derivative of sin(x)?

2. What is the second derivative of A sin(ωx)?

3. What is the value of x ?

4. What is the magnitude of the gravitational force between two planetsof mass M and mass m separated by a distance r?

5. Calculate limx→3−

1x− 3

.

6. Solve for t in:7(3 + 4t) = 11(6t− 4).

7. What is the component of the weight �W acting in the x direction?

8. A mass-spring system is undergoing simple harmonic motion. Its posi-tion function is x(t) = A sin(ωt). What is its maximum acceleration?

1Ans: 1. cos(x), 2.−Aω2 sin(ωx), 3.√

32 , 4. |�Fg| = GMm

r2 , 5.−∞, 6. 6538 , 7.+mg sin θ,

8. Aω2. Key: If you didn’t get Q3, Q6 right, you should read the book starting fromChapter 1. If you are mystified by Q1, Q2, Q5, read Chapter 5. If you want to learnhow to solve Q4, Q7 and Q8, read Chapter 4.

iv

Concept map

Figure 1: Each concept in this diagram corresponds to one section in the book.

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