Table Of ContentFeynman Simplified
4A: Math
for
Physicists
Everyone’s Guide
to the
Feynman Lectures on Physics
by
Robert L. Piccioni, Ph.D.
Copyright © 2016
by
Robert L. Piccioni
Published by
Real Science Publishing
3949 Freshwind Circle
Westlake Village, CA 91361, USA
Edited by Joan Piccioni
All rights reserved, including the right of
reproduction in whole or in part, in any form.
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www.guidetothecosmos.com
Everyone’s Guide to the
Feynman Lectures on Physics
Feynman Simplified gives mere mortals access to the fabled Feynman Lectures on Physics.
This Book
Feynman Simplified: 4A explores all the math needed to understand The Feynman Lectures on
Physics and much more.
The topics we explore include:
Trigonometric Functions & Identities
Rectilinear, Polar, Cylindrical & Spherical Coordinates
Real & Complex Numbers; Scientific Notation
Quadratic & Polynomial Equations & Solutions
Dimensional Analysis & Approximation Methods
Finite & Infinite Series
Zeno’s Paradox & Mortgage Payments
Exponentials, Logarithms & Hyperbolic Functions
Permutations, Combinations & Binomial Coefficients
Discrete & Continuous Probabilities
Poisson, Gaussian, and Chi-Squared Distributions
Rotation & Velocity Transformations
Vector Algebra, Identities & Theorems
Differential, Integral & Variational Calculus
Differential Equations
Tensors & Matrices
Numerical Integration & Data Fitting
Transforms & Fourier Series
Monte Carlo & Advanced Data Analysis
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Table of Contents
Chapter 1: Review of Basic Math
Geometry in 2 and 3 Dimensions
Functions
Basic Trig Functions
Inverse Trig Functions
Sine & Cosine Laws
Chapter 2: Coordinate Systems
Rectilinear Coordinates in 1-D, 2-D & 3-D
Polar Coordinates in 2-D & 3-D
Cylindrical Coordinates, 3-D
Spherical Coordinates, 3-D
4-D Spacetime
Chapter 3: Numbers
Integer, Rational & Irrational Numbers
Real, Imaginary & Complex Numbers
Scientific Notation
Chapter 4: Advanced Algebra
Absolute Value
Factorials
Polynomials & Quadratic Equations
Conditions for Circular Orbits
Chapter 5: Dimensional Analysis
Units of Measure
mks / SI System of Units
Matching Units
Restoring Missing Constants
Chapter 6: Infinite Series
Zeno’s Paradox
Infinite Series with Finite Sums
Finite Series, Bessel Functions
Monthly Loan Payments
Chapter 7: Exponentials
Exponential Grow & Decay
Definition of e & Natural Logarithm
Exponential & Trig Series
Hyperbolic sinh, cosh, tanh
Chapter 8: Approximation Techniques
Approximate or Perish
Taylor Series
Interpolation & Extrapolation
Functions at Extrema
Chapter 9: Probability & Statistics
Permutations & Combinations
Binomials Coefficients
Discrete & Continuous Probabilities
Poisson & Gaussian Distributions, Error Function
Combining Uncertainties
Chi-Square Analysis & Degrees of Freedom
Chapter 10: Rotation & Velocity Transformations
Rotations in 2-D & 3-D
Euler Angle Rotations
Relativistic Boosts
Rotating Quantum Spins
Chapter 11: Vector Algebra
Rotational Invariance
Dot & Cross Products
Right Hand Rule
Polar & Axial Vectors
Chapter 12: Differential Calculus
Need for Speed
Taking the Limit
Differentiation
Partial & Directional Derivatives
Higher Order Derivatives
Vector Operators in All Coordinate Systems
Chapter 13: Integral Calculus
It All Adds Up
Area Under The Curve
Definite & Indefinite Integrals
Integration by Parts
Chapter 14: More Calculus
Path & Loop Integrals
Area & Volume Integrals
Integrals in Non-Rectilinear Coordinates
Variational Calculus
Divide and Conquer
Chapter 15: Differential Equations
Linear Differential Equations
Quasi-Linear Equations
Dividing Coupled Differential Equations
Separation of Variables by Axes
Separation of Variables by Scale
Solutions of Laplace’s 2-D Equation
Chapter 16: Tensors & Matrices
Matrix Algebra
Determinants & Inverse Matrices
Eigenvalues & Eigenvectors
Matrix Characteristic Polynomials
Tensor Invariance, Tensor Calculus
Tensor Ranks & Indices
Covariant & Contravariant Indices
Cross Products as Tensors
Free Indices & Einstein Summation
Tensor Calculus in Curved Spacetime
Chapter 17: Numerical Integration
Summing Rectangles & Trapezoids
Romberg Integration: Get More for Less
Chapter 18: Data Fitting
Fitting Polynomials to Data
Comparing Data to Theory
Comparing Data to Data
Curve Fitting Cautions
Chapter 19: Transforms & Fourier Series
Fourier Series & Transforms
Fourier Transform of a Gaussian
Green’s Functions
Spherical Harmonics
Chapter 20: Advanced Data Analysis
Monte Carlo Methods & Example
Searching for Optima
Edge Degradation & Recovery
Ask: “How Could I Be Wrong?”
Appendix 1: Trigonometric Identities
Appendix 2: Sums of Common Series
Appendix 3: Tables of Gaussian Probability
Appendix 4: χ2 & Degrees of Freedom
Appendix 5: Vector Identities & Theorems
Appendix 6: Table of Common Derivatives
Appendix 7: Table of Common Integrals
Appendix 8: Principal Physical Constants
Chapter 1
Review of Basic Math
I presume you have previously studied geometry, trigonometry, and basic algebra. The first few
chapters of this eBook provide a quick review of that material and a definitions of key terms.
Trigonometry quantifies the geometric relationships among angles and distances, and is most often
employed in analyzing triangles.
Let’s begin by reviewing some basic shapes of Euclidean geometry and their key properties. Figure
1-1 shows five two-dimensional shapes.
In the upper left is a triangle, a figure bounded by three straight line segments; its three internal
angles sum to π radians (180 degrees), and its area equals hw/2, where h is its height, and w is its
width; h and w are mutually perpendicular.
In the upper right is a square, a figure bounded by four line segments of equal length, with four
internal angles that are each 90 degrees (π/2 radians); its area equals h2 (h=w).
Figure 1-1 Two-Dimensional Shapes
In the middle is a rectangle, a figure bounded by four line segments with opposite sides of equal
length, and four internal angles that are each 90 degrees; its area equals hw. (All squares are
rectangles, but not all rectangles are squares.)
In the lower left is a circle of radius R and diameter D=2R, which is the locus (collection) of all
points that are a distance R from the circle’s center. The circle’s circumference (length of its
perimeter) equals πD, and its area equals πR2. The length of the bolded arc that subtends angle θ
equals θR, when θ is measured in radians. This makes sense: for θ=2π, the arc becomes the circle’s
circumference whose length is 2πR. (This is why God invented radians.)
The area enclosed by a circle is called a disk.
Lastly, in the lower right is a parallelogram, a figure bounded by four line segments with opposite
sides of equal length, and opposite angles equal; its area equals hw, the product of its height and
width. (All rectangles are parallelograms, but not all parallelograms are rectangles.)
Moving to three dimensions, Figure 1-2 shows three common shapes.
In the upper right is a cuboid bounded by six rectangles, with all internal angles being equal. Its
height h, width w, and length L may be different. Its enclosed volume equals hwL, and its surface area
equals 2(hw+hL+wL). In the upper left is a cube, a cuboid in which h=w=L; its enclosed volume
equals h3 and its surface area equals 6h2, where h is the length of any side. (All cubes are cuboids, but
not all cuboids are cubes.)
Figure 1-2 Three-dimensional Shapes
Lastly, in the lower image, is a sphere, the locus of all points that are a distance R from the sphere’s
center. The sphere’s area equals 4πR2, and its enclosed volume equals 4πR3/3. Proper mathematical
terminology defines a sphere as the 2-D surface that encloses a 3-D volume called a ball.
We wish to examine some quantitative relationships established by trigonometry. But first, we must
discuss functions.
Functions
