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Accelerated Reference Frames - Rindler Coordinates

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Introduction to Conformal Field Theory

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Polar Coordinates

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Polynomial Division

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Regular Polygons

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Sets, Groups, Modules, Rings and Vector Spaces

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Similar Matrices and Diagonalization

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Spherical Trigonometry

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Stirling's Approximation

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Sum and Differences of Squares and Cubes

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Symmetric Groups

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Tangent and Normal Line

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Taylor and Maclaurin Series

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The Essential Mathematics of Lie Groups

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The Integers Modulo n Under + and x

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The Limit Definition of the Exponential Function

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Tic-Tac-Toe Factoring

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Trapezoidal Rule

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Vector Calculus

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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Field Dimensions

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Scattering - Mandelstam Variables

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Spin 1 Eigenvectors

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Chebyshev's Theorem

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Factor Analysis

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Hypothesis Testing

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Pearson Correlation

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Permutations and Combinations

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Pooled Variance and Standard Error

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How this site works ...

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Quantum Field Theory

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Creation and Annihilation Operators

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Field Operators for Bosons and Fermions

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Lagrangians in Quantum Field Theory

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Path Integral Formulation

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Relativistic Quantum Field Theory

Quantum Mechanics

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Basic Relationships

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Clebsch-Gordan Coefficients

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Electron Orbital Angular Momentum and Spin

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Entangled States

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Heisenberg Uncertainty Principle

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Ladder Operators

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Multi Electron Wavefunctions

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Pauli Exclusion Principle

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Pauli Spin Matrices

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Photoelectric Effect

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Position and Momentum States

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Probability Current

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Schrodinger Equation for Hydrogen Atom

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Schrodinger Wave Equation

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Schrodinger Wave Equation (continued)

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Spin 1/2 Eigenvectors

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The Differential Operator

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The Essential Mathematics of Quantum Mechanics

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The Observer Effect

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The Schrodinger, Heisenberg and Dirac Pictures

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The WKB Approximation

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Time Dependent Perturbation Theory

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Time Evolution and Symmetry Operations

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Time Independent Perturbation Theory

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Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids

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Fermi-Dirac Statistics

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Intrinsic and Extrinsic Semiconductors

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The P-N Junction

Special Relativity

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Electromagnetic 4 - Potential

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Energy and Momentum, E = mc²

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Lorentz Invariance

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Lorentz Transform

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Lorentz Transformation of the EM Field

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Newton versus Einstein

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Spinors - Part 1

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Spinors - Part 2

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The Lorentz Group

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Velocity Addition

Statistical Mechanics

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Black Body Radiation

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Entropy and the Partition Function

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The Harmonic Oscillator

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String Theory

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Bosonic Strings

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Extra Dimensions

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Introduction to String Theory

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Kaluza-Klein Compactification of Closed Strings

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Strings in Curved Spacetime

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Toroidal Compactification

Superconductivity

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Introduction to Superconductors

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Superconductivity (Lectures 1 - 10)

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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields

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Generators of a Supergroup

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Grassmann Numbers

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Introduction to Supersymmetry

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The Gauge Hierarchy Problem

test

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The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)

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Gauge Theories (Yang-Mills)

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Gravitational Force and the Planck Scale

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Introduction to the Standard Model

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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge

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Quantum Flavordynamics and Quantum Chromodynamics

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Special Unitary Groups and the Standard Model - Part 1

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Special Unitary Groups and the Standard Model - Part 2

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Special Unitary Groups and the Standard Model - Part 3

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Standard Model Lagrangian

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The Higgs Mechanism

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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Last modified: January 26, 2018


Polar Coordinates
-----------------

                         P
                  ^ y   (x,y)           P = (r,θ) 
                  |    /                  ≡ (r,θ + 2nπ)
                  |   /                 x = rcosθ
                  |  /r                 y = rsinθ
                  | /
                  |/θ
    --------------+------------------>x
                 /|  
                / |
               /  |                    
              /   |
             Q    |   Q = (r,θ + π) 
          (-x,-y)       ≡ (-r,θ)
                        ≡ (-r,θ + (2n + 1)π)

Slope of Tangent Line at θ
---------------------------

        dy/dθ
dy/dx = -----  ... CHAIN RULE
        dx/dθ


Horizontal tangent line:  dy/dθ = 0

Vertical tangent line:  dx/dθ = 0

Example:

r = 2sinθ

x = rcosθ = 2sinθcosθ

y = rsinθ = 2sin²θ

dx/dθ = 2cos²θ - 2sin²θ = 0

∴ θ = π/4, 3π/4, 5π/4, 7π/4

dy/dθ = 4sinθcosθ = 0

∴ θ = 0, π/2, π, 3π/2

Not all of these are θs lines correspond to tangent 
lines.  To determine the valid ones it is necessary 
to look at the polar plot.



We can now see that:

π/4 and 3π/4 for vertical tangent lines.

0 and π/2 for horizontal tangent lines.

The r coordinate is found by solving the original polar
equation.  Therefore,

r = 2sin(π/4) = √3

r = 2sin(3π/4) = √3

r = 2sin(0) = 0

r = 2sin(π/2) = 2

Therefore, the polar coordinates of the tangent lines 
are:

(√3,π/4) (√3,3π/4) (0,0) (2,π/2)