Skip to Content
OpenStax Logo
University Physics Volume 3

E | Mathematical Formulas

University Physics Volume 3E | Mathematical Formulas
Buy book
  1. Preface
  2. Unit 1. Optics
    1. 1 The Nature of Light
      1. Introduction
      2. 1.1 The Propagation of Light
      3. 1.2 The Law of Reflection
      4. 1.3 Refraction
      5. 1.4 Total Internal Reflection
      6. 1.5 Dispersion
      7. 1.6 Huygens’s Principle
      8. 1.7 Polarization
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 Geometric Optics and Image Formation
      1. Introduction
      2. 2.1 Images Formed by Plane Mirrors
      3. 2.2 Spherical Mirrors
      4. 2.3 Images Formed by Refraction
      5. 2.4 Thin Lenses
      6. 2.5 The Eye
      7. 2.6 The Camera
      8. 2.7 The Simple Magnifier
      9. 2.8 Microscopes and Telescopes
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 3 Interference
      1. Introduction
      2. 3.1 Young's Double-Slit Interference
      3. 3.2 Mathematics of Interference
      4. 3.3 Multiple-Slit Interference
      5. 3.4 Interference in Thin Films
      6. 3.5 The Michelson Interferometer
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 Diffraction
      1. Introduction
      2. 4.1 Single-Slit Diffraction
      3. 4.2 Intensity in Single-Slit Diffraction
      4. 4.3 Double-Slit Diffraction
      5. 4.4 Diffraction Gratings
      6. 4.5 Circular Apertures and Resolution
      7. 4.6 X-Ray Diffraction
      8. 4.7 Holography
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Modern Physics
    1. 5 Relativity
      1. Introduction
      2. 5.1 Invariance of Physical Laws
      3. 5.2 Relativity of Simultaneity
      4. 5.3 Time Dilation
      5. 5.4 Length Contraction
      6. 5.5 The Lorentz Transformation
      7. 5.6 Relativistic Velocity Transformation
      8. 5.7 Doppler Effect for Light
      9. 5.8 Relativistic Momentum
      10. 5.9 Relativistic Energy
      11. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Photons and Matter Waves
      1. Introduction
      2. 6.1 Blackbody Radiation
      3. 6.2 Photoelectric Effect
      4. 6.3 The Compton Effect
      5. 6.4 Bohr’s Model of the Hydrogen Atom
      6. 6.5 De Broglie’s Matter Waves
      7. 6.6 Wave-Particle Duality
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    3. 7 Quantum Mechanics
      1. Introduction
      2. 7.1 Wave Functions
      3. 7.2 The Heisenberg Uncertainty Principle
      4. 7.3 The Schrӧdinger Equation
      5. 7.4 The Quantum Particle in a Box
      6. 7.5 The Quantum Harmonic Oscillator
      7. 7.6 The Quantum Tunneling of Particles through Potential Barriers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Atomic Structure
      1. Introduction
      2. 8.1 The Hydrogen Atom
      3. 8.2 Orbital Magnetic Dipole Moment of the Electron
      4. 8.3 Electron Spin
      5. 8.4 The Exclusion Principle and the Periodic Table
      6. 8.5 Atomic Spectra and X-rays
      7. 8.6 Lasers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    5. 9 Condensed Matter Physics
      1. Introduction
      2. 9.1 Types of Molecular Bonds
      3. 9.2 Molecular Spectra
      4. 9.3 Bonding in Crystalline Solids
      5. 9.4 Free Electron Model of Metals
      6. 9.5 Band Theory of Solids
      7. 9.6 Semiconductors and Doping
      8. 9.7 Semiconductor Devices
      9. 9.8 Superconductivity
      10. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Nuclear Physics
      1. Introduction
      2. 10.1 Properties of Nuclei
      3. 10.2 Nuclear Binding Energy
      4. 10.3 Radioactive Decay
      5. 10.4 Nuclear Reactions
      6. 10.5 Fission
      7. 10.6 Nuclear Fusion
      8. 10.7 Medical Applications and Biological Effects of Nuclear Radiation
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Particle Physics and Cosmology
      1. Introduction
      2. 11.1 Introduction to Particle Physics
      3. 11.2 Particle Conservation Laws
      4. 11.3 Quarks
      5. 11.4 Particle Accelerators and Detectors
      6. 11.5 The Standard Model
      7. 11.6 The Big Bang
      8. 11.7 Evolution of the Early Universe
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  12. Index

Quadratic formula

If ax2+bx+c=0,ax2+bx+c=0, then x=b±b24ac2ax=b±b24ac2a

Triangle of base bb and height hh Area =12bh=12bh
Circle of radius rr Circumference =2πr=2πr Area =πr2=πr2
Sphere of radius rr Surface area =4πr2=4πr2 Volume =43πr3=43πr3
Cylinder of radius rr and height hh Area of curved surface =2πrh=2πrh Volume =πr2h=πr2h
Table E1 Geometry

Trigonometry

Trigonometric Identities

  1. sinθ=1/cscθsinθ=1/cscθ
  2. cosθ=1/secθcosθ=1/secθ
  3. tanθ=1/cotθtanθ=1/cotθ
  4. sin(900θ)=cosθsin(900θ)=cosθ
  5. cos(900θ)=sinθcos(900θ)=sinθ
  6. tan(900θ)=cotθtan(900θ)=cotθ
  7. sin2θ+cos2θ=1sin2θ+cos2θ=1
  8. sec2θtan2θ=1sec2θtan2θ=1
  9. tanθ=sinθ/cosθtanθ=sinθ/cosθ
  10. sin(α±β)=sinαcosβ±cosαsinβsin(α±β)=sinαcosβ±cosαsinβ
  11. cos(α±β)=cosαcosβsinαsinβcos(α±β)=cosαcosβsinαsinβ
  12. tan(α±β)=tanα±tanβ1tanαtanβtan(α±β)=tanα±tanβ1tanαtanβ
  13. sin2θ=2sinθcosθsin2θ=2sinθcosθ
  14. cos2θ=cos2θsin2θ=2cos2θ1=12sin2θcos2θ=cos2θsin2θ=2cos2θ1=12sin2θ
  15. sinα+sinβ=2sin12(α+β)cos12(αβ)sinα+sinβ=2sin12(α+β)cos12(αβ)
  16. cosα+cosβ=2cos12(α+β)cos12(αβ)cosα+cosβ=2cos12(α+β)cos12(αβ)

Triangles

  1. Law of sines: asinα=bsinβ=csinγasinα=bsinβ=csinγ
  2. Law of cosines: c2=a2+b22abcosγc2=a2+b22abcosγ
    Figure shows a triangle with three dissimilar sides labeled a, b and c. All three angles of the triangle are acute angles. The angle between b and c is alpha, the angle between a and c is beta and the angle between a and b is gamma.
  3. Pythagorean theorem: a2+b2=c2a2+b2=c2
    Figure shows a right triangle. Its three sides are labeled a, b and c with c being the hypotenuse. The angle between a and c is labeled theta.

Series expansions

  1. Binomial theorem: (a+b)n=an+nan1b+n(n1)an2b22!+n(n1)(n2)an3b33!+···(a+b)n=an+nan1b+n(n1)an2b22!+n(n1)(n2)an3b33!+···
  2. (1±x)n=1±nx1!+n(n1)x22!±···(x2<1)(1±x)n=1±nx1!+n(n1)x22!±···(x2<1)
  3. (1±x)n=1nx1!+n(n+1)x22!···(x2<1)(1±x)n=1nx1!+n(n+1)x22!···(x2<1)
  4. sinx=xx33!+x55!···sinx=xx33!+x55!···
  5. cosx=1x22!+x44!···cosx=1x22!+x44!···
  6. tanx=x+x33+2x515+···tanx=x+x33+2x515+···
  7. ex=1+x+x22!+···ex=1+x+x22!+···
  8. ln(1+x)=x12x2+13x3···(|x|<1)ln(1+x)=x12x2+13x3···(|x|<1)

Derivatives

  1. ddx[af(x)]=addxf(x)ddx[af(x)]=addxf(x)
  2. ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)ddx[f(x)+g(x)]=ddxf(x)+ddxg(x)
  3. ddx[f(x)g(x)]=f(x)ddxg(x)+g(x)ddxf(x)ddx[f(x)g(x)]=f(x)ddxg(x)+g(x)ddxf(x)
  4. ddxf(u)=[dduf(u)]dudxddxf(u)=[dduf(u)]dudx
  5. ddxxm=mxm1ddxxm=mxm1
  6. ddxsinx=cosxddxsinx=cosx
  7. ddxcosx=sinxddxcosx=sinx
  8. ddxtanx=sec2xddxtanx=sec2x
  9. ddxcotx=csc2xddxcotx=csc2x
  10. ddxsecx=tanxsecxddxsecx=tanxsecx
  11. ddxcscx=cotxcscxddxcscx=cotxcscx
  12. ddxex=exddxex=ex
  13. ddxlnx=1xddxlnx=1x
  14. ddxsin−1x=11x2ddxsin−1x=11x2
  15. ddxcos−1x=11x2ddxcos−1x=11x2
  16. ddxtan−1x=11+x2ddxtan−1x=11+x2

Integrals

  1. af(x)dx=af(x)dxaf(x)dx=af(x)dx
  2. [f(x)+g(x)]dx=f(x)dx+g(x)dx[f(x)+g(x)]dx=f(x)dx+g(x)dx
  3. xmdx=xm+1m+1(m1)=lnx(m=−1)xmdx=xm+1m+1(m1)=lnx(m=−1)
  4. sinxdx=cosxsinxdx=cosx
  5. cosxdx=sinxcosxdx=sinx
  6. tanxdx=ln|secx|tanxdx=ln|secx|
  7. sin2axdx=x2sin2ax4asin2axdx=x2sin2ax4a
  8. cos2axdx=x2+sin2ax4acos2axdx=x2+sin2ax4a
  9. sinaxcosaxdx=cos2ax4asinaxcosaxdx=cos2ax4a
  10. eaxdx=1aeaxeaxdx=1aeax
  11. xeaxdx=eaxa2(ax1)xeaxdx=eaxa2(ax1)
  12. lnaxdx=xlnaxxlnaxdx=xlnaxx
  13. dxa2+x2=1atan−1xadxa2+x2=1atan−1xa
  14. dxa2x2=12aln|x+axa|dxa2x2=12aln|x+axa|
  15. dxa2+x2=sinh−1xadxa2+x2=sinh−1xa
  16. dxa2x2=sin−1xadxa2x2=sin−1xa
  17. a2+x2dx=x2a2+x2+a22sinh−1xaa2+x2dx=x2a2+x2+a22sinh−1xa
  18. a2x2dx=x2a2x2+a22sin−1xaa2x2dx=x2a2x2+a22sin−1xa
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/university-physics-volume-3/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/university-physics-volume-3/pages/1-introduction
Citation information

© Sep 29, 2016 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.