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Calculus Volume 3

Key Equations

Calculus Volume 3Key Equations
  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Key Equations

Double integral Rf(x,y)dA=limm,ni=1mj=1nf(xij*,yij*)ΔARf(x,y)dA=limm,ni=1mj=1nf(xij*,yij*)ΔA
Iterated integral abcdf(x,y)dxdy=ab[cdf(x,y)dy]dxabcdf(x,y)dxdy=ab[cdf(x,y)dy]dx
or
cdbaf(x,y)dxdy=cd[abf(x,y)dx]dycdbaf(x,y)dxdy=cd[abf(x,y)dx]dy
Average value of a function of two variables fave=1AreaRRf(x,y)dxdyfave=1AreaRRf(x,y)dxdy
Iterated integral over a Type I region Df(x,y)dA=Df(x,y)dydx=ab[g1(x)g2(x)f(x,y)dy]dxDf(x,y)dA=Df(x,y)dydx=ab[g1(x)g2(x)f(x,y)dy]dx
Iterated integral over a Type II region Df(x,y)dA=Df(x,y)dxdy=cd[h1(y)h2(y)f(x,y)dx]dyDf(x,y)dA=Df(x,y)dxdy=cd[h1(y)h2(y)f(x,y)dx]dy
Double integral over a polar rectangular region RR Rf(r,θ)dA=limm,ni=1mj=1nf(rij*,θij*)ΔA=limm,ni=1mj=1nf(rij*,θij*)rij*ΔrΔθRf(r,θ)dA=limm,ni=1mj=1nf(rij*,θij*)ΔA=limm,ni=1mj=1nf(rij*,θij*)rij*ΔrΔθ
Double integral over a general polar region Df(r,θ)rdrdθ=θ=αθ=βr=h1(θ)r=h2(θ)f(r,θ)rdrdθDf(r,θ)rdrdθ=θ=αθ=βr=h1(θ)r=h2(θ)f(r,θ)rdrdθ
Triple integral liml,m,ni=1lj=1mk=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz=Bf(x,y,z)dVliml,m,ni=1lj=1mk=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz=Bf(x,y,z)dV
Triple integral in cylindrical coordinates Bg(x,y,z)dV=Bg(rcosθ,rsinθ,z)rdrdθdz=Bf(r,θ,z)rdrdθdzBg(x,y,z)dV=Bg(rcosθ,rsinθ,z)rdrdθdz=Bf(r,θ,z)rdrdθdz
Triple integral in spherical coordinates Bf(ρ,θ,φ)ρ2sinφdρdφdθ=φ=γφ=ψθ=αθ=βρ=aρ=bf(ρ,θ,φ)ρ2sinφdρdφdθBf(ρ,θ,φ)ρ2sinφdρdφdθ=φ=γφ=ψθ=αθ=βρ=aρ=bf(ρ,θ,φ)ρ2sinφdρdφdθ
Mass of a lamina m=limk,li=1kj=1lmij=limk,li=1kj=1lρ(xij*,yij*)ΔA=Rρ(x,y)dAm=limk,li=1kj=1lmij=limk,li=1kj=1lρ(xij*,yij*)ΔA=Rρ(x,y)dA
Moment about the x-axis Mx=limk,li=1kj=1l(yij*)mij=limk,li=1kj=1l(yij*)ρ(xij*,yij*)ΔA=Ryρ(x,y)dAMx=limk,li=1kj=1l(yij*)mij=limk,li=1kj=1l(yij*)ρ(xij*,yij*)ΔA=Ryρ(x,y)dA
Moment about the y-axis My=limk,li=1kj=1l(xij*)mij=limk,li=1kj=1l(xij*)ρ(xij*,yij*)ΔA=Rxρ(x,y)dAMy=limk,li=1kj=1l(xij*)mij=limk,li=1kj=1l(xij*)ρ(xij*,yij*)ΔA=Rxρ(x,y)dA
Center of mass of a lamina x=Mym=Rxρ(x,y)dARρ(x,y)dAx=Mym=Rxρ(x,y)dARρ(x,y)dA and y=Mxm=Ryρ(x,y)dARρ(x,y)dAy=Mxm=Ryρ(x,y)dARρ(x,y)dA
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