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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

Vertical trace f(a,y)=zf(a,y)=z for x=ax=a or f(x,b)=zf(x,b)=z for y=by=b
Level surface of a function of three variables f(x,y,z)=cf(x,y,z)=c
Partial derivative of ff with respect to xx fx=limh0f(x+h,y)f(x,y)hfx=limh0f(x+h,y)f(x,y)h
Partial derivative of ff with respect to yy fy=limk0f(x,y+k)f(x,y)kfy=limk0f(x,y+k)f(x,y)k
Tangent plane z=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)z=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)
Linear approximation L(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)L(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)
Total differential dz=fx(x0,y0)dx+fy(x0,y0)dy.dz=fx(x0,y0)dx+fy(x0,y0)dy.
Differentiability (two variables) f(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)+E(x,y),f(x,y)=f(x0,y0)+fx(x0,y0)(xx0)+fy(x0,y0)(yy0)+E(x,y),
where the error term EE satisfies
lim(x,y)(x0,y0)E(x,y)(xx0)2+(yy0)2=0.lim(x,y)(x0,y0)E(x,y)(xx0)2+(yy0)2=0.
Differentiability (three variables) f(x,y)=f(x0,y0,z0)+fx(x0,y0,z0)(xx0)+fy(x0,y0,z0)(yy0)+fz(x0,y0,z0)(zz0)+E(x,y,z),f(x,y)=f(x0,y0,z0)+fx(x0,y0,z0)(xx0)+fy(x0,y0,z0)(yy0)+fz(x0,y0,z0)(zz0)+E(x,y,z),
where the error term EE satisfies
lim(x,y,z)(x0,y0,z0)E(x,y,z)(xx0)2+(yy0)2+(zz0)2=0.lim(x,y,z)(x0,y0,z0)E(x,y,z)(xx0)2+(yy0)2+(zz0)2=0.
Chain rule, one independent variable dzdt=zx·dxdt+zy·dydtdzdt=zx·dxdt+zy·dydt
Chain rule, two independent variables dzdu=zxxu+zyxudzdu=zxxu+zyxu
dzdv=zxxv+zyyvdzdv=zxxv+zyyv
Generalized chain rule wtj=wx1x1tj+wx2x1tj++wxmxmtjwtj=wx1x1tj+wx2x1tj++wxmxmtj
directional derivative (two dimensions) Duf(a,b)=limh0f(a+hcosθ,b+hsinθ)f(a,b)hDuf(a,b)=limh0f(a+hcosθ,b+hsinθ)f(a,b)h
or
Duf(x,y)=fx(x,y)cosθ+fy(x,y)sinθDuf(x,y)=fx(x,y)cosθ+fy(x,y)sinθ
gradient (two dimensions) f(x,y)=fx(x,y)i+fy(x,y)jf(x,y)=fx(x,y)i+fy(x,y)j
gradient (three dimensions) f(x,y,z)=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)kf(x,y,z)=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k
directional derivative (three dimensions) Duf(x,y,z)=f(x,y,z)·u=fx(x,y,z)cosα+fy(x,y,z)cosβ+fx(x,y,z)cosγDuf(x,y,z)=f(x,y,z)·u=fx(x,y,z)cosα+fy(x,y,z)cosβ+fx(x,y,z)cosγ
Discriminant D=fxx(x0,y0)fyy(x0,y0)(fxy(x0,y0))2D=fxx(x0,y0)fyy(x0,y0)(fxy(x0,y0))2
Method of Lagrange multipliers, one constraint f(x0,y0)=λg(x0,y0)g(x0,y0)=0f(x0,y0)=λg(x0,y0)g(x0,y0)=0
Method of Lagrange multipliers, two constraints f(x0,y0,z0)=λ1g(x0,y0,z0)+λ2h(x0,y0,z0)g(x0,y0,z0)=0h(x0,y0,z0)=0f(x0,y0,z0)=λ1g(x0,y0,z0)+λ2h(x0,y0,z0)g(x0,y0,z0)=0h(x0,y0,z0)=0
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