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Calculus Volume 3

Key Concepts

Calculus Volume 3Key Concepts
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  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. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter 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. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter 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. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter 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. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter 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

5.1 Double Integrals over Rectangular Regions

  • We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. By taking the limit, this becomes a double integral representing the volume of the solid.
  • Properties of double integral are useful to simplify computation and find bounds on their values.
  • We can use Fubini’s theorem to write and evaluate a double integral as an iterated integral.
  • Double integrals are used to calculate the area of a region, the volume under a surface, and the average value of a function of two variables over a rectangular region.

5.2 Double Integrals over General Regions

  • A general bounded region DD on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.
  • To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.
  • We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.
  • We can use Fubini’s theorem for improper integrals to evaluate some types of improper integrals.

5.3 Double Integrals in Polar Coordinates

  • To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.
  • The area dAdA in polar coordinates becomes rdrdθ.rdrdθ.
  • Use x=rcosθ,x=rcosθ, y=rsinθ,y=rsinθ, and dA=rdrdθdA=rdrdθ to convert an integral in rectangular coordinates to an integral in polar coordinates.
  • Use r2=x2+y2r2=x2+y2 and θ=tan−1(yx)θ=tan−1(yx) to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.
  • To find the volume in polar coordinates bounded above by a surface z=f(r,θ)z=f(r,θ) over a region on the xyxy-plane, use a double integral in polar coordinates.

5.4 Triple Integrals

  • To compute a triple integral we use Fubini’s theorem, which states that if f(x,y,z)f(x,y,z) is continuous on a rectangular box B=[a,b]×[c,d]×[e,f],B=[a,b]×[c,d]×[e,f], then
    Bf(x,y,z)dV=efcdabf(x,y,z)dxdydzBf(x,y,z)dV=efcdabf(x,y,z)dxdydz

    and is also equal to any of the other five possible orderings for the iterated triple integral.
  • To compute the volume of a general solid bounded region EE we use the triple integral
    V(E)=E1dV.V(E)=E1dV.
  • Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.
  • To compute the average value of a function over a general three-dimensional region, we use
    fave=1V(E)Ef(x,y,z)dV.fave=1V(E)Ef(x,y,z)dV.

5.5 Triple Integrals in Cylindrical and Spherical Coordinates

  • To evaluate a triple integral in cylindrical coordinates, use the iterated integral
    θ=αθ=βr=g1(θ)r=g2(θ)z=u1(r,θ)z=u2(r,θ)f(r,θ,z)rdzdrdθ.θ=αθ=βr=g1(θ)r=g2(θ)z=u1(r,θ)z=u2(r,θ)f(r,θ,z)rdzdrdθ.
  • To evaluate a triple integral in spherical coordinates, use the iterated integral
    θ=αθ=βρ=g1(θ)ρ=g2(θ)φ=u1(r,θ)φ=u2(r,θ)f(ρ,θ,φ)ρ2sinφdφdρdθ.θ=αθ=βρ=g1(θ)ρ=g2(θ)φ=u1(r,θ)φ=u2(r,θ)f(ρ,θ,φ)ρ2sinφdφdρdθ.

5.6 Calculating Centers of Mass and Moments of Inertia

Finding the mass, center of mass, moments, and moments of inertia in double integrals:

  • For a lamina RR with a density function ρ(x,y)ρ(x,y) at any point (x,y)(x,y) in the plane, the mass is m=Rρ(x,y)dA.m=Rρ(x,y)dA.
  • The moments about the x-axisx-axis and y-axisy-axis are
    Mx=Ryρ(x,y)dAandMy=Rxρ(x,y)dA.Mx=Ryρ(x,y)dAandMy=Rxρ(x,y)dA.
  • The center of mass is given by x=Mym,y=Mxm.x=Mym,y=Mxm.
  • The center of mass becomes the centroid of the plane when the density is constant.
  • The moments of inertia about the xaxis,xaxis, yaxis,yaxis, and the origin are
    Ix=Ry2ρ(x,y)dA,Iy=Rx2ρ(x,y)dA,andI0=Ix+Iy=R(x2+y2)ρ(x,y)dA.Ix=Ry2ρ(x,y)dA,Iy=Rx2ρ(x,y)dA,andI0=Ix+Iy=R(x2+y2)ρ(x,y)dA.

Finding the mass, center of mass, moments, and moments of inertia in triple integrals:

  • For a solid object QQ with a density function ρ(x,y,z)ρ(x,y,z) at any point (x,y,z)(x,y,z) in space, the mass is m=Qρ(x,y,z)dV.m=Qρ(x,y,z)dV.
  • The moments about the xy-plane,xy-plane, the xz-plane,xz-plane, and the yz-planeyz-plane are
    Mxy=Qzρ(x,y,z)dV,Mxz=Qyρ(x,y,z)dV,Myz=Qxρ(x,y,z)dV.Mxy=Qzρ(x,y,z)dV,Mxz=Qyρ(x,y,z)dV,Myz=Qxρ(x,y,z)dV.
  • The center of mass is given by x=Myzm,y=Mxzm,z=Mxym.x=Myzm,y=Mxzm,z=Mxym.
  • The center of mass becomes the centroid of the solid when the density is constant.
  • The moments of inertia about the yz-plane,yz-plane, the xz-plane,xz-plane, and the xy-planexy-plane are
    Ix=Q(y2+z2)ρ(x,y,z)dV,Iy=Q(x2+z2)ρ(x,y,z)dV,Iz=Q(x2+y2)ρ(x,y,z)dV.Ix=Q(y2+z2)ρ(x,y,z)dV,Iy=Q(x2+z2)ρ(x,y,z)dV,Iz=Q(x2+y2)ρ(x,y,z)dV.

5.7 Change of Variables in Multiple Integrals

  • A transformation TT is a function that transforms a region GG in one plane (space) into a region RR in another plane (space) by a change of variables.
  • A transformation T:GRT:GR defined as T(u,v)=(x,y)T(u,v)=(x,y) (orT(u,v,w)=(x,y,z))(orT(u,v,w)=(x,y,z)) is said to be a one-to-one transformation if no two points map to the same image point.
  • If ff is continuous on R,R, then Rf(x,y)dA=Sf(g(u,v),h(u,v))|(x,y)(u,v)|dudv.Rf(x,y)dA=Sf(g(u,v),h(u,v))|(x,y)(u,v)|dudv.
  • If FF is continuous on R,R, then
    RF(x,y,z)dV=GF(g(u,v,w),h(u,v,w),k(u,v,w))|(x,y,z)(u,v,w)|dudvdw=GH(u,v,w)|J(u,v,w)|dudvdw.RF(x,y,z)dV=GF(g(u,v,w),h(u,v,w),k(u,v,w))|(x,y,z)(u,v,w)|dudvdw=GH(u,v,w)|J(u,v,w)|dudvdw.
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