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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

1.1 Parametric Equations

  • Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity.
  • It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve.
  • There is always more than one way to parameterize a curve.
  • Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates.

1.2 Calculus of Parametric Curves

  • The derivative of the parametrically defined curve x=x(t)x=x(t) and y=y(t)y=y(t) can be calculated using the formula dydx=y(t)x(t).dydx=y(t)x(t). Using the derivative, we can find the equation of a tangent line to a parametric curve.
  • The area between a parametric curve and the x-axis can be determined by using the formula A=t1t2y(t)x(t)dt.A=t1t2y(t)x(t)dt.
  • The arc length of a parametric curve can be calculated by using the formula s=t1t2(dxdt)2+(dydt)2dt.s=t1t2(dxdt)2+(dydt)2dt.
  • The surface area of a volume of revolution revolved around the x-axis is given by S=2πaby(t)(x(t))2+(y(t))2dt.S=2πaby(t)(x(t))2+(y(t))2dt. If the curve is revolved around the y-axis, then the formula is S=2πabx(t)(x(t))2+(y(t))2dt.S=2πabx(t)(x(t))2+(y(t))2dt.

1.3 Polar Coordinates

  • The polar coordinate system provides an alternative way to locate points in the plane.
  • Convert points between rectangular and polar coordinates using the formulas
    x=rcosθandy=rsinθx=rcosθandy=rsinθ

    and
    r=x2+y2andtanθ=yx.r=x2+y2andtanθ=yx.
  • To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.
  • Use the conversion formulas to convert equations between rectangular and polar coordinates.
  • Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.

1.4 Area and Arc Length in Polar Coordinates

  • The area of a region in polar coordinates defined by the equation r=f(θ)r=f(θ) with αθβαθβ is given by the integral A=12αβ[f(θ)]2dθ.A=12αβ[f(θ)]2dθ.
  • To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.
  • The arc length of a polar curve defined by the equation r=f(θ)r=f(θ) with αθβαθβ is given by the integral L=αβ[f(θ)]2+[f(θ)]2dθ=αβr2+(drdθ)2dθ.L=αβ[f(θ)]2+[f(θ)]2dθ=αβr2+(drdθ)2dθ.

1.5 Conic Sections

  • The equation of a vertical parabola in standard form with given focus and directrix is y=14p(xh)2+ky=14p(xh)2+k where p is the distance from the vertex to the focus and (h,k)(h,k) are the coordinates of the vertex.
  • The equation of a horizontal ellipse in standard form is (xh)2a2+(yk)2b2=1(xh)2a2+(yk)2b2=1 where the center has coordinates (h,k),(h,k), the major axis has length 2a, the minor axis has length 2b, and the coordinates of the foci are (h±c,k),(h±c,k), where c2=a2b2.c2=a2b2.
  • The equation of a horizontal hyperbola in standard form is (xh)2a2(yk)2b2=1(xh)2a2(yk)2b2=1 where the center has coordinates (h,k),(h,k), the vertices are located at (h±a,k),(h±a,k), and the coordinates of the foci are (h±c,k),(h±c,k), where c2=a2+b2.c2=a2+b2.
  • The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0.
  • The polar equation of a conic section with eccentricity e is r=ep1±ecosθr=ep1±ecosθ or r=ep1±esinθ,r=ep1±esinθ, where p represents the focal parameter.
  • To identify a conic generated by the equation Ax2+Bxy+Cy2+Dx+Ey+F=0,Ax2+Bxy+Cy2+Dx+Ey+F=0, first calculate the discriminant D=4ACB2.D=4ACB2. If D>0D>0 then the conic is an ellipse, if D=0D=0 then the conic is a parabola, and if D<0D<0 then the conic is a hyperbola.
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