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Calculus Volume 3

Key Concepts

Calculus Volume 3Key Concepts
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Table of contents
  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 Concepts

3.1 Vector-Valued Functions and Space Curves

  • A vector-valued function is a function of the form r(t)=f(t)i+g(t)jr(t)=f(t)i+g(t)j or r(t)=f(t)i+g(t)j+h(t)k,r(t)=f(t)i+g(t)j+h(t)k, where the component functions f, g, and h are real-valued functions of the parameter t.
  • The graph of a vector-valued function of the form r(t)=f(t)i+g(t)jr(t)=f(t)i+g(t)j is called a plane curve. The graph of a vector-valued function of the form r(t)=f(t)i+g(t)j+h(t)kr(t)=f(t)i+g(t)j+h(t)k is called a space curve.
  • It is possible to represent an arbitrary plane curve by a vector-valued function.
  • To calculate the limit of a vector-valued function, calculate the limits of the component functions separately.

3.2 Calculus of Vector-Valued Functions

  • To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function.
  • Many of the properties of differentiation from the Introduction to Derivatives also apply to vector-valued functions.
  • The derivative of a vector-valued function r(t)r(t) is also a tangent vector to the curve. The unit tangent vector T(t)T(t) is calculated by dividing the derivative of a vector-valued function by its magnitude.
  • The antiderivative of a vector-valued function is found by finding the antiderivatives of the component functions, then putting them back together in a vector-valued function.
  • The definite integral of a vector-valued function is found by finding the definite integrals of the component functions, then putting them back together in a vector-valued function.

3.3 Arc Length and Curvature

  • The arc-length function for a vector-valued function is calculated using the integral formula s(t)=∫at‖r′(u)‖du.s(t)=∫at‖r′(u)‖du. This formula is valid in both two and three dimensions.
  • The curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that point. The arc-length parameterization is used in the definition of curvature.
  • There are several different formulas for curvature. The curvature of a circle is equal to the reciprocal of its radius.
  • The principal unit normal vector at t is defined to be
    N(t)=T′(t)‖T′(t)‖.N(t)=T′(t)‖T′(t)‖.
  • The binormal vector at t is defined as B(t)=T(t)×N(t),B(t)=T(t)×N(t), where T(t)T(t) is the unit tangent vector.
  • The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector.
  • The osculating circle is tangent to a curve at a point and has the same curvature as the tangent curve at that point.

3.4 Motion in Space

  • If r(t)r(t) represents the position of an object at time t, then r′(t)r′(t) represents the velocity and r″(t)r″(t) represents the acceleration of the object at time t. The magnitude of the velocity vector is speed.
  • The acceleration vector always points toward the concave side of the curve defined by r(t).r(t). The tangential and normal components of acceleration aTaT and aNaN are the projections of the acceleration vector onto the unit tangent and unit normal vectors to the curve.
  • Kepler’s three laws of planetary motion describe the motion of objects in orbit around the Sun. His third law can be modified to describe motion of objects in orbit around other celestial objects as well.
  • Newton was able to use his law of universal gravitation in conjunction with his second law of motion and calculus to prove Kepler’s three laws.
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