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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

4.1 Functions of Several Variables

  • The graph of a function of two variables is a surface in 33 and can be studied using level curves and vertical traces.
  • A set of level curves is called a contour map.

4.2 Limits and Continuity

  • To study limits and continuity for functions of two variables, we use a δδ disk centered around a given point.
  • A function of several variables has a limit if for any point in a δδ ball centered at a point P,P, the value of the function at that point is arbitrarily close to a fixed value (the limit value).
  • The limit laws established for a function of one variable have natural extensions to functions of more than one variable.
  • A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.

4.3 Partial Derivatives

  • A partial derivative is a derivative involving a function of more than one independent variable.
  • To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.
  • Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.

4.4 Tangent Planes and Linear Approximations

  • The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.
  • Tangent planes can be used to approximate values of functions near known values.
  • A function is differentiable at a point if it is ”smooth” at that point (i.e., no corners or discontinuities exist at that point).
  • The total differential can be used to approximate the change in a function z=f(x0,y0)z=f(x0,y0) at the point (x0,y0)(x0,y0) for given values of ΔxΔx and Δy.Δy.

4.5 The Chain Rule

  • The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.
  • Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables.

4.6 Directional Derivatives and the Gradient

  • A directional derivative represents a rate of change of a function in any given direction.
  • The gradient can be used in a formula to calculate the directional derivative.
  • The gradient indicates the direction of greatest change of a function of more than one variable.

4.7 Maxima/Minima Problems

  • A critical point of the function f(x,y)f(x,y) is any point (x0,y0)(x0,y0) where either fx(x0,y0)=fy(x0,y0)=0,fx(x0,y0)=fy(x0,y0)=0, or at least one of fx(x0,y0)fx(x0,y0) and fy(x0,y0)fy(x0,y0) do not exist.
  • A saddle point is a point (x0,y0)(x0,y0) where fx(x0,y0)=fy(x0,y0)=0,fx(x0,y0)=fy(x0,y0)=0, but (x0,y0)(x0,y0) is neither a maximum nor a minimum at that point.
  • To find extrema of functions of two variables, first find the critical points, then calculate the discriminant and apply the second derivative test.

4.8 Lagrange Multipliers

  • An objective function combined with one or more constraints is an example of an optimization problem.
  • To solve optimization problems, we apply the method of Lagrange multipliers using a four-step problem-solving strategy.
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