Chapters 14, 15, and 16.

Chapters 14, 15, and 16.的思维导图，介绍了Chapter 14 Partial Derivatives、Chapter 15 Multiple Integrals、Chapter 16 Integration in Vector Fields的内容。

Chapters 14, 15, and 16.

• Chapter 14 Partial Derivatives
  • 14.1 Fuctions of Several Variables
    • Bounded and Unbounded Regions in the Plane
      • A region in the plane is bounded if it lies inside a disk of fixed radius. A region is unbounded if it is not bounded.
    • Level Curve, Graph, Surface
      • The set of points in the plane where a function ƒ(x, y) has a constant value ƒ(x,y)=c is called a level curve of ƒ. The set of all points (x, y, ƒ(x, y)) in space, for (x, y) in the domain of ƒ, is called the graph of ƒ. The graph of ƒ is also called the surface z=f(x,y).
    • Level Surface
      • The set of points (x, y, z) in space where a function of three independent variables has a constant value f(x,y,z)=c is called a level surface of ƒ.
  • 14.2 Limits and Continuity in Higher Dimensions
    • Limit of a Function of Two Variables
    • Properties of Limits of Functions of Two Variables
      • 1. Sum Rule 2. Difference Rule 3. Product Rule 4. Constant Multiple Rule 5. Quotient Rule 6. Power Rule
    • Continuous Function of Two Variables
      • Definition:
        • A function ƒ(x, y) is continuous at the point (a, b) if:
      • Properties:
        • ƒ(a,b) exists
        • Limit of ƒ(x,y) as (x,y) approaches (a,b) exists
        • Limit of ƒ(x,y) as (x,y) approaches (a,b) is equal to ƒ(a,b)
  • 14.3 Partial Derivatives
    • Partial Derivative with Respect to x
    • The Mixed Derivative Theorem
    • Differentiability Implies Continuity
      • If a function ƒ(x, y) is differentiable at（xo,yo) then ƒ is continuous at (xo,yo).
  • 14.4 The Chain Rule
    • Main Fuctions
  • 14.5 Directional Derivatives and Gradient Vectors
    • Directional Derivative
    • Gradient Vector
    • The Dot Product of Directional Derivative
    • Algebra Rules for Gradients
      • Constant Multiple Rule
      • Sum Rule
      • Difference Rule
      • Product Rule
      • Quotient Rule
  • 14.6 Tangent Planes and Differentials
    • Tangent Plane
    • Normal Line
    • Linearization
  • 14.7 Extreme Values and Saddle Points
    • Points
      • Critical Point
      • Saddle Point
    • Local Maximum and Local Minimum
• Chapter 15 Multiple Integrals
  • 15.1 Double Integrals
    • Fubini’s Theorem
    • Properties of Double Integrals
  • 15.2 Areas,Moments,and Centers of Mass
    • Area
    • Average value of ƒ over R
    • Centers of Mass
    • Moments
  • 15.3 Double Integrals in Polar Form
    • Area in Polar Coordinates
  • 15.4 Triple Integrals in Rectangular Coordinates
    • Volume
  • 15.5 Masses and Moments in Three Dimensions
    • Mass and moment formulas for solid objects in space
  • 15.6 Triple Integrals in Cylindrical and Spherical Coordinates
    • Cylindrical Coordinates
    • Equations Relating Spherical Coordinates to Cartesian and Cylindrical Coordinates
• Chapter 16 Integration in Vector Fields
  • 16.1 Line Integrals
    • Mass and Moments
  • 16.2 Vector Fields,Work,Circulation,and Flux
    • Gradient Field
    • Six different ways to write the work integral
      • The definition
      • Compact differential form
      • Expanded to include dt; emphasizes the parameter t and velocity vector dr/dt
      • Emphasizes the component functions
      • Abbreviates the components of r
      • dt’s canceled; the most common form
    • Flux Across a Closed Curve in the Plane
    • Calculating Flux Across a Smooth Closed Plane Curve
  • 16.3 Path Independence,Potential Functions,and Conservative Fields
    • The Fundamental Theorem of Line Integrals
    • Component Test
      • Exactness of Mdx+Ndy+Pdz
      • Conservative Fields
  • 16.4 Green's Theorem in the Plane
    • Divergence (Flux Density)
    • k-Component of Curl (Circulation Density)
    • Green’s Theorem (Flux-Divergence or Normal Form)
    • Green’s Theorem (Circulation-Curl or Tangential Form)
  • 16.5 Surface Area and Surface Integrals
    • Formula for Surface Area
    • Surface Integral
    • Flux
  • 16.6 Parametrized Surfaces
    • Area of a Smooth Surface
    • Parametric Surface Integral
• Mass and moment formulas for very thin shells
• The partial derivative of ƒ(x, y) with respect to y at the point is similar