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Multivariable Calculus

Math Academy official course import: Multivariable Calculus

185 节课

课程大纲

  1. 1.Vector Functions and Vector Fields
    1. 1.1.1. Vector-Valued Functions
      1. 1
        The Domain of a Vector-Valued Function
      2. 2
        Limits of Vector-Valued Functions
      3. 3
        Continuity and Differentiability of Vector-Valued Functions
      4. 4
        Differentiation Rules for Vector-Valued Functions
      5. 5
        Integration Rules for Vector-Valued Functions
      6. 6
        The Arc Length of a Vector-Valued Function
      7. 7
        Tangent Vectors and Tangent Lines to Curves
      8. 8
        Unit Tangent Vectors
      9. 9
        Principal Normal Vectors
      10. 10
        Binormal Vectors
      11. 11
        The Osculating Plane
      12. 12
        Parameterization by Arc Length
    2. 2.1.2. Curvature
      1. 1
        Introduction to Curvature
      2. 2
        Finding Curvature Using the Cross Product
      3. 3
        Radius of Curvature
      4. 4
        The Curvature of a Plane Curve
    3. 3.1.3. Vector Fields
      1. 1
        Vector Fields
      2. 2
        Visualizing Vector Fields
      3. 3
        Gradient Vector Fields
      4. 4
        Conservative Vector Fields in the Cartesian Plane
      5. 5
        Calculating Potential Functions
      6. 6
        Connected and Simply-Connected Regions
      7. 7
        Stream Functions
    4. 4.1.4. Divergence and Curl
      1. 1
        The Divergence of a Vector Field
      2. 2
        Properties of the Divergence Operator
      3. 3
        The Curl of a Vector Field
      4. 4
        Properties of the Curl Operator
  2. 2.Multivariable Functions
    1. 1.2.1. Introduction to Multivariable Functions
      1. 1
        Interior and Boundary Points
      2. 2
        Interiors and Boundaries of Sets
      3. 3
        Open and Closed Sets
      4. 4
        The Domain of a Multivariable Function
      5. 5
        Level Curves
      6. 6
        Level Surfaces
      7. 7
        Limits and Continuity of Multivariable Functions
    2. 2.2.2. Quadric Surfaces and Cylinders
      1. 1
        Ellipsoids
      2. 2
        Hyperboloids
      3. 3
        Paraboloids
      4. 4
        Elliptic Cones
      5. 5
        Cylinders
      6. 6
        Intersections of Lines and Planes With Surfaces
      7. 7
        Identifying Quadric Surfaces
    3. 3.2.3. Partial Derivatives
      1. 1
        Introduction to Partial Derivatives
      2. 2
        Computing Partial Derivatives Using the Rules of Differentiation
      3. 3
        Geometric Interpretations of Partial Derivatives
      4. 4
        Partial Differentiability of Multivariable Functions
      5. 5
        Higher-Order Partial Derivatives
      6. 6
        Equality of Mixed Partial Derivatives
      7. 7
        Tangent Planes to Surfaces
      8. 8
        Linearization of Multivariable Functions
    4. 4.2.4. The Gradient Vector
      1. 1
        The Gradient Vector
      2. 2
        The Gradient as a Normal Vector
      3. 3
        Tangent Lines to Level Curves
      4. 4
        Tangent Planes to Level Surfaces
      5. 5
        Directional Derivatives
      6. 6
        The Multivariable Mean-Value Theorem
    5. 5.2.5. The Multivariable Chain Rule
      1. 1
        The Multivariable Chain Rule
      2. 2
        The Multivariable Chain Rule With Polar Coordinates
      3. 3
        The Multivariable Chain Rule in Vector Form
      4. 4
        Differentials
    6. 6.2.6. Plane Transformations
      1. 1
        Affine Transformations
      2. 2
        The Image of an Affine Transformation
      3. 3
        The Inverse of an Affine Transformation
      4. 4
        Nonlinear Transformations of Plane Regions
      5. 5
        Polar Coordinate Transformations
      6. 6
        Transformations of Regions Between Curves
    7. 7.2.7. Differentiation
      1. 1
        The Jacobian
      2. 2
        The Inverse Function Theorem
      3. 3
        The Jacobian of a Three-Dimensional Transformation
      4. 4
        The Derivative of a Multivariable Function
      5. 5
        The Second Derivative of a Multivariable Function
      6. 6
        Second-Degree Taylor Polynomials of Multivariable Functions
    8. 8.2.8. Optimization
      1. 1
        Global vs. Local Extrema and Critical Points of Multivariable Functions
      2. 2
        The Second Partial Derivatives Test
      3. 3
        Calculating Global Extrema of Multivariable Functions
      4. 4
        Lagrange Multipliers With One Constraint
      5. 5
        Lagrange Multipliers With Multiple Constraints
      6. 6
        Optimizing Multivariable Functions Using Lagrange Multipliers
  3. 3.Multiple Integrals
    1. 1.3.1. Riemann Sums
      1. 1
        Double Summations
      2. 2
        Partitions of Intervals
      3. 3
        Calculating Double Summations Over Partitions
      4. 4
        Approximating Volumes Using Lower Riemann Sums
      5. 5
        Approximating Volumes Using Upper Riemann Sums
      6. 6
        Lower Riemann Sums Over General Rectangular Partitions
      7. 7
        Upper Riemann Sums Over General Rectangular Partitions
      8. 8
        Defining Double Integrals Using Lower and Upper Riemann Sums
    2. 2.3.2. Double Integrals
      1. 1
        Double Integrals Over Rectangular Domains
      2. 2
        Double Integrals Over Non-Rectangular Domains
      3. 3
        Properties of Double Integrals
      4. 4
        Type I and II Regions in Two-Dimensional Space
      5. 5
        Double Integrals Over Type I Regions
      6. 6
        Double Integrals Over Type II Regions
      7. 7
        Double Integrals Over Partitioned Regions
      8. 8
        Changing the Order of Integration in Double Integrals
    3. 3.3.3. Triple Integrals
      1. 1
        Repeated Integrals in Three Dimensions
      2. 2
        Triple Integrals Over Rectangular Domains
      3. 3
        Type I, II, and III Regions in Three-Dimensional Space
      4. 4
        Triple Integrals Over Type I Regions
      5. 5
        Triple Integrals Over Type II Regions
      6. 6
        Triple Integrals Over Type III Regions
      7. 7
        Calculating Volumes of Solids Using Triple Integrals
      8. 8
        Changing the Order of Integration in Triple Integrals: Changing Projection
      9. 9
        Changing the Order of Integration in Triple Integrals: Changing Region
    4. 4.3.4. Change of Variables for Double Integrals
      1. 1
        Double Integrals in Plane Polar Coordinates
      2. 2
        Double Integrals Between Polar Curves
      3. 3
        Computing Areas Using a Change of Variables
      4. 4
        Computing Double Integrals Using a Change of Variables
    5. 5.3.5. Change of Variables for Triple Integrals
      1. 1
        Cylindrical Polar Coordinates
      2. 2
        Surfaces in Cylindrical Polar Coordinates
      3. 3
        Spherical Polar Coordinates
      4. 4
        Surfaces in Spherical Polar Coordinates
      5. 5
        Triple Integrals in Cylindrical Polar Coordinates
      6. 6
        Triple Integrals in Spherical Polar Coordinates
      7. 7
        Computing Triple Integrals Using a Change of Variables
  4. 4.Line Integrals
    1. 1.4.1. Line Integrals of Scalar Functions
      1. 1
        Simple, Closed, and Oriented Curves
      2. 2
        Line Integrals of Scalar Functions
      3. 3
        Properties of Line Integrals of Scalar Functions
      4. 4
        Line Integrals of Scalar Functions Over Paths Expressed as Functions of X
      5. 5
        Line Integrals of Scalar Functions Over Paths Expressed as Functions of Y
    2. 2.4.2. Line Integrals of Scalar Functions Over Parametric Curves
      1. 1
        Line Integrals of Scalar Functions Over Line Segments
      2. 2
        Line Integrals of Scalar Functions Over Circles
      3. 3
        Line Integrals of Scalar Functions Over Ellipses
      4. 4
        Further Properties of Line Integrals of Scalar Functions
      5. 5
        Line Integrals of Scalar Functions Over Polar Curves
    3. 3.4.3. Line Integrals With Respect to X and Y
      1. 1
        Line Integrals With Respect to X and Y
      2. 2
        Properties of Line Integrals With Respect to X and Y
      3. 3
        Sums of Line Integrals With Respect to X and Y Over Parametric Curves
      4. 4
        Sums of Line Integrals With Respect to X and Y
    4. 4.4.4. Line Integrals of Vector-Valued Functions
      1. 1
        Line Integrals of Vector-Valued Functions Over Parametric Curves
      2. 2
        Line Integrals of Vector-Valued Functions Over General Curves
      3. 3
        Interpreting Line Integrals of Vector-Valued Functions
      4. 4
        Properties of Line Integrals of Vector-Valued Functions
      5. 5
        The Fundamental Theorem for Line Integrals
      6. 6
        Path Independence of Line Integrals
    5. 5.4.5. Circulation and Flux
      1. 1
        Outward-Pointing Unit Normal Vectors in 2D
      2. 2
        Circulation
      3. 3
        Flux in Two-Dimensional Vector Fields
      4. 4
        Calculating Flux in Two-Dimensional Vector Fields
      5. 5
        Source-Free Vector Fields
    6. 6.4.6. Green's Theorem
      1. 1
        Introduction to Green's Theorem
      2. 2
        Green's Theorem in Polar Coordinates
      3. 3
        Using Green's Theorem to Calculate Area
      4. 4
        Extending Green's Theorem
      5. 5
        Green's Theorem in Flux Form
  5. 5.Surface Integrals
    1. 1.5.1. Parametric Surfaces
      1. 1
        The Hyperbolic Functions
      2. 2
        Parametric Surfaces
      3. 3
        Tangent Planes to Parametric Surfaces
      4. 4
        Parametrizations of Ellipsoids and Cones
      5. 5
        Parametrizations of Paraboloids and Hyperboloids
      6. 6
        Parametrizations of Cylinders
    2. 2.5.2. Surface Area
      1. 1
        Surface Areas of Revolution: Rotation About the X-Axis
      2. 2
        Surface Areas of Revolution: Rotation About the Y-Axis
      3. 3
        Surface Areas of Revolution for Parametric Curves
      4. 4
        Areas of Parametric Surfaces
      5. 5
        Surfaces of Revolution
      6. 6
        Areas of Surfaces
    3. 3.5.3. Surface Integrals
      1. 1
        Surface Integrals Over Parametric Surfaces
      2. 2
        Surface Integrals Over Cartesian Surfaces
      3. 3
        Flux in Three-Dimensional Vector Fields
      4. 4
        Flux Through Closed Surfaces
      5. 5
        Calculating Flux Through Parametric Surfaces
      6. 6
        Calculating Flux Through Cartesian Surfaces
      7. 7
        Calculating Flux Through Closed Surfaces
      8. 8
        The Divergence Theorem
      9. 9
        Stokes' Theorem
  6. 6.Applications of Multivariable Calculus
    1. 1.6.1. Vector Mechanics
      1. 1
        Velocity and Acceleration as Functions of Displacement
      2. 2
        Determining Properties of Objects Described as Functions of Displacement
      3. 3
        The Components of Acceleration
      4. 4
        Newton's Second Law
      5. 5
        Applying Newton's Second Law in the Plane
      6. 6
        The Work-Energy Principle
      7. 7
        Circular Motion About the Origin
    2. 2.6.2. Applications of Multiple Integrals
      1. 1
        The Average Value of a Multivariable Function
      2. 2
        Density, Mass, and Charge of Plane Laminas
      3. 3
        Moments and Center of Mass
      4. 4
        Moments and Centers of Mass of Thin Rods
      5. 5
        Moments and Centers of Mass of Plane Laminas
      6. 6
        Moments of Inertia of Laminas About the Coordinate Axes
      7. 7
        Moments of Inertia of Laminas About Other Axes
      8. 8
        Calculating the Radius of Gyration of a Plane Lamina
      9. 9
        The Parallel Axis Theorem
    3. 3.6.3. Random Variables
      1. 1
        Probability Density Functions of Continuous Random Variables
      2. 2
        Calculating Probabilities With Continuous Random Variables
      3. 3
        Continuous Random Variables Over Infinite Domains
      4. 4
        Joint Distributions for Discrete Random Variables
      5. 5
        Joint Distributions for Continuous Random Variables