ModelCourses/Multivariate Calculus

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Multivariate Calculus Model Course Units

A user of this material should locate appropriate units below that fit their particular course in multivariate calculus.

ModelCourses/Instructions for importing problem sets

Vectors

Unit 1 - Vectors

  • Vectors in Space
    • Space Coordinates
  • The Dot Product of Two Vectors
    • Calculations
    • Parallel and geometric implications
    • Angle between vectors, orthogonality and cos(theta)
  • The Cross Product of Two Vectors in Space
    • Calculations
    • Orthogonality between three vectors

ModelCourses/Calculus/Vectors/setUnit1

Unit 2 - Vector Applications

  • Projections
  • Lines and Planes in Space
    • Relationship to dot product and cross product (normal vector)
  • Distances in Space

ModelCourses/Calculus/Vectors/setUnit2

Unit 3 - Non-rectangular coordinates

  • Surfaces in Space
    • Graphing quadric surfaces
  • Cylindrical Coordinates
    • Conversions with rectangular
  • Spherical Coordinates
    • Conversions with rectangular
  • Applications
    • Conversions between rectangular, cylindrical and spherical
    • Finding intersection of surfaces in any given coordinate system

ModelCourses/Calculus/Vectors/setUnit3

ModelCourses/Calculus/Vectors

Vector Functions

Unit 1 - Vector Functions

  • Vector Functions and Space Curves
  • Derivatives and Integrals of Vector Functions

ModelCourses/Calculus/VectorFunctions/setUnit1

Unit 2 - Vector Function Properties

  • Arc Length
  • Curvature
  • Unit Tangent and Unit Normal vectors
    • Computing T(t)
    • Computing N(t)
    • Computing T(t) and N(t) and other stuff in one problem

ModelCourses/Calculus/VectorFunctions/setUnit2

Unit 3 - Vector Function Applications

  • Computing equation of osculating circle
  • Motion in Space: Velocity and Acceleration

ModelCourses/Calculus/VectorFunctions/setUnit3

ModelCourses/Calculus/VectorFunctions

Partial Derivatives

Unit 1 - Partial Derivatives - Definition

  • Functions of Several Variables and Level Curves
  • Limits and Continuity
  • Partial Derivatives by Definition

ModelUnits/Calculus/PartialDerivatives/Unit1

Unit 2 - Partial Derivatives - Rules

  • Partial Derivatives using Rules
  • The Chain Rule
  • Directional Derivatives and the Gradient Vector

ModelUnits/Calculus/PartialDerivatives/Unit2

Unit 3 - Partial Derivatives - Applications

  • Tangent Planes and Linear and Other Approximations
  • Maximum and Minimum Values
  • Lagrange Multipliers

ModelUnits/Calculus/PartialDerivatives/Unit3

ModelCourses/Calculus/PartialDerivatives

Multiple Integrals

Unit 1 - Double Integrals Rectangular

  • Iterated Integrals
    • Simple Calculations
    • Changing the order of integration
    • Simple area questions
  • Setting up Double Integrals over General Regions
    • Setup, given a set of inequalities
  • Applications of Double Integrals in Rectangular Coordinates
    • Volume
    • Total Mass, Centroid, Moments

ModelUnits/Calculus/MultipleIntegrals/Unit1

Unit 2 - Double Integral Polar

  • Double Integrals in Polar Coordinates
  • Applications of Double Integrals in Polar Coordinates

ModelUnits/Calculus/MultipleIntegrals/Unit2

Unit 3 - Triple Integrals

  • Triple Integrals
  • Triple Integrals in Cylindrical Coordinates
  • Triple Integrals in Spherical Coordinates
  • Change of Variables in Multiple Integrals
  • Applications of Triple Integrals
    • Volume
    • Total Mass, Centroid, Moments

ModelUnits/Calculus/MultipleIntegrals/Unit3

ModelCourses/Calculus/MultipleIntegrals

Vector Calculus

Unit 1 - Vector Fields

  • Vector Fields in 2D
    • Basic Graphing
    • Gradient vector fields and tests for conservative vector fields
  • Vector Fields in 3D
    • Basic Graphing tricks and software
    • Gradient vector fields and tests for conservative vector fields

ModelUnits/Calculus/VectorCalculus/Unit1

Unit 2 - Line Integrals in 2D

  • Line Integrals of a scalar function
    • Simple computations with respect to ds, dx, dy and dz
    • Application to Total Mass and Lateral Surface Area
  • Line Integrals over a vector field
    • Simple computations
    • Application to Work
  • The Fundamental Theorem of Calculus for Line Integrals
    • Relationship with conservative fields and independence of path.
  • Green's Theorem
    • Simple calculations
    • Changing orientations, holes
    • Applications in Physics

ModelUnits/Calculus/VectorCalculus/Unit2

Unit 3 - Line Integrals in 3D

  • Parametric Surfaces and Areas (sometimes optional due to time constraints)
  • Curl and Divergence (sometimes optional due to time constraints)
  • Surface Integrals (sometimes optional due to time constraints)
  • Stokes' Theorem (often optional)
  • The Divergence Theorem (often optional)

ModelUnits/Calculus/VectorCalculus/Unit3

ModelCourses/Calculus/VectorCalculus

Packaged Courses

Stewart

Stewart_packaged

Hughes-Hallett

Hughes_Hallett_packaged

Smith and Minton

Smith_Minton_packaged

Larson

Larson_packaged



``Future Work: A rubric needs to be developed that helps instructors determine the hardness level of a particular problem.``

  • Development Workgroup: Mei Qin Chen, Dick Lane and John Travis
  • To Do:
    • Finish choosing problem sets for remaining units
    • Add features to problems to include:
      • Hints
      • Solutions
      • MetaTags
      • Improvements such as changing multiple choice problems to fill in the blank, etc.


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