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.

Instructions for importing problem sets Instructions for exporting problem sets


Within each Unit below, specific problem types should be described. Complete problem sets for each unit will eventually be collected and made available from this site (and perhaps from within the WebWork system itself) but these have not been made available yet. Also, the specific problems suggested could be directly linked if desired although this might be a bit too much!


Vectors

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Unit 1 - Vectors

  • Vectors in Space
    • Space Coordinates
    • The Dot Product of Two Vectors
    • The Cross Product of Two Vectors in Space
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Unit 2 - Vector Applications

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Unit 3 - Non-rectangular coordinates

  • Coordinate Systems
    • Surfaces in Space
    • Cylindrical Coordinates
    • Spherical Coordinates
    • Applications
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Vector Functions

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Unit 1 - Vector Functions

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Unit 2 - Vector Function Properties

  • Properties
    • Arc Length
    • Curvature
    • Unit Tangent and Unit Normal vectors
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Unit 3 - Vector Function Applications

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

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Unit 1 - Partial Derivatives - Definition

  • Definition
    • Functions of Several Variables and Level Curves
    • Limits and Continuity
    • Partial Derivatives by Definition
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Unit 2 - Partial Derivatives - Rules

  • Rules
    • Partial Derivatives using Rules
    • The Chain Rule
    • Directional Derivatives and the Gradient Vector
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Unit 3 - Partial Derivatives - Applications

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

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Unit 1 - Double Integrals Rectangular

  • Rectangular Coordinates
    • Iterated Integrals
    • Setting up Double Integrals over General Regions
    • Applications of Double Integrals in Rectangular Coordinates
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Unit 2 - Double Integral Polar

  • Polar Coordinates
    • Double Integrals in Polar Coordinates
    • Applications of Double Integrals in Polar Coordinates
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Unit 3 - Triple Integrals

  • Triple Integrals
    • Triple Integrals
    • Triple Integrals in Cylindrical Coordinates
    • Triple Integrals in Spherical Coordinates
    • Change of Variables in Multiple Integrals
    • Applications of Triple Integrals
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Vector Calculus

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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

Packaged Courses

Moodle

https://test.webwork.maa.org/moodle/

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.


[Other Webwork Course Templates]