ModelCourses/Multivariate Calculus

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

Set Definition Files for this collection

Unit 1 - Vectors

• Vectors in Space
  • Space Coordinates
  • The Dot Product of Two Vectors
  • The Cross Product of Two Vectors in Space

ModelCourses/Calculus/Vectors/setUnit1

Unit 2 - Vector Applications

• Vector Applications
  • Projections
  • Lines and Planes in Space
  • Distances in Space

ModelCourses/Calculus/Vectors/setUnit2

Unit 3 - Non-rectangular coordinates

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

ModelCourses/Calculus/Vectors/setUnit3

Vector Functions

Unit 1 - Vector Functions

• Calculus of 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

Moodle

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

Stewart

Stewart_packaged

Hughes-Hallett

Hughes_Hallett_packaged

Smith and Minton

Smith_Minton_packaged

Larson

Larson_packaged

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``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]