ModelCourses/Multivariate Calculus
Contents
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
Vectors
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
- 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
- Surfaces in Space
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
Hughes-Hallett
Smith and Minton
Larson
``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.