ModelCourses/Multivariate Calculus

Multivariate Calculus Model Course Units

• Mei Qin Chen, Dick Lane and John Travis
• Breaking "courses" first into units and finding appropriate content for them. Then, package these units as appropriate to fit various calculus breakdown models. However, it appears that most calculus courses cover similar topics in some order.
• Many software packages are available and can be used from within Webwork.
• Idea is to create a course table of content for each subject area and link problems to that table instead of particular textbooks. Then, develop textbook models that draw from those problems instead of having problems that draw from particular textbooks.
• A rubric needs to be developed that helps instructors determine the hardness level of a particular problem.

Typical Table of Contents

By this time in calculus, there is no difference between regular versus early transcendentals.

Vectors

Unit 1 - Model Course - Calculus - 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 - Model Course - Calculus - 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 - Model Course - Calculus - Non-rectangular coordinates

• Surfaces in Space

• Cylindrical Coordinates

• Spherical Coordinates

ModelCourses/Calculus/Vectors/setUnit3

Vector Functions

Unit 1 - Model Course - Calculus - Vector Functions

* Vector Functions and Space Curves
* Derivatives and Integrals of Vector Functions
* ModelCourses/Calculus/VectorFunctions/setUnit1

Unit 2 - Model Course - Calculus - 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 - Model Course - Calculus - Vector Function Applications

* Computing equation of osculating circle
* Motion in Space: Velocity and Acceleration
* ModelCourses/Calculus/VectorFunctions/setUnit3

Partial Derivatives

Unit 1 - Model Course - Calculus - Partial Derivatives - Definition

* Functions of Several Variables and Level Curves
* Limits and Continuity
* Partial Derivatives by Definition
* ModelUnits/Calculus/PartialDerivatives/Unit1

Unit 2 - Model Course - Calculus - Partial Derivatives - Rules

* Partial Derivatives using Rules
* The Chain Rule
* Directional Derivatives and the Gradient Vector
* ModelUnits/Calculus/PartialDerivatives/Unit2

Unit 3 - Model Course - Calculus - Partial Derivatives - Applications

* Tangent Planes and Linear and Other Approximations
* Maximum and Minimum Values
* Lagrange Multipliers
* ModelUnits/Calculus/PartialDerivatives/Unit3

Multiple Integrals

Unit 1 - Model Course - Calculus - Double Integrals Rectangular

* Iterated Integrals
* Setting up Double Integrals over General Regions
* Applications of Double Integrals in Rectangular Coordinates
* ModelUnits/Calculus/MultipleIntegrals/Unit1

Unit 2 - Model Course - Calculus - Double Integral Polar

* Double Integrals in Polar Coordinates
* Applications of Double Integrals in Polar Coordinates
* ModelUnits/Calculus/MultipleIntegrals/Unit2

Unit 3 - Model Course - Calculus - Triple Integrals

* Triple Integrals
* Triple Integrals in Cylindrical Coordinates
* Triple Integrals in Spherical Coordinates
* Change of Variables in Multiple Integrals
* Applications of Triple Integrals
* ModelUnits/Calculus/MultipleIntegrals/Unit3

Vector Calculus

Unit 1 - Model Course - Calculus - Vector Fields

* Vector Fields in 2D
* Vector Fields in 3D
* ModelUnits/Calculus/VectorCalculus/Unit1

Unit 2 - Model Course - Calculus - Line Integrals in 2D

* Line Integrals of a scalar function
* Line Integrals over a vector field
* The Fundamental Theorem for Line Integrals
* Green's Theorem
* ModelUnits/Calculus/VectorCalculus/Unit2

Unit 3 - Model Course - Calculus - 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

[Partial set of Course Templates]