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 and the Geometry of Space

• Space Coordinates and Vectors in Space
• The Dot Product of Two Vectors
• The Cross Product of Two Vectors in Space
• Lines and Planes in Space
• Section Project: Distances in Space
• Surfaces in Space
• Cylindrical and Spherical Coordinates

Vector Functions

• Unit 1 - Model Course - Calculus - Vector Functions

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

• 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

• Unit 3 - Model Course - Calculus - Vector Function Applications

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

Partial Derivatives

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

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

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

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

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

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

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

• Unit 2 - Model Course - Calculus - Double Integral Polar

* Double Integrals in Polar Coordinates
* Applications of Double Integrals in Polar Coordinates

• 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

Vector Calculus

• Unit 1 - Model Course - Calculus - Vector Fields

* Vector Fields in 2D
* Vector Fields in 3D

• Unit 2 - Model Course - Calculus - Line Integrals

* Line Integrals of a scalar function
* Line Integrals over a vector field
* The Fundamental Theorem for Line Integrals

• Unit 3 - Model Course - Calculus - Line Integral Applications

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

[Partial set of Course Templates]