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 - 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 - 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 - Non-rectangular coordinates

• Surfaces in Space

• Cylindrical Coordinates

• Spherical Coordinates

ModelCourses/Calculus/Vectors/setUnit3

Vector Functions

Unit 1 - 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

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

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

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

[Partial set of Course Templates]