Difference between revisions of "ModelCourses/Multivariate Calculus"
Jump to navigation
Jump to search
Line 129: | Line 129: | ||
=== Unit 2 - Line Integrals in 2D === |
=== 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 === |
=== Unit 3 - Line Integrals in 3D === |
Revision as of 10:41, 26 June 2011
Contents
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
- Setting up Double Integrals over General Regions
- 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