Difference between revisions of "ModelCourses/Multivariate Calculus"
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** Orthogonality between three vectors |
** Orthogonality between three vectors |
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− | ModelCourses/Calculus/Vectors/setUnit1 |
+ | [[ModelCourses/Calculus/Vectors/setUnit1]] |
=== Unit 2 - Vector Applications === |
=== Unit 2 - Vector Applications === |
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** Finding intersection of surfaces in any given coordinate system |
** Finding intersection of surfaces in any given coordinate system |
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− | ModelCourses/Calculus/Vectors/setUnit3 |
+ | [[ModelCourses/Calculus/Vectors/setUnit3]] |
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+ | [[ModelCourses/Calculus/Vectors]] |
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== <span style="color:blue">Vector Functions<span> == |
== <span style="color:blue">Vector Functions<span> == |
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* Vector Functions and Space Curves |
* Vector Functions and Space Curves |
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* Derivatives and Integrals of Vector Functions |
* Derivatives and Integrals of Vector Functions |
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− | * ModelCourses/Calculus/VectorFunctions/setUnit1 |
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+ | [[ModelCourses/Calculus/VectorFunctions/setUnit1]] |
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** Computing N(t) |
** Computing N(t) |
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** Computing T(t) and N(t) and other stuff in one problem |
** Computing T(t) and N(t) and other stuff in one problem |
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− | * ModelCourses/Calculus/VectorFunctions/setUnit2 |
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+ | [[ModelCourses/Calculus/VectorFunctions/setUnit2]] |
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=== Unit 3 - Vector Function Applications === |
=== Unit 3 - Vector Function Applications === |
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* Computing equation of osculating circle |
* Computing equation of osculating circle |
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* Motion in Space: Velocity and Acceleration |
* Motion in Space: Velocity and Acceleration |
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− | * ModelCourses/Calculus/VectorFunctions/setUnit3 |
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+ | [[ModelCourses/Calculus/VectorFunctions/setUnit3]] |
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+ | [[ModelCourses/Calculus/VectorFunctions]] |
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== <span style="color:blue">Partial Derivatives</span> == |
== <span style="color:blue">Partial Derivatives</span> == |
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* Limits and Continuity |
* Limits and Continuity |
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* Partial Derivatives by Definition |
* Partial Derivatives by Definition |
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− | * ModelUnits/Calculus/PartialDerivatives/Unit1 |
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+ | [[ModelUnits/Calculus/PartialDerivatives/Unit1]] |
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=== Unit 2 - Partial Derivatives - Rules === |
=== Unit 2 - Partial Derivatives - Rules === |
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* The Chain Rule |
* The Chain Rule |
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* Directional Derivatives and the Gradient Vector |
* Directional Derivatives and the Gradient Vector |
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− | * ModelUnits/Calculus/PartialDerivatives/Unit2 |
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+ | [[ModelUnits/Calculus/PartialDerivatives/Unit2]] |
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* Maximum and Minimum Values |
* Maximum and Minimum Values |
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* Lagrange Multipliers |
* Lagrange Multipliers |
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− | * ModelUnits/Calculus/PartialDerivatives/Unit3 |
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+ | |||
+ | [[ModelUnits/Calculus/PartialDerivatives/Unit3]] |
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+ | [[ModelCourses/Calculus/PartialDerivatives]] |
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== <span style="color:blue">Multiple Integrals</span> == |
== <span style="color:blue">Multiple Integrals</span> == |
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** Total Mass, Centroid, Moments |
** Total Mass, Centroid, Moments |
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− | + | [[ModelUnits/Calculus/MultipleIntegrals/Unit1]] |
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=== Unit 2 - Double Integral Polar === |
=== Unit 2 - Double Integral Polar === |
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* Applications of Double Integrals in Polar Coordinates |
* Applications of Double Integrals in Polar Coordinates |
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− | + | [[ModelUnits/Calculus/MultipleIntegrals/Unit2]] |
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=== Unit 3 - Triple Integrals === |
=== Unit 3 - Triple Integrals === |
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** Total Mass, Centroid, Moments |
** Total Mass, Centroid, Moments |
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− | + | [[ModelUnits/Calculus/MultipleIntegrals/Unit3]] |
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+ | [[ModelCourses/Calculus/MultipleIntegrals]] |
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== <span style="color:blue">Vector Calculus</span> == |
== <span style="color:blue">Vector Calculus</span> == |
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** Basic Graphing tricks and software |
** Basic Graphing tricks and software |
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** Gradient vector fields and tests for conservative vector fields |
** Gradient vector fields and tests for conservative vector fields |
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− | * ModelUnits/Calculus/VectorCalculus/Unit1 |
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+ | |||
+ | [[ModelUnits/Calculus/VectorCalculus/Unit1]] |
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=== Unit 2 - Line Integrals in 2D === |
=== Unit 2 - Line Integrals in 2D === |
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** Applications in Physics |
** Applications in Physics |
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− | + | [[ModelUnits/Calculus/VectorCalculus/Unit2]] |
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=== Unit 3 - Line Integrals in 3D === |
=== Unit 3 - Line Integrals in 3D === |
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* Stokes' Theorem (often optional) |
* Stokes' Theorem (often optional) |
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* The Divergence Theorem (often optional) |
* The Divergence Theorem (often optional) |
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− | * ModelUnits/Calculus/VectorCalculus/Unit3 |
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+ | |||
+ | [[ModelUnits/Calculus/VectorCalculus/Unit3]] |
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+ | [[ModelCourses/Calculus/VectorCalculus]] |
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---- |
---- |
Revision as of 11:01, 26 June 2011
Contents
Multivariate Calculus Model Course Units
- Mei Qin Chen, Dick Lane and John Travis
- A user of this material should locate appropriate units below that fit their particular course in multivariate calculus.
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
- 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
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
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
``Future Work: A rubric needs to be developed that helps instructors determine the hardness level of a particular problem.``