ModelCourses/Multivariate Calculus

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

ModelCourses/Calculus/Vectors

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

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``Future Work: A rubric needs to be developed that helps instructors determine the hardness level of a particular problem.``

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