Difference between revisions of "ModelCourses/Multivariate Calculus"

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= Multivariate Calculus Model Course Units =
  
= Multivariate Calculus Model Course Units =
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* Mei Qin Chen, Dick Lane and John Travis
  
  +
<span style="italics">A user of this material should locate appropriate units below that fit their particular course in multivariate calculus.</span>
−
* 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''
  
  +
[[ModelCourses/Instructions for importing problem sets|Instructions for importing problem sets]]
  +
[[ModelCourses/Instructions for exporting problem sets|Instructions for exporting problem sets]]
   
−
By this time in calculus, there is no difference between regular versus early transcendentals.
  
  +
Within each Unit below, specific problem types should be described. Detailed problem descriptions are given by clicking on the unit title.
   
−
== Vectors ==
  
  +
Complete problem sets for each unit will eventually be collected and made available from this site (and perhaps from within the WebWork system itself) but these have not been made available yet. Also, the specific problems suggested could be directly linked if desired although this might be a bit too much!
−
=== Unit 1 - Vectors ===
  
−
* Vectors in Space
  
−
** Space Coordinates
  
   
−
* The Dot Product of Two Vectors
  
  +
== [[File:setCourse_model_multivariate_calculus.def|Download this entire course of problem sets]] ==
−
** Calculations
  
−
** Parallel and geometric implications
  
−
** Angle between vectors, orthogonality and cos(theta)
  
   
−
* The Cross Product of Two Vectors in Space
  
  +
== <span style="color:blue">Vectors in Space<span> ==
−
** Calculations
  
  +
[[File:setTopic_Vectors_model_multivariate_calculus.def.txt|Download this entire collection of problem sets]]
−
** Orthogonality between three vectors
  
  +
=== Unit 1 - [[ModelCourses/Calculus/Vectors/Vectors in Space|Vector Algebra]] ===
  +
* Vector Algebra
  +
* Dot Product and Applications
  +
* Cross Product and Applications
   
−
ModelCourses/Calculus/Vectors/setUnit1
 
  +
: [[File:setTopic01_Vectors_model_multivariate_calculus.def|Download Set Definition File]]
   
−
=== Unit 2 - Vector Applications ===
  
  +
=== Unit 2 - [[ModelCourses/Calculus/Vectors/Vectors in Space|Lines and Planes]] ===
−
* Projections
  
  +
* Lines in Space
−
**
  
  +
** Two forms of a line in space: (1) parametric equations and symmetric equations
  +
*** Express a line that passes two given points.
  +
*** Express a line that passes through a given point and is parallel to a given line.
  +
*** Express a line that passes through a given point and is perpendicular to a given line.
  +
*** Express a line that passes through a given point and is perpendicular to two given lines that are not parallel.
  +
*** Determine if two given lines are parallel, perpendicular or neither.
  +
*** Determine if two given lines intersect or are skew, and find the intersection if they intersect.
  +
*** Determine if a given line passes through a given point.
   
−
* Lines and Planes in Space
 +
* Planes in Space
−
** Relationship to dot product and cross product (normal vector)
 +
** Plane formula
  +
*** Determine if a given plane contains a given point.
  +
*** Express a plane that contains a given point and is perpendicular to a given vector.
  +
*** Express a plane that contains a given point and is parallel to a given plane.
  +
*** Express a plane that contains a given point and is perpendicular to a given plane.
  +
*** Express a plane that contains a given point and is perpendicular to given two planes that are not parallel.
  +
*** Determine if two given planes are parallel, perpendicular or neither.
  +
*** Find the intersection of two planes that are not parallel.
   
  +
:[[File:setTopic02_Vectors_model_multivariate_calculus.def|Download Set Definition File]]
   
−
* Distances in Space
  
  +
=== Unit 3 - [[ModelCourses/Calculus/Vectors/Vectors in Space|Cylindrical Surfaces and Quadric surfaces]] ===
   
−
ModelCourses/Calculus/Vectors/setUnit2
 
  +
* Cylindrical Surfaces
  +
*** Sketch the curve f(x,y)=0, z=z0 (or g(x,z)=0, y=y0 or h(y,z), x=x0) in the xyz-space.
  +
*** Sketch the surface f(x,y)=0 (or g(x,z)=0, or h(y,z)=0) in the xyz-space.
   
−
=== Unit 3 - Non-rectangular coordinates ===
  
  +
* Quadric Surfaces
−
* Surfaces in Space
  
  +
** Quadratic equations for (1) sphere (2) ellipsoid (3) paraboloid (4) elliptic cone (5) hyperboloid of one sheet (6) hyperboloid of two sheet (7) hyperbolic paraboloid
   
−
* Cylindrical Coordinates
  
  +
:[[File:setTopic03_Vectors_model_multivariate_calculus.def|Download Set Definition File]]
   
  +
=== Unit 4 - [[ModelCourses/Calculus/Vectors/Coordinate Systems|Non-rectangular Coordinate Systems]] ===
  +
  +
* Cylindrical Coordinates
 
* Spherical Coordinates
  
* Spherical Coordinates
  +
* Applications
   
−
ModelCourses/Calculus/Vectors/setUnit3
  
  +
:[[File:setTopic04_Vectors_model_multivariate_calculus.def|Download Set Definition File]]
   
−
== Vector Functions ==
 +
== <span style="color:blue">Vector-valued Functions in One Variable<span> ==
  +
:[[File:setTopic_Vector_Functions_model_multivariate_calculus.def|Download this entire collection of problem sets]]
   
−
=== Unit 1 - Vector Functions ===
 +
=== Unit 1 - [[ModelCourses/Calculus/VectorFunctions/CalculusBasics|Calculus of Vector Functions]] ===
−
* Vector Functions and Space Curves
  
−
* Derivatives and Integrals of Vector Functions
  
−
* ModelCourses/Calculus/VectorFunctions/setUnit1
  
   
  +
* Calculus of Vector-valued Function in One Variable
  +
** Vector-value function r(t)
  +
*** Express the domain of r(t) a 2-D (or 3-D) vector-valued function in an interval notation.
  +
*** Determine if a given point is on the curve traced out by r(t).
  +
*** Sketch the curved traced out by r(t).
  +
*** Find all values of t at which r(t) is parallel to a given vector.
  +
*** Find all values of t at which r(t) is perpendicular to a given vector
  +
** Derivatives of Vector Functions
  +
*** Compute the tangent vector r'(t) of r(t).
  +
*** Find all values of t at which r(t) and r'(t) are perpendicular.
  +
*** Show that r(t) and r'(t) are perpendicular at every t if the magnitude of r(t) is a constant.
  +
** Integrals of Vector Functions
  +
*** Compute an indefinite integral of r(t).
  +
*** Compute a definite integral of r(t).
   
−
=== Unit 2 - Vector Function Properties ===
  
  +
:[[File:setTopic01_Vector_Functions_model_multivariate_calculus.def|Download Set Definition File]]
−
* 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 ===
 +
=== Unit 2 - [[ModelCourses/Calculus/VectorFunctions/Properties|Vector Function Applications ]] ===
−
* Computing equation of osculating circle
  
−
* Motion in Space: Velocity and Acceleration
  
−
* ModelCourses/Calculus/VectorFunctions/setUnit3
  
   
−
== Partial Derivatives ==
  
  +
* Arc Length
−
=== Unit 1 - Partial Derivatives - Definition ===
  
  +
* Curvature
−
* Functions of Several Variables and Level Curves
  
  +
* Unit Tangent and Unit Normal vectors
−
* Limits and Continuity
  
  +
* Computing the radius, center and equation of osculating circle
−
* Partial Derivatives by Definition
  
  +
* Motion in Space: Velocity and Acceleration
−
* ModelUnits/Calculus/PartialDerivatives/Unit1
  
   
−
=== Unit 2 - Partial Derivatives - Rules ===
  
  +
:[[File:setTopic02_Vector_Functions_model_multivariate_calculus.def|Download Set Definition File]]
−
* Partial Derivatives using Rules
  
−
* The Chain Rule
  
−
* Directional Derivatives and the Gradient Vector
  
−
* ModelUnits/Calculus/PartialDerivatives/Unit2
  
   
  +
== <span style="color:blue">Multi-variable Functions</span> ==
  +
:[[File:setTopic_Partial_Derivatives_model_multivariate_calculus.def|Download this entire collection of problem sets]]
  +
=== Unit 1 - [[ModelCourses/Calculus/PartialDerivatives/Definition|Multi-variable Functions ]]===
   
−
=== Unit 3 - Partial Derivatives - Applications ===
  
  +
* Domain and range
−
* Tangent Planes and Linear and Other Approximations
  
  +
** Find and sketch the domain of a given function.
−
* Maximum and Minimum Values
  
  +
** Evaluate a function at a given point.
−
* Lagrange Multipliers
  
  +
** Find the range of a given function.
−
* ModelUnits/Calculus/PartialDerivatives/Unit3
  
  +
** Sketch level curves (surfaces) of f(x,y) (f(x,y,z)).
  +
** Sketch the surface z=f(x,y) and traces f(x,y)=z0 on the surface.
  +
* Limits
  +
** Definition of limit
  +
*** Show the limit of a function does not exist by showing the function has two different limits along two distinct paths, respectively.
  +
*** Compute the limit or show limit does not exist using polar coordinates.
  +
*** Compute the limit or show limit does not exist using the definition of limit.
  +
* Continuity
  +
** Definition of continuity
  +
*** Determine if a function is continuous at a given point by checking if the limit is the same as the value of the function at this point.
  +
*** Find the region over which the function is continuous.
   
−
== Multiple Integrals ==
  
  +
:[[File:setTopic01_Partial_Derivatives_model_multivariate_calculus.def|Download Set Definition File]]
   
−
=== Unit 1 - Double Integrals Rectangular ===
  
  +
=== Unit 2 - [[ModelCourses/Calculus/PartialDerivatives/Rules|Partial Derivatives - Rules ]]===
−
* 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
  
  +
* Partial Derivatives
  +
** Definitions of Partial Derivatives
  +
*** Compute the first partial derivative of f with respect to x (y or z).
  +
*** Sketch the tangent line at a given point on the trace f(x,y)=z0.
  +
*** Compute higher order partial derivatives.
   
−
=== Unit 2 - Double Integral Polar ===
  
  +
* Partial Derivatives using Rules
−
* Double Integrals in Polar Coordinates
  
  +
** Differentiation Rule
−
* Applications of Double Integrals in Polar Coordinates
  
  +
** The Chain Rule
  +
** Implicit differentiation using partial derivatives
   
−
* ModelUnits/Calculus/MultipleIntegrals/Unit2
  
  +
* Directional Derivatives and the Gradient Vectors
  +
** Gradient Vectors
  +
*** Compute the gradient vector of f at a given point.
  +
*** Compute the gradient vector of f at any point in its domain.
  +
*** Find all points at which the gradient of f is a zero vector.
  +
*** Sketch the gradient vector of f(x,y) at (x0,y0) on the level curve f(x,y)=f(x0,y0).
  +
** Definition of Directional Derivative
  +
** Directional Derivative as the dot product of gradient vector and unit direction vector
  +
*** Compute the directional derivative of f at a point along a unit direction.
  +
*** Compute the directional derivative of f as (x,y) (or (x,y,z)) moves from point A to point B.
  +
*** Find the maximum (minimum) rate of change of f at a given point and the corresponding direction vector.
  +
*** Determine if f is increasing, decreasing or neither at a given point along a give direction.
  +
  +
:[[File:setTopic02_Partial_Derivatives_model_multivariate_calculus.def|Download Set Definition File]]
  +
  +
=== Unit 3 - [[ModelCourses/Calculus/PartialDerivatives/Applications|Partial Derivatives - Applications ]]===
  +
  +
* Tangent Planes and Linear and Quadratic Approximations
  +
** Tangent Planes and Linear Approximations
  +
*** Find the equation of the tangent plane of the surface z=f(x,y) at (x0,y0).
  +
*** Find all points (x,y) at which the tangent plane of z=f(x,y) at (x0,y0) is parallel to a given plane.
  +
*** Find the linear approximation L(x,y) of f(x,y) at (x0,y0).
  +
*** Approximate f(x1,y1) by L(x1,y1).
  +
** Quadratic Approximations
  +
*** Find the quadratic approximation Q(x,y) of f(x,y) at (x0,y0).
  +
*** Approximate f(x1,y1) by Q(x1,y1).
  +
* Maximum and Minimum Values
  +
** Critical Points
  +
*** Find all critical points of f(x,y) (or f(x,y,z)).
  +
** Second Derivative Test for Local Maximum and Minimum
  +
*** Find all local maximum and minimum of f(x,y) at the critical point (x0,y0) by the Second Derivative Test.
  +
** Absolute Maximum and Minimum
  +
*** Find the absolute maximum and minimum of f(x,y) over a region.
  +
* Lagrange Multipliers
  +
  +
  +
:[[File:setTopic03_Partial_Derivatives_model_multivariate_calculus.def|Download Set Definition File]]
  +
  +
== <span style="color:blue">Multiple Integrals</span> ==
  +
:[[File:setTopic_Multiple_Integration_model_multivariate_calculus.def|Download this entire collection of problem sets]]
  +
  +
=== Unit 1 - [[ModelCourses/Calculus/MultipleIntegrals/Rectangular|Double Integrals and Applications]]===
  +
  +
* Double Integrals
  +
** Double Integrals in the Rectangular Coordinates
  +
*** Set up and evaluate a double integral over a rectangular region.
  +
*** Set up and evaluate a double integral over a general region.
  +
*** Change the order of a given double integral.
  +
** Double Integrals in Polar Coordinates
  +
*** Set up and evaluate a double integral over a region in polar coordinates.
  +
*** Express a given double integral in rectangular coordinates in polar coordinates.
  +
* Applications of Double Integrals
  +
** Area of a region
  +
** Average value of f(x,y) over a region
  +
** Surface area of a solid
  +
** Volume of the solid under the surface z=f(x,y) over a region
  +
** Centroid of a region, center of mass
  +
  +
:[[File:setTopic01_Multiple_Integration_model_multivariate_calculus.def|Download Set Definition File]]
  +
  +
=== Unit 2 - [[ModelCourses/Calculus/MultipleIntegrals/TripleIntegrals|Triple Integrals and Applications]] ===
   
−
=== Unit 3 - Triple Integrals ===
  
 
* Triple Integrals
  
* Triple Integrals
−
* Triple Integrals in Cylindrical Coordinates
 +
** Triple Integrals in Rectangular Coordinates
−
* Triple Integrals in Spherical Coordinates
 +
** Triple Integrals in Cylindrical Coordinates
  +
** Triple Integrals in Spherical Coordinates
  +
*** Set up a triple integral for in the rectangular, cylindrical or spherical coordinates.
  +
*** Exchange the order of integration of a triple integral.
  +
*** Exchange the coordinate systems of a triple integral.
 
* Change of Variables in Multiple Integrals
  
* Change of Variables in Multiple Integrals
 
* Applications of Triple Integrals
  
* Applications of Triple Integrals
−
** Volume
 +
** Volume of a Solid
−
** Total Mass, Centroid, Moments
 +
** Average Value of f(x,y,z) over a Solid
  +
** Center of Mass
   
−
* ModelUnits/Calculus/MultipleIntegrals/Unit3
  
  +
:[[File:setTopic02_Multiple_Integration_model_multivariate_calculus.def|Download Set Definition File]]
   
−
== Vector Calculus ==
 +
== <span style="color:blue">Vector Calculus</span> ==
  +
:[[File:setTopic_Vector_Calculus_model_multivariate_calculus.def|Download this entire collection of problem sets]]
  +
=== Unit 1 - [[ModelCourses/Calculus/VectorCalculus/VectorFields|Vector Fields]] ===
   
−
=== Unit 1 - Vector Fields ===
 +
* Vector Fields
−
* Vector Fields in 2D
 +
** Vector Fields in 2D and 3D
−
** Basic Graphing
 +
** Gradient Vector Fields, and Conservative Vector Fields
−
** Gradient vector fields and tests for conservative vector fields
 +
*** For a given vector field, determine if it is conservative and find its potential function if it is conservative.
−
* Vector Fields in 3D
 +
*** Sketch a vector field at a given point.
−
** Basic Graphing tricks and software
 +
*** Sketch a trace of the flow of a velocity vector field.
−
** Gradient vector fields and tests for conservative vector fields
  
−
* ModelUnits/Calculus/VectorCalculus/Unit1
  
   
−
=== Unit 2 - Line Integrals in 2D ===
  
  +
:[[File:setTopic01_Vector_Calculus_model_multivariate_calculus.def|Download Set Definition File]]
  +
  +
=== Unit 2 - [[ModelCourses/Calculus/VectorCalculus/LineIntegrals|Line Integrals in 2D and Applications]] ===
  +
  +
* Parametric Equations for Curves (line segments, circle, ellipses)
 
* Line Integrals of a scalar function
  
* Line Integrals of a scalar function
−
** Simple computations with respect to ds, dx, dy and dz
  
  +
*** Set up a line integral of f(x,y) over a curve C.
−
** Application to Total Mass and Lateral Surface Area
  
  +
*** Compute the total mass of a wire with a given density function.
  +
* Line Integrals of a vector field
  +
*** Set up a line integral of vector field over a curve C.
  +
*** Compute the total work done by a given force vector field along a curve C.
  +
*** Compute the total circulation of a vector field along a closed curve C.
  +
* The Fundamental Theorem of Calculus for Line Integrals
  +
*** Use a potential function of a conservative vector field F to evaluate the line integral of F along a curve C.
  +
* Green's Theorem
  +
*** Use Green's Theorem to evaluate a line integral of a vector field over a closed curve when all required conditions are satisfied.
   
−
* Line Integrals over a vector field
  
  +
:[[File:setTopic02_Vector_Calculus_model_multivariate_calculus.def|Download Set Definition File]]
−
** Simple computations
  
−
** Application to Work
  
   
  +
=== Unit 3 - [[ModelCourses/Calculus/VectorCalculus/LineIntegrals3D|Line Integrals in 3D and Applications]] ===
   
−
* The Fundamental Theorem of Calculus for Line Integrals
  
  +
* Parametric Surfaces and Areas (sometimes optional due to time constraints)
−
** Relationship with conservative fields and independence of path.
  
  +
* 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)
   
−
* Green's Theorem
  
  +
:[[File:setTopic03_Vector_Calculus_model_multivariate_calculus.def|Download Set Definition File]]
−
** Simple calculations
  
  +
−
** Changing orientations, holes
  
  +
= Packaged Courses =
−
** Applications in Physics
  
  +
== Moodle ==
  +
https://test.webwork.maa.org/moodle/
  +
== Stewart ==
  +
[[Stewart_packaged]]
  +
== Hughes-Hallett ==
  +
[[Hughes_Hallett_packaged]]
  +
== Smith and Minton ==
  +
[[Smith_Minton_packaged]]
  +
== Larson ==
  +
[[Larson_packaged]]
  +
  +
= =
  +
=[[ModelCourses|Other Model Course Pages]]=
  +
  +
  +
----
  +
``Future Work: A rubric needs to be developed that helps instructors determine the hardness level of a particular problem.``
   
−
* ModelUnits/Calculus/VectorCalculus/Unit2
  
  +
* Development Workgroup: Mei Qin Chen, Dick Lane and John Travis
  +
* To Do:
  +
** Finish choosing problem sets for remaining units
  +
** Add features to problems to include:
  +
*** Hints
  +
*** Solutions
  +
*** MetaTags
  +
*** Improvements such as changing multiple choice problems to fill in the blank, etc.
   
−
=== 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
  
   
  +
[[http://webwork.maa.org/wiki/SubjectAreaTemplates#Multivariable_Differential_Calculus| Other Webwork Course Templates]]
   
−
[[http://webwork.maa.org/wiki/SubjectAreaTemplates#Multivariable_Differential_Calculus| Partial set of Course Templates]]
  
  +
[[Category:PREP 2011]]
  +
[[Category:Events]]
  +
[[Category:Model Courses]]

Latest revision as of 00:03, 3 January 2012

Multivariate Calculus Model Course Units

A user of this material should locate appropriate units below that fit their particular course in multivariate calculus.

Instructions for importing problem sets Instructions for exporting problem sets

Within each Unit below, specific problem types should be described. Detailed problem descriptions are given by clicking on the unit title.

Complete problem sets for each unit will eventually be collected and made available from this site (and perhaps from within the WebWork system itself) but these have not been made available yet. Also, the specific problems suggested could be directly linked if desired although this might be a bit too much!

Download this entire course of problem sets

Vectors in Space

Download this entire collection of problem sets

Unit 1 - Vector Algebra

  • Vector Algebra
  • Dot Product and Applications
  • Cross Product and Applications
Download Set Definition File

Unit 2 - Lines and Planes

  • Lines in Space
    • Two forms of a line in space: (1) parametric equations and symmetric equations
      • Express a line that passes two given points.
      • Express a line that passes through a given point and is parallel to a given line.
      • Express a line that passes through a given point and is perpendicular to a given line.
      • Express a line that passes through a given point and is perpendicular to two given lines that are not parallel.
      • Determine if two given lines are parallel, perpendicular or neither.
      • Determine if two given lines intersect or are skew, and find the intersection if they intersect.
      • Determine if a given line passes through a given point.
  • Planes in Space
    • Plane formula
      • Determine if a given plane contains a given point.
      • Express a plane that contains a given point and is perpendicular to a given vector.
      • Express a plane that contains a given point and is parallel to a given plane.
      • Express a plane that contains a given point and is perpendicular to a given plane.
      • Express a plane that contains a given point and is perpendicular to given two planes that are not parallel.
      • Determine if two given planes are parallel, perpendicular or neither.
      • Find the intersection of two planes that are not parallel.
Download Set Definition File

Unit 3 - Cylindrical Surfaces and Quadric surfaces

  • Cylindrical Surfaces
      • Sketch the curve f(x,y)=0, z=z0 (or g(x,z)=0, y=y0 or h(y,z), x=x0) in the xyz-space.
      • Sketch the surface f(x,y)=0 (or g(x,z)=0, or h(y,z)=0) in the xyz-space.
  • Quadric Surfaces
    • Quadratic equations for (1) sphere (2) ellipsoid (3) paraboloid (4) elliptic cone (5) hyperboloid of one sheet (6) hyperboloid of two sheet (7) hyperbolic paraboloid
Download Set Definition File

Unit 4 - Non-rectangular Coordinate Systems

  • Cylindrical Coordinates
  • Spherical Coordinates
  • Applications
Download Set Definition File

Vector-valued Functions in One Variable

Download this entire collection of problem sets

Unit 1 - Calculus of Vector Functions

  • Calculus of Vector-valued Function in One Variable
    • Vector-value function r(t)
      • Express the domain of r(t) a 2-D (or 3-D) vector-valued function in an interval notation.
      • Determine if a given point is on the curve traced out by r(t).
      • Sketch the curved traced out by r(t).
      • Find all values of t at which r(t) is parallel to a given vector.
      • Find all values of t at which r(t) is perpendicular to a given vector
    • Derivatives of Vector Functions
      • Compute the tangent vector r'(t) of r(t).
      • Find all values of t at which r(t) and r'(t) are perpendicular.
      • Show that r(t) and r'(t) are perpendicular at every t if the magnitude of r(t) is a constant.
    • Integrals of Vector Functions
      • Compute an indefinite integral of r(t).
      • Compute a definite integral of r(t).
Download Set Definition File

Unit 2 - Vector Function Applications

  • Arc Length
  • Curvature
  • Unit Tangent and Unit Normal vectors
  • Computing the radius, center and equation of osculating circle
  • Motion in Space: Velocity and Acceleration
Download Set Definition File

Multi-variable Functions

Download this entire collection of problem sets

Unit 1 - Multi-variable Functions

  • Domain and range
    • Find and sketch the domain of a given function.
    • Evaluate a function at a given point.
    • Find the range of a given function.
    • Sketch level curves (surfaces) of f(x,y) (f(x,y,z)).
    • Sketch the surface z=f(x,y) and traces f(x,y)=z0 on the surface.
  • Limits
    • Definition of limit
      • Show the limit of a function does not exist by showing the function has two different limits along two distinct paths, respectively.
      • Compute the limit or show limit does not exist using polar coordinates.
      • Compute the limit or show limit does not exist using the definition of limit.
  • Continuity
    • Definition of continuity
      • Determine if a function is continuous at a given point by checking if the limit is the same as the value of the function at this point.
      • Find the region over which the function is continuous.
Download Set Definition File

Unit 2 - Partial Derivatives - Rules

  • Partial Derivatives
    • Definitions of Partial Derivatives
      • Compute the first partial derivative of f with respect to x (y or z).
      • Sketch the tangent line at a given point on the trace f(x,y)=z0.
      • Compute higher order partial derivatives.
  • Partial Derivatives using Rules
    • Differentiation Rule
    • The Chain Rule
    • Implicit differentiation using partial derivatives
  • Directional Derivatives and the Gradient Vectors
    • Gradient Vectors
      • Compute the gradient vector of f at a given point.
      • Compute the gradient vector of f at any point in its domain.
      • Find all points at which the gradient of f is a zero vector.
      • Sketch the gradient vector of f(x,y) at (x0,y0) on the level curve f(x,y)=f(x0,y0).
    • Definition of Directional Derivative
    • Directional Derivative as the dot product of gradient vector and unit direction vector
      • Compute the directional derivative of f at a point along a unit direction.
      • Compute the directional derivative of f as (x,y) (or (x,y,z)) moves from point A to point B.
      • Find the maximum (minimum) rate of change of f at a given point and the corresponding direction vector.
      • Determine if f is increasing, decreasing or neither at a given point along a give direction.
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Unit 3 - Partial Derivatives - Applications

  • Tangent Planes and Linear and Quadratic Approximations
    • Tangent Planes and Linear Approximations
      • Find the equation of the tangent plane of the surface z=f(x,y) at (x0,y0).
      • Find all points (x,y) at which the tangent plane of z=f(x,y) at (x0,y0) is parallel to a given plane.
      • Find the linear approximation L(x,y) of f(x,y) at (x0,y0).
      • Approximate f(x1,y1) by L(x1,y1).
    • Quadratic Approximations
      • Find the quadratic approximation Q(x,y) of f(x,y) at (x0,y0).
      • Approximate f(x1,y1) by Q(x1,y1).
  • Maximum and Minimum Values
    • Critical Points
      • Find all critical points of f(x,y) (or f(x,y,z)).
    • Second Derivative Test for Local Maximum and Minimum
      • Find all local maximum and minimum of f(x,y) at the critical point (x0,y0) by the Second Derivative Test.
    • Absolute Maximum and Minimum
      • Find the absolute maximum and minimum of f(x,y) over a region.
  • Lagrange Multipliers


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Multiple Integrals

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Unit 1 - Double Integrals and Applications

  • Double Integrals
    • Double Integrals in the Rectangular Coordinates
      • Set up and evaluate a double integral over a rectangular region.
      • Set up and evaluate a double integral over a general region.
      • Change the order of a given double integral.
    • Double Integrals in Polar Coordinates
      • Set up and evaluate a double integral over a region in polar coordinates.
      • Express a given double integral in rectangular coordinates in polar coordinates.
  • Applications of Double Integrals
    • Area of a region
    • Average value of f(x,y) over a region
    • Surface area of a solid
    • Volume of the solid under the surface z=f(x,y) over a region
    • Centroid of a region, center of mass
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Unit 2 - Triple Integrals and Applications

  • Triple Integrals
    • Triple Integrals in Rectangular Coordinates
    • Triple Integrals in Cylindrical Coordinates
    • Triple Integrals in Spherical Coordinates
      • Set up a triple integral for in the rectangular, cylindrical or spherical coordinates.
      • Exchange the order of integration of a triple integral.
      • Exchange the coordinate systems of a triple integral.
  • Change of Variables in Multiple Integrals
  • Applications of Triple Integrals
    • Volume of a Solid
    • Average Value of f(x,y,z) over a Solid
    • Center of Mass
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Vector Calculus

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Unit 1 - Vector Fields

  • Vector Fields
    • Vector Fields in 2D and 3D
    • Gradient Vector Fields, and Conservative Vector Fields
      • For a given vector field, determine if it is conservative and find its potential function if it is conservative.
      • Sketch a vector field at a given point.
      • Sketch a trace of the flow of a velocity vector field.
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Unit 2 - Line Integrals in 2D and Applications

  • Parametric Equations for Curves (line segments, circle, ellipses)
  • Line Integrals of a scalar function
      • Set up a line integral of f(x,y) over a curve C.
      • Compute the total mass of a wire with a given density function.
  • Line Integrals of a vector field
      • Set up a line integral of vector field over a curve C.
      • Compute the total work done by a given force vector field along a curve C.
      • Compute the total circulation of a vector field along a closed curve C.
  • The Fundamental Theorem of Calculus for Line Integrals
      • Use a potential function of a conservative vector field F to evaluate the line integral of F along a curve C.
  • Green's Theorem
      • Use Green's Theorem to evaluate a line integral of a vector field over a closed curve when all required conditions are satisfied.
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Unit 3 - Line Integrals in 3D and Applications

  • 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)
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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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