Difference between revisions of "ModelCourses/Multivariate Calculus"

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[[ModelCourses/Instructions for exporting problem sets|Instructions for exporting problem sets]]
  
[[ModelCourses/Instructions for exporting 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.
   
−
Within each Unit below, specific problem types should be described. 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!
 +
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!
   
  +
== [[File:setCourse_model_multivariate_calculus.def|Download this entire course of problem sets]] ==
   
−
== <span style="color:blue">Vectors<span> ==
 +
== <span style="color:blue">Vectors in Space<span> ==
−
:[[ModelCourses/Calculus/Vectors|Download this entire collection of problem sets]]
 +
[[File:setTopic_Vectors_model_multivariate_calculus.def.txt|Download this entire collection of problem sets]]
−
=== Unit 1 - Vectors ===
 +
=== Unit 1 - [[ModelCourses/Calculus/Vectors/Vectors in Space|Vector Algebra]] ===
−
* [[ModelCourses/Calculus/Vectors/Vectors in Space|Vectors in Space]]
 +
* Vector Algebra
−
** Space Coordinates
 +
* Dot Product and Applications
−
** The Dot Product of Two Vectors
 +
* Cross Product and Applications
−
** The Cross Product of Two Vectors in Space
  
   
−
: [[ModelCourses/Calculus/Vectors/setUnit1|Download Set Definition File]]
 +
: [[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]] ===
−
* [[ModelCourses/Calculus/Vectors/Applications|Vector Applications]]
  
  +
* Lines in Space
−
** Projections
  
  +
** Two forms of a line in space: (1) parametric equations and symmetric equations
−
** Lines and Planes in Space
  
  +
*** Express a line that passes two given points.
−
** Distances in Space
  
  +
*** 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.
   
−
:[[ModelCourses/Calculus/Vectors/setUnit2|Download Set Definition File]]
  
  +
* 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.
   
−
=== Unit 3 - Non-rectangular coordinates ===
  
  +
:[[File:setTopic02_Vectors_model_multivariate_calculus.def|Download Set Definition File]]
−
*[[ModelCourses/Calculus/Vectors/Coordinate Systems|Coordinate Systems]]
  
−
** Surfaces in Space
  
−
** Cylindrical Coordinates
  
−
** Spherical Coordinates
  
−
** Applications
  
   
−
:[[ModelCourses/Calculus/Vectors/setUnit3|Download Set Definition File]]
 +
=== Unit 3 - [[ModelCourses/Calculus/Vectors/Vectors in Space|Cylindrical Surfaces and Quadric surfaces]] ===
   
−
== <span style="color:blue">Vector Functions<span> ==
  
  +
* Cylindrical Surfaces
−
:[[ModelCourses/Calculus/VectorFunctions|Download this entire collection of problem sets]]
  
  +
*** 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 1 - Vector Functions ===
  
  +
* Quadric Surfaces
−
* [[ModelCourses/Calculus/VectorFunctions/CalculusBasics|Calculus of Vector Functions]]
  
  +
** 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
−
** Vector Functions and Space Curves
  
−
** Derivatives and Integrals of Vector Functions
  
−
:[[ModelCourses/Calculus/VectorFunctions/setUnit1|Download Set Definition File]]
  
   
−
=== Unit 2 - Vector Function Properties ===
  
  +
:[[File:setTopic03_Vectors_model_multivariate_calculus.def|Download Set Definition File]]
−
* [[ModelCourses/Calculus/VectorFunctions/Properties|Properties]]
  
−
** Arc Length
  
−
** Curvature
  
−
** Unit Tangent and Unit Normal vectors
  
   
−
:[[ModelCourses/Calculus/VectorFunctions/setUnit2|Download Set Definition File]]
  
  +
=== Unit 4 - [[ModelCourses/Calculus/Vectors/Coordinate Systems|Non-rectangular Coordinate Systems]] ===
   
−
=== Unit 3 - Vector Function Applications ===
  
  +
* Cylindrical Coordinates
−
* [[ModelCourses/Calculus/VectorFunctions/Applications|Applications]]
  
  +
* Spherical Coordinates
−
** Computing equation of osculating circle
  
  +
* Applications
−
** Motion in Space: Velocity and Acceleration
  
   
−
:[[ModelCourses/Calculus/VectorFunctions/setUnit3|Download Set Definition File]]
 +
:[[File:setTopic04_Vectors_model_multivariate_calculus.def|Download Set Definition File]]
   
−
== <span style="color:blue">Partial Derivatives</span> ==
 +
== <span style="color:blue">Vector-valued Functions in One Variable<span> ==
−
:[[ModelCourses/Calculus/PartialDerivatives|Download this entire collection of problem sets]]
 +
:[[File:setTopic_Vector_Functions_model_multivariate_calculus.def|Download this entire collection of problem sets]]
−
=== Unit 1 - Partial Derivatives - Definition ===
  
−
* [[ModelCourses/Calculus/PartialDerivatives/Definition|Definition]]
  
−
** Functions of Several Variables and Level Curves
  
−
** Limits and Continuity
  
−
** Partial Derivatives by Definition
  
   
−
:[[ModelUnits/Calculus/PartialDerivatives/Unit1|Download Set Definition File]]
  
  +
=== Unit 1 - [[ModelCourses/Calculus/VectorFunctions/CalculusBasics|Calculus of Vector Functions]] ===
   
−
=== Unit 2 - Partial Derivatives - Rules ===
  
  +
* Calculus of Vector-valued Function in One Variable
−
* [[ModelCourses/Calculus/PartialDerivatives/Rules|Rules]]
  
  +
** Vector-value function r(t)
−
** Partial Derivatives using Rules
  
  +
*** 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).
  +
  +
:[[File:setTopic01_Vector_Functions_model_multivariate_calculus.def|Download Set Definition File]]
  +
  +
=== Unit 2 - [[ModelCourses/Calculus/VectorFunctions/Properties|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
  +
  +
:[[File:setTopic02_Vector_Functions_model_multivariate_calculus.def|Download Set Definition File]]
  +
  +
== <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 ]]===
  +
  +
* 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.
  +
  +
:[[File:setTopic01_Partial_Derivatives_model_multivariate_calculus.def|Download Set Definition File]]
  +
  +
=== Unit 2 - [[ModelCourses/Calculus/PartialDerivatives/Rules|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
  
** The Chain Rule
−
** Directional Derivatives and the Gradient Vector
  
  +
** 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.
  +
  +
:[[File:setTopic02_Partial_Derivatives_model_multivariate_calculus.def|Download Set Definition File]]
  +
  +
=== Unit 3 - [[ModelCourses/Calculus/PartialDerivatives/Applications|Partial Derivatives - Applications ]]===
   
−
:[[ModelUnits/Calculus/PartialDerivatives/Unit2|Download Set Definition File]]
  
  +
* 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
   
−
=== Unit 3 - Partial Derivatives - Applications ===
  
−
* [[ModelCourses/Calculus/PartialDerivatives/Applications|Applications]]
  
−
** Tangent Planes and Linear and Other Approximations
  
−
** Maximum and Minimum Values
  
−
** Lagrange Multipliers
  
   
−
:[[ModelUnits/Calculus/PartialDerivatives/Unit3|Download Set Definition File]]
 +
:[[File:setTopic03_Partial_Derivatives_model_multivariate_calculus.def|Download Set Definition File]]
   
 
== <span style="color:blue">Multiple Integrals</span> ==
  
== <span style="color:blue">Multiple Integrals</span> ==
−
:[[ModelCourses/Calculus/MultipleIntegrals|Download this entire collection of problem sets]]
 +
:[[File:setTopic_Multiple_Integration_model_multivariate_calculus.def|Download this entire collection of problem sets]]
   
−
=== Unit 1 - Double Integrals Rectangular ===
 +
=== Unit 1 - [[ModelCourses/Calculus/MultipleIntegrals/Rectangular|Double Integrals and Applications]]===
−
* 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]]
  
  +
* 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
   
−
=== Unit 2 - Double Integral Polar ===
  
  +
:[[File:setTopic01_Multiple_Integration_model_multivariate_calculus.def|Download Set Definition File]]
−
* Double Integrals in Polar Coordinates
  
−
* Applications of Double Integrals in Polar Coordinates
  
   
−
:[[ModelUnits/Calculus/MultipleIntegrals/Unit2]]
 +
=== 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]]
   
 
== <span style="color:blue">Vector Calculus</span> ==
  
== <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]]
  
  +
:[[File:setTopic01_Vector_Calculus_model_multivariate_calculus.def|Download Set Definition File]]
   
−
=== Unit 2 - Line Integrals in 2D ===
 +
=== 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 over a vector field
 +
* Line Integrals of a vector field
−
** Simple computations
 +
*** Set up a line integral of vector field over a curve C.
−
** Application to Work
 +
*** 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
  
* The Fundamental Theorem of Calculus for Line Integrals
−
** Relationship with conservative fields and independence of path.
  
  +
*** Use a potential function of a conservative vector field F to evaluate the line integral of F along a curve C.
 
* Green's Theorem
  
* Green's Theorem
−
** Simple calculations
  
  +
*** Use Green's Theorem to evaluate a line integral of a vector field over a closed curve when all required conditions are satisfied.
−
** Changing orientations, holes
  
−
** Applications in Physics
  
   
−
[[ModelUnits/Calculus/VectorCalculus/Unit2]]
  
  +
:[[File:setTopic02_Vector_Calculus_model_multivariate_calculus.def|Download Set Definition File]]
  +
  +
=== Unit 3 - [[ModelCourses/Calculus/VectorCalculus/LineIntegrals3D|Line Integrals in 3D and Applications]] ===
   
−
=== Unit 3 - Line Integrals in 3D ===
  
 
* Parametric Surfaces and Areas (sometimes optional due to time constraints)
  
* Parametric Surfaces and Areas (sometimes optional due to time constraints)
 
* Curl and Divergence (sometimes optional due to time constraints)
  
* Curl and Divergence (sometimes optional due to time constraints)
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* The Divergence Theorem (often optional)
  
* The Divergence Theorem (often optional)
   
−
[[ModelUnits/Calculus/VectorCalculus/Unit3]]
  
  +
:[[File:setTopic03_Vector_Calculus_model_multivariate_calculus.def|Download Set Definition File]]
−  
−
[[ModelCourses/Calculus/VectorCalculus]]
  
   
 
= Packaged Courses =
  
= Packaged Courses =
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== Larson ==
  
== Larson ==
 
[[Larson_packaged]]
  
[[Larson_packaged]]
  +
  +
= =
  +
=[[ModelCourses|Other Model Course Pages]]=
   
   
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[[Category:PREP 2011]]
  
[[Category:PREP 2011]]
 
[[Category:Events]]
  
[[Category:Events]]
  +
[[Category:Model Courses]]

Latest revision as of 23:03, 2 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.``

  • 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
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      • MetaTags
      • Improvements such as changing multiple choice problems to fill in the blank, etc.


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