20804233Calculus 3
Course Information
Description
Designed for students of mathematics, science, and engineering. Topics include differentiation and integration of vector functions, space curves and curvature, motion in space, scalar functions of more than one variable, level curves and level surfaces, limits and continuity, partial derivatives, total differential, tangent planes, the gradient operator, the directional derivative, multivariable forms of the chain rule, locating maxima, minima, and saddle points, the method of Lagrange multipliers, multiple integrals in rectangular, polar, cylindrical and spherical coordinates, transformations of multiple integrals and the Jacobian, surface area, applications of multiple integrals to geometry and mechanics, line integrals in two and three dimensions, vector fields, circulation and flux in two dimensions, Green's & Theorem, the curl and divergence operators, surfaces and surface area defined parametrically, Gauss's and Stokes' Theorems, applications of vector calculus to geometry, mechanical work, fluid mechanics and electromagnetic fields.
Total Credits
5
Course Competencies
  1. Operate with and on space curves defined as vector functions of one variable
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    set up and/or compute the arc length integral of a vector function describing a space curve
    set up and/or compute the curvature of a vector function describing a space curve
    set up and/or compute the torsion of a vector function describing a space curve
    set up and/or compute the unit tangent vector of a vector function describing a space curve
    set up and/or compute the unit normal vector of a vector function describing a space curve
    set up and/or compute the unit binomial vector of a vector function describing a space curve
    compute the velocity vector of a position vector given as a function of time
    compute the acceleration vector of a position vector given as a function of time
    determine the normal and tangential components of the acceleration from a position vector given as a function of time
    make appropriate connections between the kinematic description of a space curve and the unit tangent vector
    make appropriate connections between the kinematic description of a space curve and the normal vector
    make appropriate connections between the kinematic description of a space curve and the binomial vector

  2. Interpret three-dimensional coordinates
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    recognize rectangular coordinate systems
    recognize cylindrical coordinate systems
    recognize spherical coordinate systems
    distinguish between rectangular, cylindrical and spherical coordinate systems
    transform coordinates from any one of these coordinate systems to any other
    make the connection between the equation of a three-dimensional space curve and its graph in any one of these coordinate systems
    construct the graph of a three-dimensional space curve described by an equation in any one of these coordinate systems
    use the appropriate coordinate system in setting up a problem
    use the appropriate coordinate system in solving a problem

  3. Use and interpret Kepler's three laws of planetary motion
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    demonstrate the connection between Kepler's laws of planetary motion and an attractive central inverse square law force field
    use Kepler's laws to solve applied problems in planetary motion

  4. Generate level curves and level surfaces
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    sketch the surfaces and level curves of a variety of functions of two variables
    interpret properties about a function of two variables given the graph of its surface and/or level curves
    connect the information found in the graph of the surface, the graphs of the level curves, and the formula for the surface
    generate the surface and level curves of a function of two variables using numerical or graphical methods

  5. Compute limits of multivariate functions
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    compute the limits of multivariable functions
    determine whether a multivariable function is continuous at a point

  6. Compute partial derivatives of multivariate functions
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    compute symbolically the partial derivatives of multivariable functions
    relate this new process of differentiation to the earlier rules of differentiation
    compute symbolically higher order partial derivatives of multivariable functions
    compute symbolically mixed partial derivatives of multivariable functions
    demonstrate the property of the total differential of a function
    connect the property of the total differential of a function to the equation of a tangent plane

  7. Use the gradient of a multivariate function
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    compute the gradient of a multivariable function
    relate the gradient of a multivariable function vector to the total differential
    relate the gradient of a multivariable function vector to the equations of the tangent plane and normal line
    relate the gradient of a multivariable function vector to the directional derivative
    demonstrate the connection between the equation of the tangent plane and the linear approximation to a function
    demonstrate the connections between the gradient vector and the geometry of the level curves of the function under study
    demonstrate the connections between the gradient vector and the geometry of the level surfaces of the function under study
    demonstrate the connections between the directional derivative and the geometry of the level curves of the function under study
    demonstrate the connections between the directional derivative and the geometry of the level surfaces of the function under study
    observe and record the properties about a function and its gradient given a graph of the level curves or level surfaces
    compute derivatives using a multivariable form of the chain rule
    connect the chain rule to the total differential and the gradient

  8. Locate the maximum and minimum values of a multivariate function
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    analyze multivariable functions for maxima
    analyze multivariable functions for minima
    analyze multivariable functions for saddle points
    construct a Taylor series for a function of two variables to determine whether a critical point is an extremum or a saddle
    locate the extremum of a function subject to constraints using Lagrange multipliers
    demonstrate the connection between the method of Lagrange multipliers and the geometry of the problem
    solve problems of finding the extremum in a variety of verbally stated applications

  9. Evaluate double and triple integrals
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    recognize double integrals in rectangular coordinates
    recognize double integrals in polar coordinates
    formulate double integrals in rectangular coordinates
    formulate double integrals in polar coordinates
    evaluate double integrals in rectangular coordinates
    evaluate double integrals in polar coordinates
    recognize triple integrals in rectangular coordinates
    recognize triple integrals in cylindrical coordinates
    recognize triple integrals in spherical coordinates
    formulate triple integrals in rectangular coordinates
    formulate triple integrals in cylindrical coordinates
    formulate triple integrals in spherical coordinates
    evaluate triple integrals in rectangular coordinates
    evaluate triple integrals in cylindrical coordinates
    evaluate triple integrals in spherical coordinates
    use Fubini's Theorem to simplify multiple integrals
    use Fubini's Theorem to evaluate multiple integrals

  10. Transform double and triple integrals
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    transform double integrals between rectangular and polar coordinates
    transform triple integrals between rectangular, cylindrical and spherical coordinates
    properly map the limits of integration when a transformation is used to evaluate a multiple integral
    demonstrate the connection in two dimensions between the Jacobian determinant and the area of a parallelogram
    demonstrate the connection in three dimensions between the Jacobian determinant and the volume of a parallelepiped

  11. Solve applications using double and triple integrals
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    set up and/or evaluate the double integral for the surface area of a surface defined as an explicit function of two variables
    set up multiple integrals which occur in applications taken from geometry, statistics, dynamics, probability, etc
    evaluate multiple integrals which occur in applications taken from geometry, statistics, dynamics, probability, etc
    interpret multiple integrals which occur in applications taken from geometry, statistics, dynamics, probability, etc

  12. Evaluate line integrals in two and three dimensions
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    recognize line integrals in two and three dimensions
    formulate line integrals in two and three dimensions
    evaluate line integrals in two and three dimensions
    recognize line integrals representing the flow of vector fields in two dimensions
    recognize line integrals representing the circulation of vector fields in two dimensions
    recognize line integrals representing the flux of vector fields in two dimensions
    formulate line integrals representing the flow of vector fields in two dimensions
    formulate line integrals representing the circulation of vector fields in two dimensions
    formulate line integrals representing the flux of vector fields in two dimensions
    evaluate line integrals representing the flow of vector fields in two dimensions
    evaluate line integrals representing the circulation of vector fields in two dimensions
    evaluate line integrals representing the flux of vector fields in two dimensions

  13. Evaluate surface integrals
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    produce parametric equations which generate a given surface
    set up and/or evaluate the double integral for the surface area
    set up and/or evaluate the double integral for the flux of a three dimensional vector field across a surface

  14. Use the curl and divergence of a vector field
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    compute the curl and divergence of three-dimensional vector fields
    demonstrate the connections between the curl and the circulation of a vector field
    demonstrate the connections between the divergence and the flux of a vector field

  15. Use Green's, Gauss's and Stokes' theorems
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    Green's Theorem to simplify circulation and flux integrals in two dimensions
    use Green's Theorem to evaluate circulation and flux integrals in two dimensions
    use Stokes' Theorem to simplify circulation integrals in three dimensions
    use Stokes' Theorem to evaluate circulation integrals in three dimensions
    uses auss's Theorem to simplify flux integrals in three dimensions
    use Gauss's Theorem to evaluate flux integrals in three dimensions
    demonstrate the connections between Green's Theorem in two dimensions, Stokes' Theorem and Gauss's Theorem in three dimensions and the Fundamental Theorem of calculus in one dimension

  16. Apply vector calculus
    Assessment Strategies
    Quiz, Exam, Written Product, and/or Projects
    Criteria
    determine whether a vector field is conservative
    you demonstrate the connections between conservative vector fields, the gradient operator, the curl operator, and Stokes' Theorem
    set up line integrals which calculate mechanical work
    evaluate line integrals which calculate mechanical work
    interpret line integrals which calculate mechanical work
    set up line and surface integrals in applications to fluid mechanics
    set up line and surface integrals in applications to electromagnetic fields
    evaluate line and surface integrals in applications to fluid mechanics
    evaluate line and surface integrals in applications to electromagnetic fields
    interpret line and surface integrals in applications to fluid mechanics
    interpret line and surface integrals in applications to electromagnetic fields