ModelCourses/Multivariate Calculus
Revision as of 18:28, 28 December 2011 by Chenm (talk | contribs) (→Unit 2 - Partial Derivatives - Rules)
Contents
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
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.
- Two forms of a line in space: (1) parametric equations and symmetric equations
- 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.
- Plane formula
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
Unit 4 - Non-rectangular Coordinate Systems
- Cylindrical Coordinates
- Spherical Coordinates
- Applications
Vector-valued Functions in One Variable
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).
- Vector-value function r(t)
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
Multi-variable Functions
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.
- 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.
- Definition of continuity
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.
- Definitions of 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.
- Gradient Vectors
Unit 3 - Partial Derivatives - Applications
- Tangent Planes and Linear and Other Approximations
- Maximum and Minimum Values
- Lagrange Multipliers
Multiple Integrals
Unit 1 - Double Integrals Rectangular
- Iterated Integrals
- Setting up Double Integrals over General Regions
- Applications of Double Integrals in Rectangular Coordinates
Unit 2 - Double Integral Polar Coordinates
- Double Integrals in Polar Coordinates
- Applications of Double Integrals in Polar Coordinates
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
Vector Calculus
Unit 1 - Vector Fields
- Vector Fields in 2D
- Vector Fields in 3D
Unit 2 - Line Integrals in 2D
- Line Integrals of a scalar function
- Line Integrals over a vector field
- The Fundamental Theorem of Calculus for Line Integrals
- Green's Theorem
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)
Packaged Courses
Moodle
https://test.webwork.maa.org/moodle/
Stewart
Hughes-Hallett
Smith and Minton
Larson
Other Model Course Pages
``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
- Solutions
- MetaTags
- Improvements such as changing multiple choice problems to fill in the blank, etc.
