Portland Community College | Portland, Oregon

Upon completion of the course students should be able to:

• Application – Analyze real world scenarios to: recognize when vectors, geometry of space, multivariate and vector functions, partial differentiation, and multiple integration are appropriate, formulate and model these scenarios (using technology, if appropriate) in order to find solutions using multiple approaches, judge if the results are reasonable, and then interpret these results.

• Concept – Recognize the underlying mathematical concepts of vectors, geometry of space, multivariate and vector functions, partial differentiation, and multiple integration.

• Computation – Use vectors, geometry of space, multivariate and vector functions, partial differentiation, and multiple integration with correct mathematical terminology, notation, and symbolic processes.

• Communication – Communicate mathematical applications, concepts, computations, and results with classmates and colleagues in the fields of science, technology, engineering, and mathematics.

Enjoy a life enriched by exposure to one of humankind's great achievements.

1. Demonstrate an understanding of the concepts of multivariate and vector-valued functions and their application to real world problems in:

   • at least two proctored exams, one of which is a comprehensive final that is worth at least 25% of the overall grade

   • proctored exams should be worth at least 50% of the overall grade

   • and at least one of the following:

     • Take-home examinations

     • Graded homework problems

     • Quizzes

2. Consistently demonstrate proper notation, documentation, and use of language throughout all assessments and assignments. For proper documentation standards see Addendum.

3. Demonstrate an ability to work and communicate with colleagues, on the topics of multivariate and vector valued functions, in at least two of the following:

   • A team project with a written report and/or in-class presentation

   • Participation in discussions

   • In-class group activities

• Students will learn to visualize and manipulate multivariable and vector valued functions presented in graphical, numeric, and symbolic form.

• Students will learn to differentiate multivariate functions in all directions and learn several applications of multivariate derivatives.

• Students will learn to evaluate multiple integrals.

• Students will learn to graph, differentiate, integrate, and solve applied problems involving parametric equations and vector-valued functions.

• Classroom activities will include lecture/discussion and group work.

• Students will communicate their results in oral and written form.

• Students will apply concepts to real world problems.

• The use of technology will be demonstrated and encouraged by the instructor where appropriate.

1. Geometry of Two- and Three-Space The goal is to use vectors in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) to represent quantities that have direction as well as magnitude.

   1. Define a vector.

      1. Represent a vector graphically.

      2. Represent a vector using component notation.

      3. Represent a vector in terms of its unit vectors i, j, k.

      4. Define the magnitude and direction of a vector in terms of the above representations.

   2. Define and apply operations of vectors to perform addition, subtraction, and scalar multiplication graphically and symbolically.

   3. Define and use the dot product geometrically and symbolically.

      1. Define work using vector notation.

      2. Define parallel and orthogonal vectors.

      3. Calculate projections of vectors.

   4. Define and use the cross product geometrically and symbolically.

      1. Calculate the area of a parallelogram.

   5. Lines and planes.

      1. Write parametric equations and symmetric equations for a line.

      2. Write the equation for a plane given a variety of conditions such as:

         1. Three points on the plane

         2. A point and a line contained within the plane.

         3. Two lines contained within the plane.

   6. Two-variable functions.

      1. Three-dimensional graphs of two-variable functions.

         1. Plot points on a three-dimensional axes system.

         2. Sketch planes on a three-dimensional axes system.

         3. Sketch quadric surfaces on a three-dimensional axes system using a CAS.

         4. Match equations of quadric surfaces to their graphs.

      2. Two-variable functions presented in symbolic form.

         1. Evaluate a two-variable function given its formula.

         2. Find the domain of a two-variable function given its formula.

      3. Two-variable functions presented in tabular form.

         1. Visualize and describe a three dimensional function presented in tabular form.

   7. Cylindrical and spherical coordinates.

      1. Plot points in cylindrical and spherical coordinates.

      2. Convert coordinates and expressions from rectangular form to cylindrical and spherical forms.

2. Parametric Equations and Vector-Valued Functions

   1. Graphs of parametric equations.

      1. Draw and/or describe a curve in three-space given a set of parametric equations.

      2. Find parametric equations for a given curve (e.g. the intersection of two surfaces) in two-space or three-space.

   2. Calculus of vector valued functions.

      1. Differentiate and anti-differentiate a vector-valued function presented in symbolic form.

      2. Establish the relationship between position functions, velocity functions, acceleration functions, and speed functions.

      3. Project the path followed by a particle over a given velocity vector field.

      4. Find the arc length of a given curve defined by a vector-valued function.

      5. Find the normal and tangential components of acceleration.

      6. Find the curvature at a given point of a function represented in symbolic form.

3. The Algebra of Multivariate Functions

   1. Visualize and describe three-dimensional surfaces given a set of level curves.

   2. Multivariate functions presented in symbolic form.

      1. Evaluate a multivariate function given its formula.

      2. Find the domain of a multivariate function given its formula.

      3. Sketch the level curves of a three dimensional surface given the formula of the surface.

      4. Sketch and/or describe the level surfaces of a four dimensional object given its formula.

   3. Multivariate functions presented in tabular form.

      1. Visualize and describe a three dimensional function presented in tabular form.

      2. Recognize the level curves of a function presented in tabular form.

      3. Recognize the relationship between the slope of a line in two-space and the slope of a plane in the directions parallel to the \(x\)-axis and the \(y\)-axis.  

4. Differentiation of Multivariate Functions

   1. First order partial derivatives.

      1. Explore the relationship between partial derivatives and the slope of a surface in directions parallel to the \(x\)-axis and the \(y\)-axis.   

      2. Estimate partial derivatives of three-dimensional functions presented as level curves or presented in tabular form.

      3. Find partial derivatives formulas for functions presented in symbolic form.   

   2. Apply the chain rule for multivariate and vector valued functions.

   3. Directional derivatives.

      1. Explore the different rates of change of a surface as dependent upon the direction of motion in the  \(xy\)-plane.

      2. Estimate directional derivatives given a set of level curves.

      3. Define the gradient and establish its relationship to a surface at a given point and the level curve at that point.

      4. Find the directional derivative at a given point in a given direction for a function presented in symbolic form.

   4. Write the equation of the tangent plane to a surface at a point on the surface.

   5. Find the linear approximation to a multivariate function at a point.

   6. Second order partial derivatives.

      1. Explore the relationship between \(f_{xx}\) and \(f_{yy}\) and the concavity of two-dimensional curves.

      2. Determine the signs on all four second-order partial derivatives given a set of level curves.

      3. Find all four second order partial derivatives given a function in symbolic form.

   7. Optimization of three-dimensional functions.

      1. Find all relative extrema and saddle points for a function presented in symbolic form.

      2. Estimate relative extrema and saddle points given a set of level curves.

      3. Find absolute extrema for a function over a constrained region of the \(xy\)-plane.

      4. Solve applied problems involving the optimization of three-dimensional functions.

      5. Use Lagrange multipliers to optimize a multivariable function provided a constraint.

   8. Investigate limits and continuity for multivariate functions.

5. Multiple Integrals

   1. Set up Riemann sums over a region of the \(xy\)-plane to estimate double integrals for functions presented in symbolic, tabular, or level curve formats.

   2. Double integrals in rectangular coordinates.

      1. Find the limits of integration for a given region in the \(xy\)-plane.

      2. Change the order of integration for a given double integral.

      3. Use the Fundamental Theorem of Calculus to evaluate a double integral.

      4. Use a graphing calculator and/or computer to evaluate a double integral.

      5. Write a double integral to evaluate the area of a given region.

      6. Write a double integral to evaluate the volume of a given solid.

   3. Double integrals in polar coordinates.

      1. Find the polar limits of integration for a given region in the \(xy\)-plane.

      2. Convert back and forth between polar iterated integrals and rectangular iterated integrals.

      3. Use the Fundamental Theorem of Calculus to evaluate polar double integrals.

      4. Use a graphing calculator and/or computer to evaluate a polar double integral.

      5. Write a polar double integral to evaluate the area of a given region.

      6. Write a polar double integral to evaluate the volume of a given solid.

   4. Triple integrals in rectangular coordinates.

      1. Write a triple integral to evaluate the volume of a given solid.

      2. Use the Fundamental Theorem of Calculus to evaluate a triple integral.

      3. Use a graphing calculator and/or computer to evaluate a triple integral.

   5. Triple integrals in cylindrical and spherical coordinates.

      1. Use the Fundamental Theorem of Calculus to evaluate cylindrical and spherical triple integrals.

      2. Find a cylindrical and/or spherical triple integral to evaluate the volume of a given solid.

6. Optional Topics
   1. Parametric Surfaces

All work in this course will be evaluated for your ability to meet the following writing objectives as well as for “mathematical content.”

1. Every solution must be written in such a way that the question that was asked is clear simply by reading the submitted solution.
2. Any table or graph that appears in the original problem must also appear somewhere in your solution.
3. All graphs that appear in your solution must contain axis names and scales. All graphs must be accompanied by a figure number and caption. When the graph is referenced in your written work, the reference must be by figure number. Additionally, graphs for applied problems must have units on each axis and the explicit meaning of each axis must be self-apparent either by the axis names or by the figure caption.
4. All tables that appear in your solution must have well defined column headings as well as an assigned table number accompanied by a brief caption (description). When the table is referenced in your written work, the reference must be by table number.
5. A brief introduction to the problem is almost always appropriate.
6. In applied problems, all variables and constants must be defined.
7. If you used the graph or table feature of your calculator in the problem solving process, you must include the graph or table in your written solution.
8. If you used some other non-trivial feature of your calculator (e.g., SOLVER), you must state this in your solution.
9. All (relevant) information given in the problem must be stated somewhere in your solution.
10. A sentence that orients the reader to the purpose of the mathematics should usually precede symbol pushing.
11. Your conclusion shall not be encased in a box, but rather stated at the end of your solution in complete sentence form.
12. Remember to line up your equal signs.
13. If work is word-processed, all mathematical symbols must be generated with a math equation editor.

Emphasis should be placed on using technology such as Desmos and GeoGebra appropriately; such as when computing approximations, graphing curves, or visualizing or checking answers. Technology should not be used as a substitute for meeting the outcomes and skills for the course that are expected to be done by hand.