Portland Community College | Portland, Oregon

Upon completion of the course students should be able to:

• Application – Analyze real world scenarios to: recognize when multiple integration and vector calculus 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 multiple integration and vector calculus.

• 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 vector fields and scalar fields presented in graphical and symbolic form.

• Students will learn to find the gradient of a scalar field, and find the curl and divergence of a vector field.

• Students will learn to evaluate triple integrals, line integrals, and surface integrals.

• Students will learn to apply Green's Theorem, Stokes' Theorem, and the Divergence Theorem while evaluating integrals, when appropriate.

• 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 Three-Dimensional Space in Spherical and Cylindrical Coordinates

   1. Plot points in cylindrical and spherical coordinates.

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

   3. Find parametric representations of surfaces in three-space.

2. The Method of Lagrange Multipliers

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

3. Applications of Double Integrals

   1. Use a double integral to find the area of a surface.

4. Triple Integrals

   1. Evaluate triple integrals geometrically, symbolically, and with the use of technology.

   2. Evaluate triple integrals in the rectangular, cylindrical, and spherical coordinate systems.

   3. Find a triple integral in either the rectangular, cylindrical, or spherical coordinate system to evaluate the volume of a given solid.

5. Vector Calculus

   1. Vector Fields

      1. Draw two-dimensional vector fields.

      2. Find the gradient vector field of a scalar function.

      3. Determine if a vector field is conservative, and find potential functions for conservative vector fields.

      4. Find the curl and divergence of a vector field.

   2. Line Integrals

      1. Find a line integral over a scalar field.

      2. Find the line integral of a vector field along a curve in two- and three-dimensions (also called flow, or if the curve is closed, circulation).

      3. Express line integrals in several forms, including writing line integrals with respect to arc length, \(\int\limits_C \nabla f \cdot d\mathbf{r}\), and \(\int\limits_C P\ dx + Q\ dy\).  

      4. Use the Fundamental Theorem for Line Integrals to evaluate \(\int\limits_C \nabla f \cdot d\mathbf{r}\) in one-, two-, and three-dimensions.

      5. Explore path-independence of line integrals and its relationship to conservative vector fields.

      6. Find the flux of a vector field across a curve in two-dimensions.

   3. Green’s Theorem

      1. Use Green’s Theorem to evaluate the line integral of a vector field along a boundary and the corresponding double integral.

      2. Use Green’s Theorem to determine whether a vector field is conservative.

   4. Surface Integrals

      1. Find the integral of a continuous function over a surface in three-dimensions.

      2. Find the flux of a vector field across a surface in three-dimensions.

      3. Express surface integrals in various forms. 

   5. Stokes’ Theorem

      1. Use Stokes’ Theorem to evaluate the line integral of a vector field along a boundary and the corresponding integral of the curl across the surface.

   6. The Divergence Theorem

      1. Use the Divergence Theorem to evaluate the triple integral of the divergence of a vector field over a solid and the corresponding flux of the vector field along the boundary of the solid.

6. Optional Topics

   1. Applications of Vector Calculus

      1. Use integrals to find the mass and center of mass of an object.

      2. Find the electric flux of an electric field through a surface.

      3. Find the rate of heat flow across the surface of a body.

      4. Use the Divergence Theorem to identify sinks and sources in a vector field.

   2. Use the Jacobian to change the variables of a multiple integral.

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.