Portland Community College | Portland, Oregon

Upon successful completion students should be able to:

• Analyze real world scenarios to recognize when partial derivatives or multiple integrals of multivariate and vector valued functions are appropriate, formulate problems about the scenarios, creatively model these scenarios (using technology, if appropriate) in order to solve the problems using multiple approaches, judge if the results are reasonable, and then interpret and clearly communicate the results.
• Recognize partial derivative and multiple integral concepts that are encountered in the real world, understand and be able to communicate the underlying mathematics involved to help another person gain insight into the situation.
• Work with partial derivatives and multiple integrals in various situations and use correct mathematical terminology, notation, and symbolic processes in order to engage in work, study, and conversation on topics involving partial derivatives and multiple integrals with colleagues in the field of mathematics, science or engineering.

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
   • at least one of the exams should require the use of technology
   • 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

1. Context Specific Skills
   • Students will learn to visualize and manipulate multi-variable 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.
2. Learning Process Skills
   • 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 calculators and/or computers will be demonstrated and encouraged by the instructor where appropriate. Technology will be used (at least) when graphing curves, evaluating derivatives, and evaluating definite integrals.

1. Review of the Geometry of Two- and Three-Space
   1. Three-dimensional graphs of multivariate 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.
   2. Lines and Planes
      1. Find the parametric and symmetric equations of lines.
      2. Find equations for planes. 
   3. Polar Coordinates.
      1. Plot points in polar coordinates.
      2. Convert points back and forth between rectangular coordinate form and polar coordinate form.
      3. Plot polar curves by hand and with the aid of a graphing calculator.
   4. Spherical and Cylindrical coordinates.
      1. Plot points in cylindrical and spherical coordinates.
      2. Convert back and forth between rectangular form, cylindrical coordinate form, and spherical coordinate form.
   5. Graphs of parametric equations.
      1. Draw, by hand, a curve in two-space given a set of parametric equations.
      2. Use a graphing calculator to draw a curve in two-space given a set of parametric equations.
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. Find parametric representations of surfaces in three-space.
   3. 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 normal and tangential components of acceleration.
      5. 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 \(y\)-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. Second order partial derivatives.
      1. Explore the relationship between \(f_{xx}\) and \(f_{yy}\) and 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.
   6. 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.
   7. 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. 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.
   4. 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.
   5. Spherical and Cylindrical 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. Change of Variables for Multiple Integrals
   2. ​Vector Fields
   3. Line Integrals and Green's Theorem

Much of the foundational topics of vectors and \(\mathbb{R}^3\) are covered in MTH 253. MTH 253 is considered an essential pre-requisite for this course. Students who take MTH 253 at a school that does not cover these topics should be directed to resources to review these topics. Instructors are still expected to provide a brief review of vectors and \(\mathbb{R}^3\) basics to the entire class. 

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.