Students in multivariable calculus will extend the knowledge gained in the calculus curriculum to the realm of multiple inputs and outputs. An emphasis is placed on conceptual understanding, on the relationships, similariaties, and differences between one and several variables, and on geometric insight into the symbolic formulas. This is strictly a course in the calculus of several variables, and does not include differential equations or linear algebra as some second-year calculus courses do (those are distinct courses at IMSA).
Text(s)/Materials:
Colley, Susan J. (2006). Vector Calculus, 3rd ed. Upper Saddle River, NJ: Prentice Hall.
Course Description:
Students will apply and extend their knowledge of calculus to problems involving several variables. They will examine the similarities and differences between one- and several-variable situations from both a computational and theoretical point of view. The course covers the material from a traditional semester-long university course: geometry and algebra of vectors, coordinate systems, functions of several variables and their graphs and behaviors, differentiation (partial derivatives, gradients, divergence, curl, etc.), integration (multiple integrals, path, line, and surface integrals). Other topics from among applications (min/max problems, work, flux, etc.), Change-of-Variables Theorem, Green's Theorem, Stokes' Theorem.
Teaching and Learning Methodology and Philosophy:
Students should be involved in exploration of the concepts and topics through reading of the text and outside material, giving presentations to classmates, through computer work, and solving problems in individual and group settings. Students will be asked to engage in the dialogue of problem solving, and to help their classmates understand the content of the course. Learning to write and speak in clear and precise mathematical language is a major goal throughout this course.
Student Expectations:
Students are expected to come to each class prepared to contribute to the classroom learning experience. This involves reading all assigned text and attempting all assigned problems before coming to class. Students are expected to work together and independently in deepening their understanding of course concepts. Students will have to take careful and complete notes in class, as the text does not cover all required material. Students may be expected to work in groups to make classroom presentations on selected topics during the semester.
Assessment Practices, Procedures, and Processes:
Students are assessed through a variety of means including but not limited to: written in-class and out-of-class exams, quizzes, classroom presentations, and homework. Homework will constitute about 1/3 of the grade, while quizzes and exams will make up about 2/3. Late work will receive a 10% penalty per day. The final exam is worth 20% of the semester grade.
Course Resources: (homework, handouts, etc.)
Each link will open the document in a new window, or you may simply download the linked PDF document (right-click for PC, command-click for Mac).
Syllabus
HW 1: Vector Basics
HW 2: Dot Product
HW 3: Cross Product
HW 4: Lines and Planes
In-class Matrix Practice
HW 5: Curvilinear Coordinates
Spherical and Cylindrical reference sheet
HW 6: Limits and Continuity
HW 7: Partial Derivatives
HW 8: Gradients and Directional Derivatives
Differentiability examples handout
HW 9: Taylor Series
HW 10: Min/max
Langrange Multiplier Intro Sheet
HW 11: Lagrange Multipliers
HW 12: Vector Derivatives
HW 13: Div, Grad, Curl I
HW 14: Div, Grad, Curl II
HW 15: Double Integrals
HW 16: Triple Integrals and Order of Integration
HW 17: Change of Variables I
HW 18: Change of Variables II
HW 19: Path Integrals
HW 20: Line Integrals
HW 21: Green's Theorem
HW 22: Surfaces
HW 23: Surface Integrals
HW 24: Stokes' Theorem