Advanced Vector Calculus

Description of Course Goals and Curriculum

Vector spaces, limits, derivatives of vector-valued functions, Taylor's formula, Lagrange multipliers, double and triple integrals, change of coordinates, surface and line integrals, generalizations of the fundamental theorem of calculus to higher dimensions. More abstract than 201 but more concrete than 218. Recommended for prospective physics majors and others with a strong interest in applied mathematics. The goal of this class is to teach multivariable calculus. Course material builds up from partial derivatives and line, surface, and volume integrals, and culminates in the three key relationships: Green’s Theorem, Stokes’ Theorem, and Gauss’ Theorem. Another important goal of the class is to introduce students to the kind of thought required for proof-based math. In addition to being able to solve problems involving multivariable calculus, students will learn why methods work and how to logically explain what is going on.

Learning From Classroom Instruction

Lectures move very quickly, so reading the textbook section beforehand is highly recommended. This will allow students to pay more attention in class and notice details and examples that are not covered in the textbook reading. Asking questions is also a great way to temporarily pause the fire hose of information and make sure you are still on the bandwagon. Taking notes during lecture is recommended if it helps the student follow along and absorb the information, but it is important that the quality of the understanding of material always take precedence over the quality of the notes. Precepts/problem sessions are optional. On occasion, these problem sessions will reveal a helpful technique or shortcut for a certain kind of problem. However, notes will typically be posted afterwards, so students may find it just as helpful to work through the problems on their own rather than watching an instructor do them.

Learning For and From Assignments

When studying for assessments, start by taking practice tests that are posted on Blackboard. Students will be tested on their ability to apply concepts to new problems. Furthermore, doing practice problems will often reveal the key aspects of a concept much better than reading about the concept. Checking answers with the posted solutions, discussing in study groups (in a classroom with blackboards), and going to office hours are great ways to follow up on problems that were difficult. Re-taking the quizzes and midterm are also helpful; often themes recur. As with exams for any other class, DO NOT STOP while taking one. Often times just starting to write will help the student, move forward, and even if it doesn’t lead to a complete solution, generous partial credit will often be awarded. The most effective approach to problem sets is to attempt the entire problem set individually during the week, and then reserve the weekend to discuss the difficult problems in study groups. Office hours and study halls are also available, but often they are a waste of time if the student has not looked at the problem on their own yet. Throughout the semester, but particularly as assessments approach, focus on understanding rather than record. Having neat notes will not help on the test unless the information is also in your brain. Writing something down and not saving it for later is not a waste of time or paper, but an effective learning strategy.

External Resources

The old quizzes, midterms, and finals are a rich source of practice material. Keep in mind that newer practice assessments are a much more accurate representation of what the actual test will be like. Prepare well-formed questions for office hours, such as how to solve a problem from a practice test, how a concept works, etc. Individual and group study hall tutoring for this class are available for free at the McGraw Center.

What Students Should Know About This Course For Purposes Of Course Selection

Typically, students must decide between MAT 201 and MAT 203. Both teach the same essential multivariable calculus, but MAT 203 will offer much greater depth in terms of proofs and theory, whereas MAT 201 will be much more directly computational. Another consideration is which lecture section to enroll in. Aside from scheduling considerations, different professors have very different lecturing styles. Shopping different sections before settling into one will enhance the student’s learning experience considerably.