Note 2.1.
There is a high correlation between the effort and time spent fully grasping the problem sets and how well one ultimately does in the class.
Section 2 Course Content
Subsection 2.1 Course Description
Multivariable Calculus is a third-semester calculus course for students who have previously been introduced to the basic ideas of differential and integral calculus in one variable.
In this course we will take the concepts of single-variable calculus (Calculus I and II) and look at their generalizations to functions of several (but often 2 or 3) variables. Topics include:
-
Euclidean space, vectors, and vector operations
-
Calculus of curves and motion in space
-
Functions of several variables, partial derivatives, gradients, linearization, and optimization
-
Multiple integration, applications, and different coordinate systems
-
Vector calculus of curves and surfaces including the theorems of Stokes, Gauss, and Green
The material we take up in this course has applications in physics, chemistry, biology, environmental science, astronomy, economics, statistics, and just about everything else. We want you to leave the course not only with computational ability, but with the ability to use these notions in their natural scientific contexts, and with an appreciation of their mathematical beauty and power.
Subsection 2.2 Assessment
Subsubsection 2.2.1 Problem Sets
Weekly problem sets will be posted on CourseWorks. Students are to hand-write solutions and upload them to Gradescope before the due date. You are encouraged to work collaboratively with peers and/or instructional staff, but it is expected you write up the solutions on your own as both a check and a reinforcement of understanding.
The standard for a complete solution is not publication-grade mathematics, but rather Could a reasonable student follow the argument and know how to solve the problem? Credit will be given for completeness/effort over accuracy.
Subsubsection 2.2.2 Quizzes
Weekly quizzes will be given in recitation. They are based directly on the prior week’s problem set.
Subsubsection 2.2.3 Redemption
To reduce the pressure of frequent timed examinations, students can earn back missed quiz points up to 90% of each quiz’s total by visiting a CA at help room hours, explaining the error, and demonstrating knowledge of that particular topic to the CA’s satisfaction.
Note 2.2.
This is an experimental system. We will start by leaving this option open to all during any help room hour, but will implement a scheduling system should that prove necessary.
Subsubsection 2.2.4 Midterms
Two in-class midterms will be given. No calculators or other outside materials are permitted. See the Section 3 for specific dates.
Remark 2.3.
While a few older exams are provided on CourseWorks as a means of gauging format and example level, using old exams as the sole means of preparation is a recipe for poor results. The material of this class has to be consciously absorbed.
Subsubsection 2.2.5 Final
A final exam will be given during the registrar-scheduled examination time. The format is essentially a 50% longer midterm exam.
Subsection 2.3 Technology
Technology can play an important role in the learning of mathematics, and as such, graphing, scientific calculators, and computer algebra systems (CAS) are permitted for class and homework, though they will not be required. Calculators will not be permitted on tests and quizzes, and thus it is emphasized that students learn not to rely on them.
That said, students are very much encouraged to use modern tools for computation and visualization. Several such solutions will be exhibited throughout the semester. No programming skills are assumed or required, but students are encouraged to explore how the objects of calculus are represented and studied in a scientific computing environment.
Subsubsection 2.3.1 3Demos
3Demos is an open source web app authored specifically for visualizations of multivariable calculus concepts.
Subsubsection 2.3.2 Jupyter and Python
Jupyter notebooks are digital documents that combine formatted text and images with code that can be run in-place by a back-end server. Several notebooks, authored in Python with the popular NumPy and Matplotlib libraries are available in this GitHub repository. There is also an experimental JupyterLite instance available.
Subsubsection 2.3.3 Mathematica
All Columbia students are eligible for a license to run Mathematica, a powerful, but proprietary, symbolic mathematics engine. Find information on installation through the relevant CUIT software page.
