This course continues your study of calculus by focusing on the applications of integration to vector valued functions, or vector fields. These are functions that assign vectors to points in space, allowing us to develop advanced theories to then apply to real-world problems. We define line integrals, which can be used to fund the work done by a vector field. We culminate this course with Green's Theorem, which describes the relationship between certain kinds of line integrals on closed paths and double integrals. In the discrete case, this theorem is called the Shoelace Theorem and allows us to measure the areas of polygons. We use this version of the theorem to develop more tools of data analysis through a peer reviewed project.
Upon successful completion of this course, you have all the tools needed to master any advanced mathematics, computer science, or data science that builds off of the foundations of single or multivariable calculus.
Module 1: Vector Fields and Line Integrals
In this module, we define the notion of a Vector Field, which is a function that applies a vector to a given point. We then develop the notion of integration of these new functions along general curves in the plane and in space. Line integrals were developed in the early19th century initially to solve problems involving fluid flow, forces, electricity, and magnetism. Today they remain at the core of advanced mathematical theory and vector calculus.
Module 2: The Fundamental Theorem for Line Integrals
In this module, we introduce the notion of a Conservative Vector Field. In vector calculus, a conservative vector field is a vector field that is the gradient of some function f, called the potential function. Conservative vector fields have the property that the line integral is path independent, which means the choice of any path between two points does not change the value of the line integral.
Conversely, path independence of the line integral is equivalent to the vector field being conservative. We then state and formalize an important theorem about line integrals of conservative vector fields, called the Fundamental Theorem for Line Integrals. This will allow us to show that for a conservative system, the work done in moving along a path in configuration space depends only on the endpoints of the path.
Module 3: Green's Theorem
In this module we state and apply a main tool of vector calculus: Green's Theorem. Green's theorem gives a relationship between the line integral of a two-dimensional vector field over a closed path in the plane and the double integral over the region it encloses. The fact that the integral of a two-dimensional conservative field over a closed path is zero is a special case of Green's theorem.