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.

This course is part of the Integral Calculus through Data and Modeling Specialization

Offered By

## About this Course

Working knowledge of differentiable calculus and some integral calculus of a single variable function.

Working knowledge of differentiable calculus and some integral calculus of a single variable function.

## Offered by

#### Johns Hopkins University

The mission of The Johns Hopkins University is to educate its students and cultivate their capacity for life-long learning, to foster independent and original research, and to bring the benefits of discovery to the world.

## Syllabus - What you will learn from this course

**1 hour to complete**

## 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.

**1 hour to complete**

**1 hour to complete**

## 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.

**1 hour to complete**

**2 hours to complete**

## 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.

**2 hours to complete**

## About the Integral Calculus through Data and Modeling Specialization

This specialization builds on topics introduced in single and multivariable differentiable calculus to develop the theory and applications of integral calculus. , The focus on the specialization is to using calculus to address questions in the natural and social sciences. Students will learn to use the techniques presented in this class to process, analyze, and interpret data, and to communicate meaningful results, using scientific computing and mathematical modeling. Topics include functions as models of data, differential and integral calculus of functions of one and several variables, differential equations, and optimization and estimation techniques.

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