Calculus through Data & Modelling: Vector Calculus from Coursera

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.

What's inside

Syllabus

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.

Good to know

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Explores vector analysis, a component of advanced mathematics and computer science

Develops line integrals, a key concept in studying vector fields, fluid flow, and forces

Introduces the Fundamental Theorem for Line Integrals, a theorem used to calculate work done by a vector field

Covers Green's Theorem, an essential tool in vector calculus for understanding the relationship between line integrals and double integrals

Teaches techniques and theorems for analyzing vector fields, applicable in various fields such as engineering, physics, and computer science

Provides a solid foundation for further exploration in advanced mathematics and scientific fields

Reviews summary

Advanced calculus concepts

Learners say this advanced calculus course provides strong conceptual coverage of vector calculus topics. Students say lectures are clear and engaging, and appreciate that the instructor demonstrates practical applications of complex formulas. The course also builds confidence in the subject.

Concepts are explained clearly.

"Very well taught!"

"the instructor also was clear in the delivery of the content"

"the instructor shows how to pick a proper method for a given problem"

Lectures are engaging and practical.

"I love the video lecture."

"the course progressed smoothly"

"shows how to pick a proper method for a given problem"

Instructor provides formulas without detailed proofs.

"Unfortunately, the instructor gives us formulas but doesn't explain where they come from."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Calculus through Data & Modelling: Vector Calculus with these activities:

Read Spivak's Calculus

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This book will give you a deeper understanding of the broader subject of Calculus and provide additional examples and problems than what is covered in this class.

View Calculus on Amazon

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Buy or rent a copy of the book
Choose a chapter that overlaps with what we are studying in class and read it
Do some of the practice problems at the end of the chapter

Form a study group with at least one other student

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Working with other students will help you to learn through teaching, solidify your knowledge, and ask questions.

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Find at least one other student in the course
Set up a regular time to meet
Choose a topic or problem to work on together
Discuss the material and work on the problem together

Practice line integral integrals

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Solving basic line integral problems will give you more practice working through the formulas.

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Find multiple line integral examples online
Solve multiple line integral problems
Check your answers or explain your work to someone else

Four other activities

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Basic vector field problems

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Working through basic vector field problems will help you become more comfortable working through the calculations.

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Find basic vector field problems with step by step solutions
Do several vector field problems without using a step by step guide
Check your answers or explain your work to someone else

Green's Theorem video series

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This series will walk you through some more advanced Green's Theorem problems

Browse courses on Green's Theorem

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Find a video series on Green's theorem
Watch at least half of the videos in the series
Try to solve some problems using the material from the videos

Create a vector field visualization

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Creating a visualization will help you think about vector fields in a new way and help you better understand their properties.

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Decide on the type of vector field you want to visualize
Choose a programming language and visualization library
Write code to generate the vector field and create the visualization
Share your visualization with others and get feedback

Find an online workshop on vector calculus

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This workshop will provide a great opportunity to ask questions, learn from experts, and connect with other students.

Browse courses on Vector Calculus

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Search online for upcoming workshops on vector calculus
Choose a workshop and sign up
Attend the workshop and participate actively

Career center

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