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Joseph W. Cutrone, PhD
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.
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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.
Read more
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.
Register for this course and see more details by visiting:
OpenCourser.com/course/sj0pke/calculus
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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.
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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.
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Traffic lights
Traffic lights
Read about what's good
what should give you pause and
possible dealbreakers
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
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Reviews summary
Vector calculus with data focus
According to students, "Calculus through Data & Modelling: Vector Calculus" offers a largely positive and highly relevant approach to advanced calculus, particularly for those in data science and computer science. Learners praise the course's focus on real-world applications, making complex concepts like vector fields and line integrals more accessible. The peer-reviewed project is frequently highlighted as a valuable, hands-on experience that reinforces learning. While some find the material challenging due to its advanced nature and the need for a strong foundation, many appreciate the clear explanations and the direct applicability of the content to further studies or careers. It effectively bridges theoretical calculus with practical data analysis.
Pacing can be fast, requiring additional self-study.
"I felt the course moved a bit too quickly through some advanced concepts, requiring extra study on my part."
"The lectures were concise and to the point, which I appreciated given my busy schedule and limited time."
"While comprehensive, I sometimes wished for more examples or a slower pace in certain modules for deeper understanding."
Complex topics are presented with clarity and insight.
"The instructor's explanations made challenging topics much easier to grasp, even the most abstract parts."
"I appreciated the clear breakdown of concepts like conservative vector fields and line integrals."
"Even when I struggled, the explanations were well-structured and highly helpful in overcoming difficulties."
The hands-on project is a key learning experience.
"The peer-reviewed project on the Shoelace Theorem was a fantastic way to apply what I learned and solidify understanding."
"I enjoyed the practical application of Green's theorem through the data analysis project, it was truly engaging."
"The project truly solidified my understanding and was one of the strongest parts of the course for me."
Integrates calculus with real-world data and modeling.
"I really valued how the course applied vector calculus directly to data science, making it immediately useful for my work."
"This course helped me see the relevance of abstract math concepts in actual modeling scenarios."
"I found the connection to real-world problems and data analysis extremely beneficial for my career development."
Demanding; requires a solid foundation in prior calculus.
"I needed a strong background in multivariable calculus to keep up with the pace and complexity of the material."
"The concepts are inherently difficult, but the explanations helped a lot, even if I had to rewatch lectures."
"This course is definitely challenging, but it's rewarding if you put in the necessary effort and time."
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
Show steps
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
Show steps
-
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
Show steps
Working with other students will help you to learn through teaching, solidify your knowledge, and ask questions.
Show steps
-
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
Show steps
Solving basic line integral problems will give you more practice working through the formulas.
Show steps
-
Find multiple line integral examples online
-
Solve multiple line integral problems
-
Check your answers or explain your work to someone else
Four other activities
Expand to see all activities and additional details
Show all seven activities
Basic vector field problems
Show steps
Working through basic vector field problems will help you become more comfortable working through the calculations.
Show steps
-
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
Show steps
This series will walk you through some more advanced Green's Theorem problems
Browse courses on
Green's Theorem
Show steps
-
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
Show steps
Creating a visualization will help you think about vector fields in a new way and help you better understand their properties.
Show steps
-
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
Show steps
This workshop will provide a great opportunity to ask questions, learn from experts, and connect with other students.
Browse courses on
Vector Calculus
Show steps
-
Search online for upcoming workshops on vector calculus
-
Choose a workshop and sign up
-
Attend the workshop and participate actively
Read Spivak's Calculus
Show steps
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
Show steps
Show steps
Show steps
- 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
Show steps
Working with other students will help you to learn through teaching, solidify your knowledge, and ask questions.
Show steps
Show steps
Show steps
- 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
Show steps
Solving basic line integral problems will give you more practice working through the formulas.
Show steps
Show steps
Show steps
- Find multiple line integral examples online
- Solve multiple line integral problems
- Check your answers or explain your work to someone else
Four other activities
Expand to see all activities and additional details
Show all seven activities
Basic vector field problems
Show steps
Working through basic vector field problems will help you become more comfortable working through the calculations.
Show steps
Show steps
Show steps
- 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
Show steps
This series will walk you through some more advanced Green's Theorem problems
Browse courses on
Green's Theorem
Show steps
Show steps
Show steps
- 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
Show steps
Creating a visualization will help you think about vector fields in a new way and help you better understand their properties.
Show steps
Show steps
Show steps
- 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
Show steps
This workshop will provide a great opportunity to ask questions, learn from experts, and connect with other students.
Browse courses on
Vector Calculus
Show steps
Show steps
Show steps
- Search online for upcoming workshops on vector calculus
- Choose a workshop and sign up
- Attend the workshop and participate actively
Career center
Learners who complete Calculus through Data & Modelling: Vector Calculus will develop knowledge and skills
that may be useful to these careers:
Data Analyst
Data Analyst
Data Analysts are responsible for collecting, cleaning, and analyzing data to help businesses make informed decisions. This course can help you develop the skills you need to succeed in this role, including data analysis techniques, statistical modeling, and data visualization.
Data Scientist
Data Scientist
Data Scientists use their skills in data analysis, machine learning, and statistics to solve business problems. This course can help you build a foundation in these areas, and prepare you for a career as a Data Scientist.
Machine Learning Engineer
Machine Learning Engineer
Machine Learning Engineers design and build machine learning models to solve business problems. This course can help you develop the skills you need for this role, including machine learning algorithms, data analysis techniques, and software engineering.
Quantitative Analyst
Quantitative Analyst
Quantitative Analysts use mathematical and statistical models to analyze financial data and make investment decisions. This course can help you develop the skills you need for this role, including financial modeling, data analysis, and statistics.
Software Developer
Software Developer
Software Developers design, build, and maintain software applications. This course can help you develop the skills you need for this role, including software engineering, data structures, and algorithms.
Actuary
Actuary
Actuaries use mathematical and statistical models to assess risk and make financial decisions. This course can help you develop the skills you need for this role, including financial modeling, data analysis, and statistics.
Biostatistician
Biostatistician
Biostatisticians use statistical methods to design and analyze medical studies. This course can help you develop the skills you need for this role, including data analysis, statistical modeling, and epidemiology.
Economist
Economist
Economists use economic theory and data analysis to study the economy and make policy recommendations. This course can help you develop the skills you need for this role, including economic modeling, data analysis, and statistics.
Financial Analyst
Financial Analyst
Financial Analysts use financial data to make investment decisions. This course can help you develop the skills you need for this role, including financial modeling, data analysis, and statistics.
Market Researcher
Market Researcher
Market Researchers use data to understand consumer behavior and make marketing decisions. This course can help you develop the skills you need for this role, including data analysis, statistical modeling, and consumer behavior.
Operations Research Analyst
Operations Research Analyst
Operations Research Analysts use mathematical and statistical models to solve business problems. This course can help you develop the skills you need for this role, including mathematical modeling, data analysis, and optimization.
Risk Analyst
Risk Analyst
Risk Analysts use mathematical and statistical models to assess risk and make business decisions. This course can help you develop the skills you need for this role, including financial modeling, data analysis, and statistics.
Statistician
Statistician
Statisticians use statistical methods to collect, analyze, and interpret data. This course can help you develop the skills you need for this role, including data analysis, statistical modeling, and probability.
Teacher
Teacher
Teachers use a variety of methods to educate students. This course can help you develop the skills you need for this role, including lesson planning, classroom management, and assessment.
Writer
Writer
Writers use words to communicate ideas and stories. This course can help you develop the skills you need for this role, including writing, editing, and research.
For more career information including salaries, visit:
OpenCourser.com/course/sj0pke/calculus
Reading list
We've selected 12 books
that we think will supplement your
learning. Use these to
develop background knowledge, enrich your coursework, and gain a
deeper understanding of the topics covered in
Calculus through Data & Modelling: Vector Calculus.
Provides a comprehensive and rigorous introduction to vector calculus, linear algebra, and differential forms, with a focus on their applications in mathematics, physics, and engineering. It valuable resource for students and practitioners who wish to deepen their understanding of these fundamental concepts.
This classic textbook provides a comprehensive and well-written introduction to calculus, including vector calculus. It valuable resource for students who wish to review the basics of calculus or learn about vector calculus for the first time.
Provides a comprehensive and accessible introduction to linear algebra, which prerequisite for vector calculus. It valuable resource for students who wish to strengthen their understanding of linear algebra concepts.
Provides a concise and rigorous introduction to differential forms, which are a powerful tool in vector calculus. It valuable resource for students who wish to learn more about differential forms and their applications.
Save
Provides a clear and concise introduction to vector calculus, with a focus on its applications in physics and engineering. It valuable resource for students who wish to learn about vector calculus in the context of real-world problems.
Provides a comprehensive and rigorous introduction to calculus on manifolds, which generalization of vector calculus to higher dimensions. It valuable resource for students who wish to learn about the foundations of differential geometry.
Provides a practical and accessible introduction to linear algebra, with a focus on its applications in data science and machine learning. It valuable resource for students who wish to learn about linear algebra in the context of real-world problems.
Provides a comprehensive and up-to-date introduction to numerical linear algebra, which is the study of algorithms for solving linear algebra problems. It valuable resource for students who wish to learn about the numerical aspects of vector calculus.
Save
Provides a comprehensive and rigorous introduction to tensor calculus, which generalization of vector calculus to higher dimensions. It valuable resource for students who wish to learn about the foundations of differential geometry.
Provides a comprehensive and rigorous introduction to differential geometry, which is the study of smooth manifolds and their properties. It valuable resource for students who wish to learn about the foundations of differential geometry.
Save
Provides a comprehensive and rigorous introduction to Riemannian geometry, which is the study of Riemannian manifolds. It valuable resource for students who wish to learn about the foundations of differential geometry.
Provides a comprehensive and rigorous introduction to symplectic geometry, which is the study of symplectic manifolds. It valuable resource for students who wish to learn about the foundations of differential geometry.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/sj0pke/calculus
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Intermediate
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Johns Hopkins University
Instructor
Joseph W. Cutrone, PhD
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This course belongs to a
collection
called
Integral Calculus through Data and Modeling. It's comprised of four courses:
- Calculus through Data & Modelling: Techniques of Integration
- Calculus through Data & Modelling: Vector Calculus (viewing)
- Calculus through Data & Modelling: Series and Integration
- Calculus through Data & Modelling: Integration Applications
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