Learn Multivariable Calculus through Incredible Visualizations with Wolfram Language—Wolfram Blog


Learn Multivariable Calculus through Incredible Visualizations with Wolfram Language

Multivariable calculus extends calculus ideas to functions of a number of variables and is an important tool for modeling and regression analysis in economics, engineering, information science and other fields. Knowing multivariable calculus is likewise the initial step towards innovative calculus and follows single-variable calculus courses. Wolfram Language supplies first-rate performance for the calculation and visualization of ideas, that makes this classy body of mathematical understanding simple and enjoyable to discover!

Today, I am happy to reveal a complimentary interactive course, Introduction to Multivariable Calculus, that will assist trainees all over the world discover to master this extremely appropriate however hard branch of mathematics. This extensive multivariable calculus course extends the concepts of limitations, integrals and derivatives to greater measurements. It likewise thinks about constrained and unconstrained optimization issues and checks out the “3 fantastic theorems” of multivariable calculus. For anybody who has actually not yet taken a single-variable calculus course, the popular Wolfram U course Introduction to Calculus covers comparable product to an initial college-level calculus course and the AP Calculus AB course.

Ready to start? Click the following image to start checking out the interactive course before checking out the rest of this post.

Go to free course


Students taking this course will get an extensive intro to multivariable calculus. This consists of basic subjects like vectors, limitations and partial derivatives along with a few of the fantastic theorems of calculus. Intro to Multivariable Calculus will likewise go over some advanced subjects, like triple integrals, curl and divergence, and parametric surface areas.

Here is a preview at a few of the course subjects (displayed in the left-hand column):

Course topics

Multivariable calculus is a huge topic, however I wished to keep the course at a sensible length. I anticipate trainees can end up seeing the 36 videos and finish the 5 tests in about 9 hours.

This course constructs on the structures of single-variable calculus, so it is advised to have a strong understanding of that topic. Anybody who has actually finished an AP Calculus class will currently be well geared up for this course.

The next couple of areas of this post will explain the various parts of the course in higher information.


The core of the course is a set of 36 lessons, starting with “What Is Multivariable Calculus?” This initial lesson consists of a conversation about what multivariable calculus is and an introduction of what is covered in the course. This is followed by a couple of basic examples comparing single-variable ideas to their analogs in multivariable calculus, a list of multivariable calculus’s applications and a quick account of the historic roots of multivariable calculus in the works of Gauss, Maxwell, Gibbs and others. The lesson concludes with a conversation about who this course is for and how to approach it.

Later lessons start with an evaluation of how each brand-new subject suits the general course along with the field of calculus, followed by a conversation of the essential ideas, with sprinkled examples showing the essences. Wolfram Language functions are utilized throughout to confirm options or envision ideas.

The videos vary from 5 to 24 minutes in length, and each video is accompanied by a records note pad showed on the right-hand side of the screen. Paste and copy Wolfram Language input straight from the records note pad to the ingrained scratch note pad to attempt the examples on your own.


Each lesson features a set of 6 workouts to put the ideas found out in the lesson into practice. Workouts are not graded, and each features a comprehensive option. The following reveals a workout from lesson 35 on Stokes’s theorem:

Sample exercise
Sample exercise

The workout note pads are interactive, so trainees can explore each issue in theWolfram Cloud They are motivated to try the issue themselves and can copy the Wolfram Language code in order to control variables or utilize the code for other comparable issues.

Problem Sets

Each area consists of a set of 14 practice issues with comprehensive options. These example issues utilize the tools and ideas found out in the area. They are suggested to strengthen the product, so trainees can utilize the ideas to fix genuine issues:

Sample problem set


Each area of the course ends with a brief, multiple-choice test with 10 issues. The test issues are approximately at the exact same level as those gone over in the lessons, and a trainee who examines the area thoroughly must be well prepared to pass the test:

Sample quiz problem

Students get instantaneous feedback about their test concern actions, and they are motivated to attempt any approach, consisting of hand or computer system computations, to fix them.

Illuminating Illustrations with the Power of Wolfram

One of the greatest benefits of composing this course in Wolfram Notebooks is the capability to make three-dimensional, interactive visualizations that intuitively describe subjects like round collaborates, curl and divergence, vector fields, flux integrals and more. Here’s a choice of a few of the visualizations in this course.

An interactive figure concurrently revealing curves of steepest climb on a shape map and an interactive three-dimensional plot:

Curves of steepest ascent on a contour map

A user interface that reveals a change in between 2 various coordinate systems:

Transformation between two different coordinate systems

A visualization of the volume aspect tm110623img9 in round collaborates:

Visualization of volume derivative element in spherical coordinates

Visualizing the procedure of iterated combination in round collaborates:

Iterated integration in spherical coordinates

A visualization that assists to comprehend the concept of curve orientation:

Iterated integration in spherical coordinates

What Our Students Are Saying

Wolfram U just recently shared an early variation of this course with a group of 185 trainees in a Daily Study Group series. In addition to the important ideas and feedback we got, trainees had a great deal of favorable things to state about this course– here is a quick choice of their remarks:

  • ” Wolfram U has actually actually raised the bar this year, however this class has actually set the requirement– in every which method. It actually showcases the power of Mathematica, has beneficial and great coding examples with great deals of applications, and– by the method– teaches multivariable calculus. A refresher course in the method I want I might have studied it the very first time.”
  • ” This is most likely the best-looking course out there. The graphic examples are interesting and fantastic. The examples are a terrific mentor resource for me too.”
  • ” These are the very best lectures with fantastic visualizations that make it a lot simpler to understand the product. Thanks to all those who are making it readily available.”
  • ” I believe this is a premium course. The visuals are crisp, useful and tidy.”
  • ” The course is fascinating and well structured; it is really beneficial to have practice issue sets before the tests. The visualizations/graphics are actually remarkable.”
  • ” Great. Actually fantastic. This is a workout in comprehending the huge image and after that using it in the trenches. There are more juicy examples in this course than the majority of. You certainly wish to take Tim’s product apart and see how he does those things.”
  • ” Best course I’ve handled Wolfram U, and I’ve taken several!” Due to the fact that course subjects typically construct on earlier strategies and ideas,

    Completion certificate

    Course Certificate

    Students must view all the lessons and try the tests in the advised series. At the end of the course, trainees can ask for a certificate of conclusion. A course certificate is made after seeing all the lessons and passing all the tests. It represents efficiency in the basics of multivariable calculus and includes worth to a résumé or social networks profile.

    Also, an optional last examination is readily available at the end of the course. Trainees can get an innovative level of accreditation by passing this examination.

    A Building Block for Success

    Mastering the strategies of multivariable calculus is essential for students and trainees in fields including information, consisting of company, science and maker knowing. It is essential to establish a strong instinct and understanding of the ideas. That is why I have actually consisted of a great deal of specific graphics that are created to make hard ideas more available and more remarkable. I hope that Introduction to Multivariable Calculus will assist students accomplish proficiency and add to success in their picked fields. I delighted in teaching this course and developing and invite any remarks relating to the present course along with ideas for future courses.

    About the AuthorWolfram Research This course is taught by Timothy McDevitt, who teaches multivariable calculus at Elizabethtown College practically every term, and it is his preferred course to teach. McDevitt has bachelor’s degrees in mathematics and physics along with a master’s and doctorate in used mathematics from the University of Virginia. 2021–2022 Congressional Fellow In addition to being a visualization designer at American Statistical Association given that 2018, he formerly dealt with NASA and the United States Navy and and was a

    representing the


Acknowledgements(*) This course is the outcome of the work of the Wolfram U and Algorithms R&D groups. I wish to thank Devendra Kapadia, Brett Champion, Joaquin Orozco, Anisha Basil, Joyce Tracewell, Abrita Chakravarty, Arben Kalziqi, Luke Titus, Matt Coleman, Mariah Laugesen, Laura Crawford and Marco Saragnese for all the work they take into getting this course up and running.(*)


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