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HomeMathBe taught Multivariable Calculus via Unimaginable Visualizations with Wolfram Language—Wolfram Weblog

Be taught Multivariable Calculus via Unimaginable Visualizations with Wolfram Language—Wolfram Weblog

Learn Multivariable Calculus through Incredible Visualizations with Wolfram Language

Multivariable calculus extends calculus ideas to capabilities of a number of variables and is a necessary software for modeling and regression evaluation in economics, engineering, knowledge science and different fields. Studying multivariable calculus can be step one towards superior calculus and follows single-variable calculus programs. Wolfram Language gives world-class performance for the computation and visualization of ideas, which makes this elegant physique of mathematical information simple and enjoyable to be taught!

In the present day, I’m proud to announce a free interactive course, Introduction to Multivariable Calculus, that may assist college students everywhere in the world be taught to grasp this troublesome however extremely relevant department of arithmetic. This complete multivariable calculus course extends the notions of limits, derivatives and integrals to increased dimensions. It additionally considers constrained and unconstrained optimization issues and explores the “three nice theorems” of multivariable calculus. For anybody who has not but taken a single-variable calculus course, the favored Wolfram U course Introduction to Calculus covers equal materials to an introductory college-level calculus course and the AP Calculus AB course.

Able to get began? Click on the next picture to start exploring the interactive course earlier than studying the remainder of this weblog publish.

Go to free course


College students taking this course will obtain an intensive introduction to multivariable calculus. This consists of normal subjects like vectors, limits and partial derivatives in addition to a few of the nice theorems of calculus. Introduction to Multivariable Calculus may even focus on some extra superior subjects, like triple integrals, curl and divergence, and parametric surfaces.

Here’s a sneak peek at a few of the course subjects (proven within the left-hand column):

Course topics

Multivariable calculus is an enormous topic, however I wished to maintain the course at an affordable size. I anticipate college students can end watching the 36 movies and full the 5 quizzes in about 9 hours.

This course builds on the foundations of single-variable calculus, so it is strongly recommended to have a powerful understanding of that topic. Anybody who has accomplished an AP Calculus class will already be effectively outfitted for this course.

The subsequent few sections of this weblog publish will describe the totally different parts of the course in larger element.


The core of the course is a set of 36 classes, starting with “What Is Multivariable Calculus?” This introductory lesson features a dialogue about what multivariable calculus is and an summary of what’s coated within the course. That is adopted by a number of easy examples evaluating single-variable ideas to their analogs in multivariable calculus, a listing of multivariable calculus’s functions and a short account of the historic roots of multivariable calculus within the works of Gauss, Maxwell, Gibbs and others. The lesson concludes with a dialogue about who this course is for and how you can method it.

Later classes start with a overview of how every new matter matches into the general course in addition to the sector of calculus, adopted by a dialogue of the important thing ideas, with interspersed examples illustrating the primary concepts. Wolfram Language capabilities are used all through to confirm options or visualize ideas.

The movies vary from 5 to 24 minutes in size, and every video is accompanied by a transcript pocket book displayed on the right-hand facet of the display. Copy and paste Wolfram Language enter immediately from the transcript pocket book to the embedded scratch pocket book to attempt the examples for your self.


Every lesson comes with a set of six workouts to place the ideas realized within the lesson into follow. Workouts will not be graded, and every comes with an in depth resolution. The next reveals an train from lesson 35 on Stokes’s theorem:

Sample exercise
Sample exercise

The train notebooks are interactive, so college students can experiment with every downside within the Wolfram Cloud. They’re inspired to aim the issue themselves and may copy the Wolfram Language code in an effort to manipulate variables or use the code for different comparable issues.

Drawback Units

Every part incorporates a set of 14 follow issues with detailed options. These instance issues make use of the ideas and instruments realized within the part. They’re meant to bolster the fabric, so college students can use the ideas to unravel actual issues:

Sample problem set


Every part of the course ends with a brief, multiple-choice quiz with 10 issues. The quiz issues are roughly on the identical degree as these mentioned within the classes, and a pupil who critiques the part rigorously ought to be effectively ready to cross the quiz:

Sample quiz problem

College students obtain on the spot suggestions about their quiz query responses, and they’re inspired to attempt any methodology, together with hand or laptop calculations, to unravel them.

Illuminating Illustrations with the Energy of Wolfram

One of many largest benefits of penning this course in Wolfram Notebooks is the flexibility to make three-dimensional, interactive visualizations that intuitively clarify subjects like spherical coordinates, curl and divergence, vector fields, flux integrals and extra. Right here’s a number of a few of the visualizations on this course.

An interactive determine concurrently displaying curves of steepest ascent on a contour map and an interactive three-dimensional plot:

Curves of steepest ascent on a contour map

An interface that reveals a change between two totally different coordinate programs:

Transformation between two different coordinate systems

A visualization of the amount ingredient in spherical coordinates:

Visualization of volume derivative element in spherical coordinates

Visualizing the method of iterated integration in spherical coordinates:

Iterated integration in spherical coordinates

A visualization that helps to know the thought of curve orientation:

Iterated integration in spherical coordinates

What Our College students Are Saying

Wolfram U just lately shared an early model of this course with a gaggle of 185 college students in a Each day Examine Group collection. Along with the precious ideas and suggestions we acquired, college students had a whole lot of optimistic issues to say about this course—here’s a transient number of their feedback:

  • “Wolfram U has actually raised the bar this yr, however this class has set the usual—in each which approach. It actually showcases the facility of Mathematica, has incredible and helpful coding examples with numerous functions, and—by the way in which—teaches multivariable calculus. A refresher course in the way in which I want I might have studied it the primary time.”
  • “That is in all probability the best-looking course on the market. The graphic examples are nice and interesting. The examples are an ideal educating useful resource for me as effectively.”
  • “These are one of the best lectures with fantastic visualizations which make it a lot simpler to know the fabric. Because of all those that are making it accessible.”
  • “I believe it is a first-rate course. The visuals are crisp, clear and informative.”
  • “The course is fascinating and effectively structured; it is rather helpful to have follow downside units earlier than the quizzes. The visualizations/graphics are actually spectacular.”
  • “Nice. Actually nice. That is an train in understanding the massive image after which making use of it within the trenches. There are extra juicy examples on this course than most. You undoubtedly need to take Tim’s materials aside and see how he does these issues.”
  • “Finest course I’ve taken on Wolfram U, and I’ve taken fairly a number of!”

Course Certificates

College students ought to watch all the teachings and try the quizzes within the advisable sequence as a result of course subjects typically construct on earlier ideas and methods. On the finish of the course, college students can request a certificates of completion. A course certificates is earned after watching all the teachings and passing all of the quizzes. It represents proficiency within the fundamentals of multivariable calculus and provides worth to a résumé or social media profile.

Completion certificate

Additionally, an elective ultimate examination is on the market on the finish of the course. College students can obtain a sophisticated degree of certification by passing this examination.

A Constructing Block for Success

Mastering the methods of multivariable calculus is vital for learners and college students in fields involving knowledge, together with science, enterprise and machine studying. You will need to develop a powerful instinct and understanding of the ideas. That’s the reason I’ve included a whole lot of specialised graphics which might be designed to make troublesome ideas extra accessible and extra memorable. I hope that Introduction to Multivariable Calculus will assist learners obtain mastery and contribute to success of their chosen fields. I loved creating and educating this course and welcome any feedback concerning the present course in addition to ideas for future programs.

In regards to the Creator

This course is taught by Timothy McDevitt, who teaches multivariable calculus at Elizabethtown School nearly each semester, and it’s his favourite course to show. McDevitt has bachelor’s levels in math and physics in addition to a grasp’s and doctorate in utilized arithmetic from the College of Virginia.

Along with being a visualization developer at Wolfram Analysis since 2018, he beforehand labored with NASA and the US Navy and and was a 2021–2022 Congressional Fellow representing the American Statistical Affiliation.


This course is the results 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 of the work they put into getting this course up and operating.



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