Skip to main content

Upper Division Mathematics: Computational Differential Geometry

Differential Geometry is the study of curves and surfaces in 2D, 3D, and higher dimensions - the higher-dimensional objects being ones we can't really see too well without first projecting them down into 3D or 2D.

We study curves first - one-dimensional objects living in either 2D or 3D. Differential Geometry uses the derivative to analyze how a curve moves, how to classify it, and how to understand its structure by analyzing how its derivatives change. After a thorough study of curves, we move to surfaces in 3D - and sometimes higher-dimensional surfaces that require a projection down into 3D to visualize.

This is different from Algebraic Geometry, which doesn't really engage the derivative - it stays in the algebraic realm, asking different questions. But as you can see from the animations below, the subject is very rich in visual wonder.

Explore these videos below, along with the rest of the course description, and see if Differential Geometry should be the course you take after Multivariable Calculus.

Double Torus (Genus 2)
Double Torus (Genus 2)
Torus
Torus


Bumpy Surface
Bumpy Surface
Course
DMAT 451 - Computational Differential Geometry
Credits
4 Semester Credit Hours
Delivery
Fully Online, Asynchronous, Self-Paced
Schwarz Minimal Surface
Schwarz Minimal Surface

RWU Course Catalog - DMAT 451 • Computational Differential Geometry
Course
DMAT 451
Course Title
Computational Differential Geometry
Transcript Title
Differential Geometry
Credits
4 Semester Credit Hours
Description
A first course in differential geometry from a computational and graphical standpoint. Topics include a comprehensive study of curves and surfaces with emphasis on exploring a catalog of named geometrical objects, curvature and other metrics, orientable, non-orientable, ruled, and minimal surface, culminating with an introduction to the Gauss-Bonnet Theorem.
Prerequisite
Successful completion (C- or higher) of Multivariable Calculus or equivalent, or consent of instructor.
E-Textbook
Differential Geometry of Curves and Surfaces using Mathematica, 3rd Edition, by Gray et al.
Software
LiveMath & Mathematica

DMAT 451 - Learning Outcomes

  1. To use computer graphing tools to visualize 2D and 3D curves and surfaces
  2. To understand and compute various metrics about parametric and non-parametric curves and surfaces
  3. To understand and compute the key concept of curvature, and understand its relationship to derivatives and differential equations
  4. To understand the role of motions in geometry
  5. To understand and compute the Frenet frames for curves
  6. To understand and compute the concept of the derivative for vector fields
  7. To understand and compute the local Gauss map for surfaces
  8. To understand and compute the concept of orientability of surfaces
  9. To understand and compute Gaussian and Mean curvature
  10. To understand and compute Ruled and Minimal Surfaces
  11. To be introduced to the Gauss-Bonnet theorem

DMAT 451 - Syllabus of Topics

1. Getting Started
1.1 Email and Chat
1.2 Learning About the Course
1.3 Required Hardware
1.4 Software Fundamentals
2. Curves
2.1 Euclidean Spaces
2.2 2D and 3D Parametric Curves
2.3 Arclength
2.4 Curvature vs. Derivative
2.5 Angles
2.6 Catalog of Famous Curves
3. Alternative Ways of Plotting Curves
3.1 Implicit Curves
3.2 Contour Plots
3.3 Polar Coordinates
3.4 New Curves from Old
4. Solving Curvature Equations
4.1 Euclidean Motions
4.2 Intrinsic Equations
4.3 Assigned Curvature
5. Global Properties of Plane Curves
5.1 Total Signed Curvature
5.2 Turning Numbers
5.3 Rotation Index
5.4 Convexity
5.5 Constant Width
5.6 Support Functions
6. Space Curves
6.1 Tangent, Normal, Binormal Frames
6.2 Curvature and Torsion
6.3 Frenet Formulas
6.4 Arbitrary Speed Curves
6.5 Tubes and Tori
7. Fundamental Theorem of Space Curves
7.1 Assigned Curvature and Torsion
7.2 Contact
7.3 Curves That Lie on a Sphere
7.4 Curves of Constant Slope
8. Calculus of Euclidean Space
8.1 Tangent Vectors and Directional Derivatives
8.2 Tangential Maps
8.3 Vector Fields
8.4 Derivatives of Vector Fields
9. Surfaces in Euclidean Space
9.1 Patches
9.2 Local Gauss Map
9.3 Regular Surfaces
9.4 Level Surfaces
9.5 Catalog of Famous Surfaces
10. Non-Orientable Surfaces
10.1 Orientability
10.2 Mobius Strip and Klein Bottle
10.3 Projective Planes
11. Surface Metrics
11.1 Distance
11.2 Isometries
11.3 Conformal Maps
11.4 Gaussian and Mean Curvature
11.5 Non-Parametrically-Defined Surfaces
12. Ruled and Other Surfaces
12.1 Examples
12.2 Curvature
12.3 Surfaces of Revolution
12.4 Examples of Minimal Surfaces
Gyroid Minimal Surface
Gyroid Minimal Surface

Distance Calculus Referenced Colleges & Universities (29 Years - 393+ Institutions)

Distance Calculus students have transferred course credits to these colleges and universities:

Agnes Scott College • Aiken Technical College • Albany College of Pharmacy and Health Science • Alma College • American Graduate University • American Public University • American University • Andrews University • Arizona State University • Armstrong Atlantic State Univeristy • Athens State University • Auburn University • Auburn University MBA Program • Augusta State University • Austin Peay State University • Azusa Pacific University • Babson College • Baruch College • Baylor University • Belmont University • Beloit College • Bentley University • Berklee College of Music • Berry College • Bethany College • Binghamton University • Bloomsburg University • Bluefield State College • Bluegrass Community and Technical College • Borough of Manhattan Community College • Boston Conservatory • Boston University • Bryant University • Buena Vista University • California Lutheran University • California Polytechnic State University, San Luis Obispo • California state University • California State University Channel Islands • California State University, Dominguez Hills • California State University, Sacramento • Carleton College • Carnegie Mellon University • Cedarville University • Central Michigan University • Central Washington University • Champlain College • Chapman University • Charter Oak State College • Chicago State University • Clark University • Clarkson University • Clemson University • Cleveland State University • Coastal Carolina University • College of Santa Fe • College of William & Mary • Colorado Mesa University • Colorado State University • Columbia University • Columbia University School of Business • Cornell Univeristy • Cornell University • Covenant College • CUNY Medgar Evers College • Denison University • DePaul University • Drexel University • Duke University - Fuqua School of Business • Duke University School of Law • Duke University, Durham NC • Duke University, Fuqua School of Business, Law School, Graduate Programs • East Stroudsburg University • Eastern Illinois University • Eastern Kentucky University • Eastern Mennonite University • Eastern Nazarene College • Elon University • Embry Riddle Aeronautical University • Embry Riddle University • Endicott College • Evangel University • Excelsior College • Fairifield University • Fairleigh Dickenson University • Ferris State University • Florida A & M University • Florida Agricultural and Mechanical University • Florida Atlantic University • Florida Institute of Technology • Florida International University • Florida State College, Jacksonville • Florida State University • Fordham University • Fox Valley Technical College • Franklin University • Freed-Hardamen University • Fresno State University • Friends University • Gannon University • George Mason university • George Washington University • George Washington University School of Business • Georgetown University • Georgia Institute of Technology • Georgia State • Georgia State University • Georgia Tech • Gordon College • Governor's State University • Green Mountain College • Griffith University • Grinnell College • Grove City College • Hamline University • Hampshire College • Hampton University • Harvard University, Kennedy School of Government • Harvard University: Kennedy School of Government, Medical Schools • Hillsdale College • Hillsdale University • Hiram College • Hofstra University • Howard University • Huntingdon College • Illinois Institute for Technology • Illinois Institute of Technology • Indiana University • Iowa State University • Jacksonville State University • James Madison University • Jeff State Community College • Johns Hopkins Univerisity • Johns Hopkins University • Kalamazoo College • Kansas State University • Kaplan University • Kennesaw State University • Kentucky State University • Kettering University • Kings College, University of London • La Sierra University • Lebanon Valley College • Lee University • LeTourneau University • Liberty University • Lincoln University of Pennsylvania • Lipscomb University • Loma Linda University • London School of Economics • Loyola Marymount University • Luther College • Macon State College • Marian University • Marquette University • Mars Hill College • Marshall University • Mary Baldwin College • Massachusetts Maritime Academy • McHenry County College • Mercer University • Mercyhurst College • Meredith College • Mesa State College • Messiah College • Miami University • Michigan State University • Michigan Technological University • Middle Tennessee State University • Middlebury College • Millersville University • Missouri University of Science and Technology • Montana State University • Montana Tech • Montclair University • Morehead State University • Murray State University • Naval Post Graduate School • New Mexico Military Institute • New Mexico State University • New York University • North Carolina Agricultural and Technical State Univerisity • Northeastern University • Northern Arizona University • Northern Michigan University • Northwest Nazarene University • Northwestern University • Oberlin College • Occidental College • Oglethorpe University • Oklahoma Baptist University • Old Dominion University • Olympic College • Orange Coast College • Oregon State University • Pacific Lutheran University • Penn State University • Pennsylvania State University • Pepperdine University • Pomona College • Portland State University • Princeton University • Purdue University • Quinnipiac University • Randolph-Macon College • Regent University • Regis University • Rensselaer Polytechnic Institute • Rhode Island School of Design • Rice University • Robert Morris University • Rochester Institute of Technology • Roger Williams Univerity • Roger Williams University • Roosevelt University • Rowan University • Rutgers University • Saint Anselm College • Saint Joseph's University • Saint Louis University • Saint Michael's College • Salve Regina University • Samford University • San Diego State University • Santa Fe Community College • Shepherd University • Smith College • South Dakota School of Mines and Technology • Southern Adventist University • Southern Methodist University • St. Anselm College • St. John's College • St. Mary's College of Maryland • Stanford University • Stanford University, MBA • State University at Buffalo Law School • State University at Buffalo, Law School • State University of New York • Stevens Institute of Technology • Strayer University • SUNY Binghamton • Swarthmore College • Syracuse University • Texas A&M University • Texas A&M • Texas A&M University • Texas Tech University • The Art Institute of Atlanta • The Catholic University of America • The Citadel • The Citadel, Military College of South Carolina • The College of New Jersey • The College of St. Scholastica • The George Washington University • The Master's College • The New England Institute of Art • The Ohio State Universtity • The University of Alabama • The University of South Carolina • The University of Texas at Austin • The University of Virginia • Thomas Edison State College • Trinity University • TUI University • Tulane University • Union University • United States Air Force Academy • United States Military Academy • Univeristy of Puget Sound • University of Alabama, Huntsville • University of Arizona • University of Arkansas, Little Rock • University of Auckland, New Zealand • University of California, Berkeley • University of California, Los Angeles • University of California, Santa Barbara • University of California, Santa Cruz • University of Central Florida • University of Central Oklahoma • University of Central Texas • University of Chicago • University of Cincinnati • University of Colorado • University of Colorado, Boulder • University of Colorado,Colorado Springs • University of Connecticut • University of Dallas • University of Findlay • University of Florida • University of Georgia • University of Hartford • University of Hawai'i-Manoa • University of Illinois • University of Kentucky • University of La Verne • University of Maine • University of Maryland • University of Massachusetts • University of Massachusetts, Amherst • University of Memphis • University of Michigan • University of Michigan: MBA, Medical Schools, Graduate Programs • University of Minnesota • University of Minnesota, School of Public Health • University of Minnesota, Twin Cities • University of Minnesota-Twin Cities • University of Mississippi • University of Missouri • University of Missouri, Columbia • University of Montana • University Of Mount Union • University of Nebraska • University of Nevada • University of New Hampshire Law School • University of New Haven • University of New Orleans • University of North Carolina • University of North Carolina at Chapel Hill • University of North Carolina, Chapel Hill • University of North Carolina, MBA • University of North Dakota • University of North Texas • University of Northern Iowa • University of Notre Dame • University of Oklahoma • University of Otago • University of Pennsylvania • University of Pennsylvania Architectural School • University of Pennsylvania, Wharton School of Business • University of Phoenix • University of Pittsburgh • University of Portland • University of Redlands • University of Richmond • University of San Francisco • University of South Carolina • University of Southern California • University of Southern Indiana • University of Sussex • University of Tampa • University of Tennessee • University of Texas • University of Texas at Austin • University of Texas, Arlington • University of Texas, Austin • University of Texas, Brownsville • University of Texas, Dallas • University of Texas, Houston • University of Utah • University of Virginia • University of Warwick • University of West Alabama • University of West Florida • University of West Georgia • University of Wisconsin • University of Wisconsin, Madison • University of Wyoming • University West Florida • US Air Force Academy • Utah State University • Utah Valley University • Valdosta State University • Valley Forge Military College • Vanderbilt University • Villanova University • Virginia Military Institute • Virginia Tech • Walla Walla University • Washing State University • Washington and Lee University • Washington State University • Webster University • West Chester University • West Virginia University • West Virginia Wesleyan College • Western Governors University • Western Kentucky University • Western Michigan University • Westminster College • Wharton School of Business, University of Pennsylvania • Wheaton College • Wheaton College (IL) • Wheaton College Illinois • Whitman College • Whittier College • Widener University • William and Mary • William Jewell College • Winthrop University • Woodbury University • Wright State University • Yale University • Yeshiva University • Yonsei University
Scherk's Fifth Surface
Scherk's Fifth Surface





Distance Calculus - Student Reviews

Michael Linton★★★★★
Posted: Jul 25, 2020
Courses Completed: Calculus I
Amazing professor, extremely helpful and graded assignments quickly. To any Cornellians out there, this is the Calculus Course to take in Summer to fulfill your reqs! I would definitely take more Calculus Classes this way in the future!
Email: mdl264@cornell.edu
Transferred Credits To: Cornell University
Aiden B.★★★★★
Posted: Jul 29, 2026
Courses Completed: Calculus II, Multivariable Calculus, Differential Equations, Linear Algebra
After completing all the math offered by my highschool, I switched over to Roger Williams for Calculus II and onward. Overall, it has been a pretty great way to understand the topics online, with transferrable college credits. The exams are not terrible, and the courses cover the vast majority (if not all) of topics covered by traditional courses, with some unique ones added to. So far, I have taken four courses, with another one or two before I graduate.

The only problem with the course is the lack of pen and paper work. This shouldn't come as a crazy surprise with an online course, but there is not much emphasis on symbolic computation. Mathematica or other software does most of the heavy computation for you. The lectures usually show how to solve the problems through paper, but those skills are rarely put into practice. Literacy Sheets act as a way to do some of that (especially with the Calculus II & III course).

All in all, this isn't a huge issue. I just memorize and practice things like double integration, Laplace, ODE solving, matrix diagonalized form, row operations, etc. With a little outside work, this resolves itself. Without Roger WIlliams, I probably would not have been able to take any of these math courses in highschool.

So overall, a great choice for highschool advanced math, especially with transferrable credits.
Email: abarger@protonmail.com
Emma C.★★★★★
Posted: Jul 22, 2021
Courses Completed: Linear Algebra
This was a great course. Flexible and informative with a great professor. It's a great option if you need to fill a prerequisite fast or if you enjoy working at your own pace.
Transferred Credits To: University of Virginia
Ryan★★★★★
Posted: Aug 28, 2026
Courses Completed: Finite Math
I will preface this review by saying that if you are looking to take the easiest course possible, cheat through it, and learn as little as possible, this is not the course for you. Below, I will provide my uncensored opinions to this course:

First, this course uses software called LiveMath. This software can be a little bit challenging to get the hang of, but once you do get the hang of it, it's super easy. Remember that for all new things, it takes a bit to get used to it. Overall, I was highly satisfied with this course. I thought that the instructions were extremely detailed, including a video and a textbook explaining every different concept. In addition, this course offered exposure to a lot of different aspects of finite math. I think that many would agree with me on the following, that, you know, when a teacher is passionate about what they are teaching, it is so much more enjoyable, and you can definitely see that in this course. This course can be challenging at times. Maybe it'll be easy for you, maybe it'll be hard, but I think it's important to remember that the things that challenge us are, you know, the things that make us learn and the things that make us better. There's also a math chat option for if you need help or if you have any questions. Overall, I would recommend this course. I actually initially went into this course looking for an easy course to just help me get the credit requirements, but it ended up being an amazing learning experience and completely shifted my perspective on math overall. And another thing that I thought was great in this course is throughout so many different lectures, the professor would explain how each different thing can be applicable in life. And I think that, you know, that is a lot of what people think about when they are learning something. They're like, will I ever use this? Do I need to learn this? And a lot of this actually is really applicable. And I think that even if, you know, you may not be doing these math equations in your future, I think that, you know, the psychology behind these and the rules can be applied to so many different fuacets of life.
Transferred Credits To: GWU
Lucas L.★★★★★
Posted: Jun 25, 2026
Courses Completed: Multivariable Calculus
The professor as well as the TAs give great feedback when you need help with problems and the videos are great at explaining concepts. Return time on work is good and the work is not too much to handle.
Transferred Credits To: University of Wisconsin
Hari K.★★★★☆
Posted: Jun 24, 2026
Courses Completed: Linear Algebra
This course gives a perspective on Linear algebra that no traditional course does. I’d say i gained much more intuition for this subject from the DC course than my friends who took traditional courses elsewhere. As a cs major, this version of learning with visualization has helped me a lot in understand ML models. However the course doesn’t have videos for the last 2 chapers so i had to self learn with the mathematica notebooks. Response times are a little slow but since it’s a remote class, i guess it’s justified. Overall amazing course and definitely take this over traditional lin alg classes.
 

Frequently Asked Questions

Are There Any Prerequisites for the Differential Geometry Course?

Yes, you need to have completed Multivariable Calculus with a grade of C- or higher, and it is a very good idea if you have completed the other Sophomore-level courses as well.

Is the Differential Geometry Course Computer-Based?

Yes, we will study a truly wonderful textbook entitled Differential Geometry of Curves and Surfaces using Mathematica, and almost all of the coursework will be in Mathematica.

Is the Differential Geometry Difficult?

Yes, Differential Geometry is a challenging course, usually reserved for the junior or senior undergraduate university level. By using Mathematica, the visualization of the subject will be made easier, but the hard mathematics are the same difficulty.

Is Differential Geometry in the lower-division Calculus sequence?

No, Differential Geometry is a junior or senior undergraduate university level course. It is a good preparation course for Real Analysis.

Will the Differential Geometry course transfer to my university?

You need to ask your university that question! See the instructions on the Transferring Credits page.

Video Player