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.
DMAT 451 - Learning Outcomes
- To use computer graphing tools to visualize 2D and 3D curves and surfaces
- To understand and compute various metrics about parametric and non-parametric curves and surfaces
- To understand and compute the key concept of curvature, and understand its relationship to derivatives and differential equations
- To understand the role of motions in geometry
- To understand and compute the Frenet frames for curves
- To understand and compute the concept of the derivative for vector fields
- To understand and compute the local Gauss map for surfaces
- To understand and compute the concept of orientability of surfaces
- To understand and compute Gaussian and Mean curvature
- To understand and compute Ruled and Minimal Surfaces
- To be introduced to the Gauss-Bonnet theorem
DMAT 451 - Syllabus of Topics
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Distance Calculus - Student Reviews
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.
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.
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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.








