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
Pros:
-- Exceptional Instruction and Support: Dr. Curtis was incredibly responsive and helpful whenever I had questions. The TA was also very supportive, and thanks to their guidance, I was proud to earn a 100% in the course—even without having taken pre-calculus beforehand.
-- Innovative Software Platform: The custom software used in the course made a huge difference for me. I found it intuitive and engaging, and it helped reinforce the concepts in a way traditional textbooks never did.
-- Thorough, Rigorous Curriculum: The structure of the course really pushed me to stay organized and plan ahead. I felt like I was being challenged in all the right ways.
Potential Considerations for Others:
-- Requires Strong Time Management: If you haven’t taken pre-calc, like me, you’ll need to be extra proactive. The course can move quickly if you need, and pacing yourself is essential.
-- Software Learning Curve: While I personally loved the software, students who aren’t comfortable adapting to new digital tools might need a bit of extra time upfront to get used to it.
-- Helpful to Have Supplementary Resources: One improvement might be to offer a short list of "starter resources" (videos, concept overviews, etc.) for students who need a broader intro to calculus before diving in.
Overall, I highly recommend this course to motivated students, especially those comfortable with self-paced learning and open to using new tools. Dr. Curtis is a fantastic instructor, and the course setup really works.
The course covered the core business calculus material thoroughly — derivatives, optimization, integration techniques including u-substitution, the Fundamental Theorem of Calculus, improper integrals, and numerical methods. The LiveMath computer algebra environment was central to the experience: it forced me to build each step explicitly rather than just arriving at an answer, which actually deepened my understanding of the mechanics.
Communication through the student portal was responsive when I had questions. For working professionals who need a rigorous, accredited calculus course on a flexible timeline, I'd recommend it.
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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.








