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
Distance Calculus Referenced Colleges & Universities (29 Years - 393+ Institutions)
Distance Calculus students have transferred course credits to these colleges and universities:
Distance Calculus - Student Reviews
More Details
- How Our Courses Work
- About the Mastery Learning Format
- Asynchronous & Self-Paced
- Computer & Software Requirements
- Maximum Course Time (1 Year)
- Completion Time Estimates
- Academics
- Course Prerequisites
- Course Syllabi
- Grading Policy
- University Accreditation (NECHE)
- Course Articulation/Transfer
- How Exams Work
- Explore
- Honors Course Track
- Student Reviews
- Introductory Videos
- Who Can Enroll?
Frequent Questions
- Enrollment
- When Can I Enroll?
- When Can I Start My Course?
- Term Dates: Enroll Anytime!
- Costs & Tuition
- Credits & Transcripts
- Will My Credits Transfer?
- Letters of Recommendation
- Is My College On Your Transfer List?
- Other Questions
- Frequently Asked Questions
- Is This the Same as AP Calculus?
- Are These Computer-Based Courses?
- Are These Online Courses?
- Financial Aid?
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.








