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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)
Torus


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

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

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

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Scherk's Fifth Surface





Distance Calculus - Student Reviews

Coury Gaffney★★★★★
Posted: Jan 18, 2021
Courses Completed: Linear Algebra
The program used gives an amazing insight into everything that's happening, that you wouldn't get in a traditional course. All of the lessons are clear and clean, and the professor is very helpful along the way. I learned a lot and am happy with taking this course
Email: courygaffney@gmail.com
Transferred Credits To: Virginia Tech
Colin Donahoe★★★★
Posted: May 17, 2025
Courses Completed: Calculus II
Dr. Curtis's lessons were well taught, and livemath wasn't too bad once you understood it. Some of the assignments were pretty difficult and required some more explanation, which was hard to get outside of the given lessons.
Email: colindonahoe4@gmail.com
Transferred Credits To: Saint Joseph's Univesity
Lauren B.★★★★★
Posted: May 5, 2025
Courses Completed: Linear Algebra, Probability Theory
You don’t just learn in this course you understand. As someone who has always struggled with math taking advanced courses was intimidating. Dr. Curtis and his TAs made math manageable and framed the questions in a way they were applicable to real world. Thanks again!
M M.★★★★★
Posted: Feb 8, 2026
Courses Completed: Precalculus, Calculus I
The courses were excellent. Very flexible and engaging and the platform offers a lot of upper-level courses. Dr. Curtis is an outstanding professor and very responsive. I would take again.
Transferred Credits To: None yet
Tanja B.★★★★★
Posted: Jan 28, 2026
Courses Completed: Calculus I
After two failed attempts at my university, this course helped me understand Calculus. The live maths tool along with Dr. Curtis were especially helpful, allowing me to visualize concepts and expand my understanding. The explanations were clear, the examples practical, and I could learn at my own pace, which built my confidence. Thank you.
Transferred Credits To: University of Namibia
Henry F.★★★★★
Posted: Dec 18, 2025
Courses Completed: Differential Equations
Transferred Credits To: Saint Joseph High School

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.

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