Upper Division Mathematics: Computational Differential Geometry

Roger Williams University Course Catalog Listing: DMAT 451 - Computational Differential Geometry

DMAT 451 - Computational Differential Geometry - 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 - Computational Differential Geometry - 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

Distance Calculus - Student Reviews

Date Posted: Dec 20, 2019
Review by: Bill K.
Courses Completed: Calculus I, Calculus II, Multivariable Calculus, Linear Algebra
Review: I took the whole calculus series and Linear Algebra via Distance Calculus. Dr. Curtis spent countless hours messaging back and forth with me, answering every question, no matter how trivial they might seem. Dr. Curtis is extremely responsive, especially if the student is curious and is willing to work hard. I don't think I ever waited much more than a day for Dr. Curtis to get a notebook back to me. Dr. Curtis would also make videos of concepts if I was really lost. The course materials are fantastic. If you are a student sitting on the fence, trying to decide between a normal classroom class or Distance Calculus classes with Livemath and Mathematica, my choice would be the Distance Calculus classes every time. The Distance Calculus classes are more engaging. The visual aspects of the class notebooks are awesome. You get the hand calculation skills you need. The best summary I can give is to say, given the opportunity, I would put my own son's math education in Dr. Curtis's hands.
Transferred Credits to: None

Date Posted: Apr 5, 2020
Review by: Catherine M.
Courses Completed: Calculus I
Review: Calculus I from Distance Calculus was wonderful! I took AB Calculus in high school, but I didn't take the AP Calc exam. Instead I took Calculus I with Distance Calculus, and it was so much better! It was a little review of topics, but not really. I really understood calculus when I finished!
Transferred Credits to: University of Chicago

Date Posted: Jul 25, 2020
Review by: Michael Linton
Student Email: mdl264@cornell.edu
Courses Completed: Calculus I
Review: 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!
Transferred Credits to: Cornell University