Math 423/673, Spring 2004

Differential Geometry

John Zweck

Course Materials

Why Take Differential Geometry?

Schedule and Homework (Updated Daily)

Background from multivariable calculus and linear algebra

Ideally, you should be very comfortable with the following topics. If needed, I will spend time in class going over topics you don't know so well.

(1) The following sections and topics from "Calculus" by James Stewart:

(13.5) Equations of lines and planes
(13.6) Quadric surfaces
(13.7) Cylindrical and spherical coordinates
(14.1-3) Vector functions and space curves; Derivatives and integrals of vector functions; Arc length and curvature
(15.1-6) Functions of several variables; Limits and continuity; Partial derivatives; Tangent planes and linear approximations; Chain rule; Directional derivatives, gradient
(16.1-3) Double integrals over rectangles and general regions
(17.1-7) Vector fields; Line integrals and Fundamental Theorem; Green's Theorem; Parametric surfaces and their areas; Surface integrals; Divergence and curl

(2) Concepts from linear algebra:

Linear independence; basis; solution of 2x2 matrix system Ax=b; eigenvalues and eigenvectors of 2x2 matrices.

A good way to test for yourself how well you know this material is to take the Final Exam (pdf) from my Math 251H course last semester.

Bonus Matlab Problem

Due: Wednesday March 3rd at start of class.

Important Note: If you want to attempt a theoretical bonus problem instead of this numerical one ask me and I'll find one for you.

Write a matlab program that takes user-supplied curvature and torsion functions, and plots a unit-speed curve which has the given curvature and torsion.
You should turn in your code (commented!) and output on paper, and email me your code so I can have the fun of trying it for myself.
The assignment is worth the equivalent of ONE homework set (ie if we have 14 hwks it will be worth 1/14 of the total hwk grade for the course, and will be for bonus points).

(1) Matlab has built-in functions for numerically solving systems of ODE's. Find out how to use them. I recommend you use the matlab function ode45.
To use these functions, you will probably have to use the Frenet formulae to write down a matrix system of the form dx/dt = A(t) x where x(t) is a vector of length 9 and A(t) is a 9x9 matrix.
You will probably have to set the options RelTol and AbsTol to be fairly small to get good accuracy. These tolerances control the time step of the numerical integration scheme. Think of ode45 as being like a very sophisticated version of Euler's method for solving ODEs in which the user specifies how much error he will tolerate in one step (RelTol) and for the whole time interval (AbsTol) and the ode solver adaptively computes how small the time steps need to be to acheive these tolerances.
If you want more info on ode45 than you can get from matlab help ask me.
(2) Check the numerical accuracy of your solution by
(a) checking that the Frenet frame you compute is actually a frame, ie T,N,B are all length 1 and are mutually orthogonal.
(b) computing the curvature and torsion of the numerically computed curve and comparing to the given curvature and torsion. One way to do this is to compute the L^2 error defined as the integral (over the interval of the reals on which the curve is defined) of the square of the difference of the given and numerically computed curvature (or torsion) function.
(3) You could allow the user to input the curvature and torsion functions as inline functions.
(4) It would be cool to make a movie of the Frenet frame moving along the curve, and simultaneously in another subplot have a point moving along graphs kappa = curvature(s) and tau = torsion(s) of the curvature and torsion.
(5) Choose the initial point of the curve to be (0,0,0) and the initial Frenet frame to be T,N,B = U1, U2, U3

Bonus Theory Problems

For those of you not doing the Bonus Matlab Problem, the following extra credit problem set on ruled surfaces is also worth the equivalent of 1 homework set. It is due one week after we cover Section 5.4.


4.2 #3,4,7; 4.5 #4; 4.8 #8; 5.4 # 12,13

Exam One

Exam One will be on Monday March 8th. It will cover Sections 1.1-1.5, 1.7, 2.1-2.6, 4.1-4.3.

Last modified Feb 25th, 2004

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