Differential Geometry MAT WEEKLY PROGRAM 1-2

Transcription

1 WEEKLY PROGRAM 1- The first week, we will talk about the contents of this course and mentioned the theory of curves and surfaces by giving the relation and differences between them with aid of some examples. We will continue with the theory of curves from the books indicated in the paper delivered to you. Mainly, we will follow the book of Pressley and we will also use (O Neill and) Do Carmo s book for the curve theory. We are planning to talk about the first four sections of the first chapter in Do Carmo s book. We will talk about reparametrization of curves followed by O Neill s book. Solve the following exercises: 1..1 ; 1.. ; 1..3 ; 1..4 ; 1..5 ; 1.3. ; ; in Do Carmo s book and also the following exercises : 1.1) Is γ(t)= (t,t 4 ) a parametrization of the parabola y=x? 1.) Find parametrizations of the following level curves : (i) y -x =1 ; (ii) x 4 + y 9 =1. 1.3) Find the cartesian equations of the following parametrized curves : (i) γ(t)=(cos t,sin t); (ii) γ(t)=(e t,t ). 1.4) Calculate the tangent vectors of the curves in Exercise ) If P is any point on the circle C in the xy-plane of radius>0 and center (0,a), let the straight line through the origin and P intersect the line y=a at Q, and let the line through P parallel to the x-axis intersect the line through Q parallel to the y-axis at R. As P moves around C, R traces out a curve called the witch of Agnesi. For this curve; find (i) a parametrization (ii) its cartesian equation. 1.6) A cycloid is the plane curve traced out by a point on the circumference of a circle as it rolls without slipping along a straight line. Show that, if the straight

2 line is the x-axis and the circle has radius a>0, the cycloid can be parametrized as γ(t)=a(t-sint,1-cost). 1.7) Generalize the previous exercise by finding parametrizations of an epicycloid ( resp. hypocycloid ), the curve traced out by a point on the circumference of a circle as it rolls without slipping around the outside (resp. inside) of a fixed circle. 1.8) Show that γ(t)=(cos t- 1,sintcost,sint) is parametrization of the curve of intersection of the circular cylinder of radius 1 and axis the z-axis with the sphere of radius 1 and center (- 1,0,0). (This called Viviani s Curve). 1.9) For the logarithmic spiral γ(t) = (e t cos t, e t sin t), show that the angle between γ(t) and the tangent vector at γ(t) is independent of t. 1.10) Calculate the arc-length of the catenary γ(t)=(t,cosht) starting at the point (0,1). 1.11) Show that the following curves are unit-speed : (i) γ(t)=( 1 3 (1 + t) 3, 1 3 (1 t) 3, t ); (ii) γ(t)=( 4 5 cost,1-sint,- 3 5 cost). 1.1) Calculate the arc-length along the cycloid in Exercise 1.6 corresponding to one complete revolution of the circle.

3 WEEKLY PROGRAM We have given level curves, parametrized curves and some examples. Furthermore, we have given vector product and its geometric interpretation. We will continue with the fifth section in Do Carmo s book which is the local property of curves parametrized by arc length. We will give the fundamental theorem of the local theory of curves by considering a rigid motion. Before talking about the canonical form of curves, we will discuss arbitrary speed curves from O Neill s book. If time permits we will end up by global properties of plane curves. Solve the following exercises: From Do Carmo: 1.3.6, 1.4.1, 1.4., 1.4.3, 1.4.8, , 1.4.1, 1.5.1, 1.5. and the ones are not solved last time. Moreover prove the followings : 1)(i) If α = α(s) is a natural representation on I s, then s s 1 is the length of the arc α = α(s) between the points corresponding to f(s 1 ) and f(s ). (ii) If α = α(s) and α = α (s ) are natural representations of the same curve, then s = ±s +constant. ) Find the equations of the tangent line and normal plane to the curve α(t)= (1+t,-t,(1 + t) 3 ) at t=1. 3) Show that a curve α = α(t) of class is a straight line if α and α are linearly dependent for all t. 4) Show that a curve is a plane curve if all osculating planes have a common point of intersection. 5) Show that a curve is a general helix if and only if τ is constant where κ 0 κ and τ=0 whenever κ=0. 6) Show that along a curve α = α(s), (α, α, α ) = κ τ.

4 WEEKLY PROGRAM We have talked about the local property of curves parametrized by arc length. We have given the fundamental theorem of the local theory of curves by considering a rigid motion. Before talking about the canonical form of curves, we discuss arbitrary speed curves from O Neill s book. We might shortly talk about global properties of plane curves from Do Carmo but I prefer you to read it for further details. YOU ARE RESPONSIBLE OF THE CHAPTER CURVES UNTIL THE CHAPTER SURFACES FOR THE MIDTERM EXAM. GOOD LUCK WITH THE EXAM. After the exam, we will start with the second chapter called Surface Theory. We will talk about the appendix Continuity and Differentiability as a preparation. Please read the appendix before the lecture:) Afterwards, we will start Regular Surfaces. Showing any space is a regular surface directly from the definition may be sometimes very tiring so we will see some propositions which make such a difficulty easier. We are going to find an answer of the question which atlas should we use to study the surface?. If time permits, we will see that the transition maps of a smooth surface are smooth and the converse of this statement which are followed from Pressley s book. Solve the following exercises from Do Carmo s book :..1,..,..3,..4,..5,..8,..10,..11,..15,..16,..17, Solve the following exercises from Pressley s book:.6,.15, , 4.5, 4.9, 4.10, 4.11, 4.1.

5 Solve the following problems which are related to the curve theory and the introduction to the surface theory : 1. Introduce arc length as a parameter along α(t) = (e t cos t, e t sin t), e t ), < t <. If the principal normal lines of a curve C are the same as the binormal lines of a curve C, show that along C, α(κ + τ ) = κ, where α is constant. 3. If two curves have the same binormal lines at corresponding points, show that the curves are plane curves. 4. Find the derivative of f : R R, f(x, y) = (x sin y, y cos x) in the direction v = ( 1 5, 5 ) at p = ( π, π) Show that the function f : R R, f(x, y) = (e x cos y, e x sin y) satisfies the conditions of the inverse function theorem on R but is not 1-1 on R. 6. Show that the ellipsoid given by the equation x a surface. + y b + z c = 1 is a regular Solve the following exercises from Do Carmo s book: 1.5.4, 1.5.5, 1.5.9, 1.5.1, , a), b) and the ones which are not solved the last time.

6 WEEKLY PROGRAM 9-10 We start the second chapter called Surface Theory. We will talk about the appendix Continuity and Differentiability as a preparation.we will continue with the second chapter called Surface Theory. We are going to find an answer of the question which atlas should we use to study the surface?. Furthermore, we will see that the transition maps of a smooth surface are smooth and the converse of this statement which are followed from Pressley s book. Furthermore, we will see the change of parameters and differentiable functions on surfaces. We will see how a regular value of a differentiable map is related to a regular surface which requires the inverse function theorem. We are going to give some examples to understand all these concepts. Solve the following exercises from Do Carmo s book :..1,..,..3,..4,..5,..8,..10,..11,..15,..16,..17,.3.1,.3.,.3.3,.3.8,.3.13,.3.14, Solve the following exercises from Pressley s book: 4.4, 4.5, 4.9, 4.10, 4.11, 4.1. Solve the following problems which are related to the curve theory and the introduction to the surface theory : 1. Introduce arc length as a parameter along α(t) = (e t cos t, e t sin t), e t ), < t <

7 . If the principal normal lines of a curve C are the same as the binormal lines of a curve C, show that along C, α(κ + τ ) = κ, where α is constant. 3. If two curves have the same binormal lines at corresponding points, show that the curves are plane curves. 4. Find the derivative of f : R R, f(x, y) = (x sin y, y cos x) in the direction v = ( 1 5, 5 ) at p = ( π, π) Show that the function f : R R, f(x, y) = (e x cos y, e x sin y) satisfies the conditions of the inverse function theorem on R but is not 1-1 on R. 6. Show that the ellipsoid given by the equation x a surface. + y b + z c = 1 is a regular