# Intrinsic Geometry. 2 + g 22. du dt. 2 du. Thus, very short distances ds on the surface can be approximated by

Save this PDF as:

Size: px
Start display at page:

Download "Intrinsic Geometry. 2 + g 22. du dt. 2 du. Thus, very short distances ds on the surface can be approximated by"

## Transcription

1 Intrinsic Geometry The Fundamental Form of a Surface Properties of a curve or surface which depend on the coordinate space that curve or surface is embedded in are called extrinsic properties of the curve. For example, the slope of a tangent line is an extrinsic property since it depends on the coordinate system in which rises and runs are measured. In contrast, intrinsic properties of surfaces are properties that can be measured within the surface itself without any reference to a larger space. For example, the length of a curve is an intrinsic property of the curve, and thus, the length of a curve (t) = r (u (t) ; v (t)) ; t in [a; b] ; on a surface r (u; v) is an intrinsic property of both the curve itself and the surface that contains it. As we saw in the last section, the square of the speed of (t) is 2 ds = g 11 dt 2 du + 2g 12 dt in terms of the metric coe cients du dt 2 dv dv + g 22 dt dt g 11 = r u r u ; g 12 = r v r u ; and g 22 = r v r v Thus, very short distances ds on the surface can be approximated by (ds) 2 = g 11 (du) 2 + 2g 12 dudv + g 22 (dv) 2 (1) That is, if du and dv are su ciently small, then ds is the length of an in nites- 1

2 imally short curve on the surface itself. Equation (1) is the fundamental form of the surface, which intrinsic to a surface because it is related to distances on the surface itself. Moreover, any properties which can be derived solely from a surface s fundamental form are also intrinsic to the surface. EXAMPLE 1 Find the fundamental form of the right circular cylinder of radius R; which can be parameterized by r (u; v) = hr cos (u) ; R sin (u) ; vi Solution: Since r u = h R sin (u) ; R cos (u) ; 0i and r v = h0; 0; 1i ; the metric coe cients are g 11 = r u r u = R 2 sin 2 (u) + R 2 cos 2 (u) = R 2 g 12 = r u r v = = 0 g 22 = r v r v = = 1 Thus, ds 2 = R 2 du 2 + dv 2 : If the parameterization is orthogonal, then g 12 = r u r v = 0; so that ds 2 = g 11 du 2 + g 22 dv 2 For example, the xy-plane is parameterized by r (u; v) = hu; v; 0i ;which implies that r u = i and r v = j and that g 11 = g 22 = 1; g 12 = 0: The fundamental form for the plane is ds 2 = du 2 + dv 2 2

3 which is, in fact, the Pythagorean theorem. Moreover, distances are not altered when a sheet of paper is rolled up into a cylinder, which means that a cylinder should have the same fundamental form as the plane. Indeed, if R = 1 in example 1, then ds 2 = du 2 + dv 2 : EXAMPLE 2 Find the fundamental form of the sphere of radius R centered at the origin in the spherical coordinate parametrization r (; ) = hr sin () cos () ; R sin () sin () ; R cos ()i Solution: To do so, we rst compute the derivatives r and r : r = hr cos () cos () ; R cos () sin () ; R sin ()i r = h R sin () sin () ; R sin () cos () ; 0i It then follows that r r = R 2 cos 2 () cos 2 () + R 2 cos 2 () sin 2 () + R 2 sin 2 () = R 2 cos 2 () + R 2 sin 2 () = R 2 and thus, g 11 = R 2 : Moreover, spherical coordinates is an orthogonal parameterization, which means that r r = 0: Thus, g 12 = 0: Finally, g 22 is given by g 22 = r r = R 2 sin 2 () sin 2 () + R 2 sin 2 () cos 2 () = R 2 sin 2 () As a result, the fundamental form of the sphere of radius R is given by ds 2 = R 2 d 2 + R 2 sin 2 () d 2 (2) 3

4 That is, the hypotenuse of a spherical right triangle corresponds to a horizontal arc of length R sin () d and a vertical arc of length Rd: Check Your Reading: Is the Pythagorean theorem intrinsic to the xy-plane? Normal Curvature If P is a point on an orientable surface and if r (u; v) is an orthogonal parameterization of a coordinate patch on containing P; then there is (p; q) such that r (p; q) = P: Thus, for each in [0; 2] ; the curves (t) = r (p + t cos () ; q + t sin ()) pass through P (i.e., (0) = r (p; q) = P ) and the tangent vectors 0 (0) 4

5 point in a di erent direction in the tangent plane to at P: Indeed, every direction in the tangent plane is parallel to 0 (0) for some : Consequently, the curvatures of (t) at P represent all the di erent ways that the surface "curves away" from the tangent plane at P: Correspondingly, we de ne n () to be the component of the curvature of (t) at P in the direction of the unit normal n = r u r v kr u r v k In particular, the normal curvature satis es v 2 n () = 00 (0) n; so that n () = 00 (0) n k 0 (0)k2 (3) The unit surface normal n is not necessarily the same as the unit normal N for the curve. They may point in opposite directions or even be orthogonal e.g., the normal N to a curve in the xy-plane is in the xy-plane itself yet n = k for the xy-plane. Thus, the normal curvature n () is a measure of how much the surface is curving rather than how much the curve is curving. 5

6 That means that n () may be positive, negative, or even 0 e.g., the normal curvature of the xy-plane is 0 even though there are curves in the xy-plane with nonzero curvature. EXAMPLE 3 Use (??) to nd the normal curvature n () for the cylinder r (u; v) = hcos (u) ; sin (u) ; vi at the point r (0; 0) = (1; 0; 0) : Solution: The curves (t) = r (t cos () ; t sin ()) are given by (t) = hcos (t cos ()) ; sin (t cos ()) ; t sin ()i Di erentiation with respect to t leads to 0 = h sin (t cos ()) cos () ; cos (t cos ()) cos () ; sin ()i 00 = cos (t cos ()) cos 2 () ; sin (t cos ()) cos 2 () ; 0 so that at t = 0 we have 0 (0) = h sin (0) cos () ; cos (0) cos () ; sin ()i = h0; cos () ; sin ()i 00 (0) = cos (0) cos 2 () ; sin (0) cos 2 () ; 0 = cos 2 () ; 0; 0 The partial derivatives of r (u; v) are r u = h sin (u) ; cos (u) ; 0i ; r v = h0; 0; 1i which are both unit vectors. and r v = h0; 1; 0i = k: Thus, It follows that r u (0; 0) = h0; 1; 0i = j n = r u (0; 0) r v (0; 0) = j k = i Since 0 (0) is a unit vector, we have n () = 00 (0) n k = 0 (0)k2 cos 2 () ; 0; 0 i 1 = cos 2 () Because the normal curvature n () is a real-valued continuous function over in [0; 2] ; there is a largest 1 and a smallest 2 curvature at each point. The numbers 1 and 2 are known as the principle curvatures of the surface r (u; v) : In example 3, the largest possible curvature is 1 = cos 2 (=2) = 0 in the vertical direction, and the smallest possible curvature is 2 = cos 2 (0) = 1 6

7 in the horizontal direction. The average of the principal curvatures is called the Mean curvature of the surface and is denoted by H: It can be shown that if C is a su ciently smooth closed curve, then the surface with C as its boundary curve that has the smallest possible area must have a mean curvature of H = 0: For example, consider two parallel circles in 3 dimensional space. 7

8 The catenoid is the surface connecting the two circles that has the least surface area, and a catenoid also has a mean curvature of H = 0. Surfaces with a mean curvature of H = 0 are called minimal surfaces because they also have the least surface area for their given boundary. For example, a soap lm spanning a wire loop is a minimal surface. Moreover, minimal surfaces have a large number of applications in architecture, mathematics, and engineering. Check your Reading: Is the normal curvature of the cylinder ever positive? Gaussian Curvature In contrast to the mean curvature of a surface, the product of the principal curvatures is known as the Gaussian curvature of the surface, which is denoted by K. For example, the Gaussian curvature of the cylinder in example 2 is K = 1 0 = 0 and the Gaussian curvature of the Enneper minimal surface is K = 2 (u 2 + v 2 + 1) 2 2 (u 2 + v 2 + 1) 2 = 4 (u 2 + v 2 + 1) 4 As another example, consider that since the geodesics of a sphere with radius R are great circles of the sphere, they each have a curvature of = 1=R: Thus, the principal curvatures must be 1 = 1=R and 2 = 1=R; so that the curvature of a sphere of radius R is K = 1 R 2 Curvature is in general an extrinsic property of a surface. For example, mean curvature H is extrinsic because it depends on how the surface is embedded in a 3 (or higher) dimensional coordinate system. In contrast, the Gaussian curvature K is an intrinsic property of the surface. This is a truly remarkable theorem, one that was proven by the mathematician Karl Gauss in In particular, if we use the comma derivative notation to denote partial derivatives of metric coe cients with respect to u and v g 11;u and g 22;v then we can state Gauss remarkable theorem as follows: Theorem Egregium: Let r (u; v) be an orthogonal parameterization of a surface and let g = (g 11 g 22 ) 1=2 be the metric discrimanent of the fundamental form of r (u; v). Then K is an instrinsic property of the surface since can be expressed in terms of the metric 8

9 coe cients by K = g11;v g22;u g (4) Since K is given in terms of the metric coe cients of the fundamental form, it is an intrinsic property of the surface. blueexample 4 blackin example 2, we showed that the fundamental form of a sphere of radius R is given by ds 2 = R 2 d 2 + R 2 sin 2 () d 2 Use the Theorem Egregium to calculate the curvature of the sphere. Solution: Since g 11 = R 2 and g 22 = R 2 sin 2 () ; the comma derivatives are g 11; = 0 and g 22; = 2R 2 sin () cos () The metric discriminant is given by q g = R 2 R 2 sin 2 = R 2 sin () so that K = = = 2R 2 sin 1 2R 2 sin () 1 2R 2 sin () [ 0 R 2 sin () 0 (2 cos 2 sin 2R 2 sin cos () R 2 sin () = 1 R 2 The Theorem Egregium is simpler still and also more easily interpreted for a parameterization r (u; v) that is conformal. Speci cally, a conformal parameterization is orthogonal and has kr u k = kr v k ; which in turn implies that g = g 11 = g 22 9

10 The formula for K reduces in this case to gv K = 1 = ln gu @ ln v ln (g) = g v =g u ln (g) = g u =g. That is, K = 2 ln (g) 2 ln 2 for a conformal surface. (5) A coordinate patch on a right circular cone is para- EXAMPLE 5 meterized by r (u; v) = D p p e u cos 2v ; e u sin 2v ; e ue Find its fundamental form and use the Theorem Egregium to calculate the curvature of the surface. Solution: Since r u = e u cos p 2v ; e u sin p 2v ; e u and D r v = the metric coe cients are e up p 2 sin 2v ; e up p E 2 cos 2v ; 0 g 11 = r u r u = e 2u cos 2 (v) + e 2u sin 2 (v) + e 2u = 2e 2u g 22 = r v r v = 2e 2u sin 2 (v) + 2e 2u cos 2 (v) = 2e 2u Thus, the fundamental form is conformal ds 2 = 2e 2u du 2 + dv 2 and g = e 2u : Thus, " 2 ln e 2u # K = 2e 2 ln e 2 2 (2u) = 2e 2 2 = 0 10

11 Because K is intrinsic, the Gaussian curvature of a surface can be measured by inhabitants of the surface without regard to the larger space the surface is embedded in. For example, a person could measure the curvature of the earth s surface even if he was blind and could not see that the earth was situated in a larger cosmos. Similarly, if one surface is mapped to another without changing distances, then the two surfaces are said to be isometric. Intrinsic properties, such as Gaussian curvature, are invariant over isometries. For example, a plane has a Gaussian curvature of 0 and a cylinder likewise has a Gaussian curvature of 0, which follows from the fact that a plane can be rolled up into a cylinder without "stretching" or "tearing" that is, without changing distances between points. Check your Reading: What is the Mean curvature of a sphere of radius R? The Poincare Half-Plane Intrinsic geometry also means that we can de ne and study abstract surfaces that cannot be embedded in 3-dimensional space; or similarly, that we can determine the intrinsic geometric properties of space-time without having "spacetime" embedded in a larger space. For example, we can de ne a new geometry on the plane by giving it a non- Euclidean fundamental form. How would we know that it was truly di erent? This is exactly what Henri Poincare did when he introduced the fundamental form ds 2 = du2 + dv 2 v 2 (6) to the upper half of the uv-plane. The result is called the Poincare half-plane and is a model of hyperbolic geometry. If we use (6) to measure distances, then the geodesics are the vertical lines and semicircles parameterized by u = R tanh (t) + p; v = R sech (t) ; t in ( 1; 1) for R and p constant. For example, because distances become shorter as v increases under the Poincare metric (6), the distance from ( 1; 1) to (1; 1) along a semi-circle of radius p 2 centered at the origin is 1:7627; which is shorter than 11

12 the distance of 2 from ( 1; 1) to (1; 1) along the line v = 1: Thus, vertical lines and semi-circles centered on the u-axis are the "straight lines" in the Poincare half-plane. Through a point P not on a semi-circle, there are in nitely many other semi-circles centered on the x-axis that pass through P: Thus, in the Poincare half plane, there are in nitely many parallel lines to a given line l through a point P not on l: Finally, we can use the Theorem Egregium to calculuate the curvature of the Poincare half-plane. In particular, since ds 2 = v 2 du 2 + v 2 dv 2 ; the metric 12

13 coe cients are g 11 = g 22 = v 2 : Thus, g 11;v = 2v 3 ; g 22;u = 0; and g = g 11 g 22 = v 4 The theorem Egregium thus yields K = 2 p v = 2v 1 = 2v 2 = 1 2v 2 2v 3 p v 2v (0) 0 p v 4 Thus, the curvature of the hyperbolic plane is K = 1: That is, the hyperbolic plane is a surface of constant negative curvature, and as a result, it cannot be studied as a surface in ordinary 3 dimensional space. Instead, all information about the hyperbolic plane must come from the intrinsic properties derived from its fundamental form. Exercises Find the rst fundamental form of the given surfaces. Explain the relationship of the fundamental form to the given surface. 1. r = hu; v; ui 2. r = hu; v; u + vi 3. r = hv sin (u) ; v cos (u) ; vi 4. r = hv sin (u) ; v; v cos (u)i 5. r = hsin (u) cos (v) ; cos (u) ; sin (u) sin (v)i 6. r = hsin (v) sin (u) ; cos (v) sin (u) ; cos (u)i 7. r = hsin (u) cosh (v) ; sinh (v) ; cos (u) cosh (v)i 8. r = hsin (u) cosh (v) ; sin (u) sinh (v) ; cos (u)i Show that each parameterization is orthogonal and then determine the normal curvature of the surface. Then determine the principal curvatures, the Mean curvature, and the Gaussian curvature of the surface. Which surfaces are minimal surfaces? Which are Gaussian at (i.e, K = 0)? 9. r (u; v) = hv; sin (u) ; cos (u)i 10. r (u; v) = hcos (u) ; v; sin (u)i 11. r (u; v) = hu; v; ui 12. r (u; v) = u; v 2 ; u 13. r (u; v) = hu cos (v) ; u sin (v) ; ui 14. r (u; v) = u 2 cos (v) ; u 2 sin (v) ; u 15. r (u; v) = he u ; e u ; vi 16. r (u; v) = e u ; e u ; v r (u; v) = he v sin (u) ; e v cos (u) ; e v i 18. r (u; v) = he u cos (v) ; e u sin (v) ; ui Use the theorem Egregium to determine the Gaussian curvature of a surface with the given fundamental form. 19. ds 2 = du 2 + e 4u dv ds 2 = v 2 du 2 + v 2 dv ds 2 = v 2 du 2 + dv ds 2 = 4u 2 du 2 + 8v 2 dv ds 2 = v 4 du 2 + 4v dv ds 2 = du 2 + sec 2 (v) dv 2 13

14 25. Find the normal curvature and principle curvatures of z = x 2 y 2 at (0; 0; 0) : Does the function have an extremum or a saddle point at (0; 0; 0)? 26. Suppose a surface has a fundamental form of ds 2 = du 2 + e 2ku dv 2 where k is a constant. What is the Gaussian curvature of the surface? 27. In latitude-longitude coordinates, a sphere of radius R centered at the origin has a fundamental form of ds 2 = R 2 d' 2 + R 2 cos 2 (') d 2 Apply the Theorem Egregium to this form to determine the curvature of the sphere intrinsically. 28. Show that the surface of revolution D r (u; v) = u; p R 2 u 2 cos (v) ; p E R 2 x 2 sin (v) is a sphere of radius R: Determine the fundamental form of the surface and use it to nd K: 29. The helicoid is the surface parametrized by What is its fundamental form? r (u; v) = hsinh (v) cos (u) ; sinh (v) sin (u) ; ui 14

15 30. A catenoid is the surface parametrized by r (u; v) = hcosh (v) cos (u) ; cosh (v) sin (u) ; vi Show that it has the same fundamental form as the helicoid (exercise 29). What does this mean? 31. Show that the helicoid is a minimal surface. 32. Show that the catenoid is a minimal surface. 33. The surface of revolution of y = f (x) about the x-axis can be parameterized by r (u; v) = hv; f (v) cos (u) ; f (v) sin (u)i Find the fundamental form and then use the Theorem Egregium to show that the curvature of a surface of revolution is given by f 00 (v) K = f (v) 1 + [f 0 (v)] The pseudosphere is a surface of revolution parameterized by D h u ie r (u; v) = sin (u) cos (v) ; sin (u) sin (v) ; cos (u) + ln tan 2 Determine the fundamental form and then use the Theorem Egregium to show that K = 1: 35. Euler s Formula: Show that if 1 occurs along r u (as it does along the cylinder, for instance), then n () = 1 cos 2 () + 2 sin 2 () 36. Write to Learn: Write an essay in which you prove mathematically that a minimal surface that is Gaussian at must be a region in a plane. Then use sketches and concepts to explain in your own words why Gaussian at minimal surfaces must be planar. 37. Write to Learn: Write a short essay in which you use the following steps to prove that the Gaussian curvature K satis es (n u n v ) n = K kr u r v k 1. (a) Explain why r u n = 0, and then show that this implies that r uu n = r u n u ; and r uv n = r u n v Also, show that r v n = 0 implies that r vu n = r v n u and r vv n = r v n v : (b) A cross product identity says that if A; B; C; and D are vectors, then (A B) (C D) = (A C) (B D) (B C) (A D) Apply this identity to the quantity (n u n v ) (r u r v ) 15

16 (c) Use (a) and (b) to show that (n u n v ) (r u r v ) = (r uu n) (r vv n) (r uv n) 2 and that (r u r v ) (r u r v ) = kr u k 2 kr v k 2 (r u r v ) 2 : (d) Conclude by explaining why (n u n v ) (r u r v ) kr u r v k 2 and then derive the desired result. = K 38. Show that if we de ne K to satisfy the relationship n u n v = K (r u r v ) then K is the Gaussian Curvature of the surface. You may assume that r (u; v) is orthogonal, and you may want to use the steps (b) and (c) in exercise (Extends Exercise 41 in section 3-2). Stereographic projecton of a sphere of radius R leads to a parameterization of the form * 2Ru r (u; v) = u 2 + v ; 2Rv u 2 + v ; R + u2 + v 2 1 u 2 + v 2 (7) + 1 Show that kr (u; v)k = R for all (u; v) : Then show that the fundamental form is conformal, and calculate the curvature of the surface. 40. Find the principle curvatures of the Enneper Minimal Surface, which is parameterized by r (u; v) = u 1 3 u3 + uv 2 ; 1 3 v3 v u 2 v; u 2 v 2 Show that it is in fact a minimal surface. Also, show that the parameterization is conformal and use the result to calculate the Gaussian curvature. 41. Any su ciently di erentiable parameterization of a surface can be transformed into a conformal parameterization. Let s explore this idea for surfaces of revolution with parameterizations of the form r (u; v) = hv; f (v) cos (u) ; f (v) sin (u)i In particular, show that if y (v) satis es the di erential equation then the parameterization dy dv = f (y) q 1 + [f 0 (y)] 2 r (u; y (v)) = hy (v) ; f (y (v)) cos (u) ; f (y (v)) sin (u)i 16

17 is a parameterization of the original surface that is also conformal. 42. Use the result in exercise 41 to nd a conformal parameterization of the right circular cone, which is a surface of revolution of the form r (u; v) = hv; v cos (u) ; v sin (u)i Then use the result in exercise 39 to compute the Gaussian curvature of the right circular cone. 17

### Surface Normals and Tangent Planes

Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to

### Components of Acceleration

Components of Acceleration Part 1: Curvature and the Unit Normal In the last section, we explored those ideas related to velocity namely, distance, speed, and the unit tangent vector. In this section,

### Parametric Equations

Parametric Equations Part 1: Vector-Valued Functions Now that we have introduced and developed the concept of a vector, we are ready to use vectors to de ne functions. To begin with, a vector-valued function

### Partial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant.

Partial Derivatives Partial Derivatives Just as derivatives can be used to eplore the properties of functions of 1 variable, so also derivatives can be used to eplore functions of 2 variables. In this

### Change of Variables in Double Integrals

Change of Variables in Double Integrals Part : Area of the Image of a egion It is often advantageous to evaluate (x; y) da in a coordinate system other than the xy-coordinate system. In this section, we

### 88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

### Coordinate Transformation

Coordinate Transformation Coordinate Transformations In this chater, we exlore maings where a maing is a function that "mas" one set to another, usually in a way that reserves at least some of the underlyign

### Figure 1: Volume between z = f(x, y) and the region R.

3. Double Integrals 3.. Volume of an enclosed region Consider the diagram in Figure. It shows a curve in two variables z f(x, y) that lies above some region on the xy-plane. How can we calculate the volume

### The Math Circle, Spring 2004

The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

### Functions of 2 Variables

Functions of 2 Variables Functions and Graphs In the last chapter, we extended di erential calculus to vector-valued functions. In this chapter, we extend calculus primarily to functions of two variables,

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

### The Cross Product. = u 1v 2 u 2 v 1 (1) ; u 1 u 2. ; u 3 u 1 v 3 v 1

The Cross Product Part 1: Determinants and the Cross Product In this section, we introduce the cross product of two vectors. However, the cross product is based on the theory of determinants, so we begin

### Parabolic surfaces in hyperbolic space with constant Gaussian curvature

Parabolic surfaces in hyperbolic space with constant Gaussian curvature Rafael López Departamento de Geometría y Topología Universidad de Granada 807 Granada (Spain) e-mail: rcamino@ugr.es eywords: hyperbolic

### The line segment can be parametrized as x = 1 + t, y = 1 + 2t, z = 1 + 3t 0 t 1. Then dx = dt, dy = 2dt, dz = 3dt so the integral becomes

MATH rd Midterm Review Solutions. Evaluate xz dx z dy + y dz where is the line segment from (,, ) to (,, ). The line segment can be parametrized as x = + t, y = + t, z = + t where t. Then dx = dt, dy =

### Vector-Valued Functions The Calculus of Vector-Valued Functions Motion in Space Curvature Tangent and Normal Vect. Calculus. Vector-Valued Functions

Calculus Vector-Valued Functions Outline 1 Vector-Valued Functions 2 The Calculus of Vector-Valued Functions 3 Motion in Space 4 Curvature 5 Tangent and Normal Vectors 6 Parametric Surfaces Vector Valued

### Lecture 8. Metrics. 8.1 Riemannian Geometry. Objectives: More on the metric and how it transforms. Reading: Hobson, 2.

Lecture 8 Metrics Objectives: More on the metric and how it transforms. Reading: Hobson, 2. 8.1 Riemannian Geometry The interval ds 2 = g αβ dx α dx β, is a quadratic function of the coordinate differentials.

### Midterm Exam I, Calculus III, Sample A

Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of

### Section 3.1 Calculus of Vector-Functions

Section 3.1 Calculus of Vector-Functions De nition. A vector-valued function is a rule that assigns a vector to each member in a subset of R 1. In other words, a vector-valued function is an ordered triple

### AB2.5: Surfaces and Surface Integrals. Divergence Theorem of Gauss

AB2.5: urfaces and urface Integrals. Divergence heorem of Gauss epresentations of surfaces or epresentation of a surface as projections on the xy- and xz-planes, etc. are For example, z = f(x, y), x =

### The Three Reflections Theorem

The Three Reflections Theorem Christopher Tuffley Institute of Fundamental Sciences Massey University, Palmerston North 24th June 2009 Christopher Tuffley (Massey University) The Three Reflections Theorem

### Geometric Modelling Summer 2016

Geometric Modelling Summer 2016 Prof. Dr. Hans Hagen http://gfx.uni-kl.de/~gm Prof. Dr. Hans Hagen Geometric Modelling Summer 2016 1 Ane Spaces, Elliptic Prof. Dr. Hans Hagen Geometric Modelling Summer

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### In this chapter we turn to surfaces in general. We discuss the following topics. Describing surfaces with equations and parametric descriptions.

Chapter 4 Surfaces In this chapter we turn to surfaces in general. We discuss the following topics. Describing surfaces with equations and parametric descriptions. Some constructions of surfaces: surfaces

### MVE041 Flervariabelanalys

MVE041 Flervariabelanalys 2015-16 This document contains the learning goals for this course. The goals are organized by subject, with reference to the course textbook Calculus: A Complete Course 8th ed.

### W CURVES IN MINKOWSKI SPACE TIME

Novi Sad J Math Vol 3, No, 00, 55-65 55 W CURVES IN MINKOWSKI SPACE TIME Miroslava Petrović Torgašev 1, Emilija Šućurović 1 Abstract In this paper we complete a classification of W curves in Minkowski

### When Is the Sum of the Measures of the Angles of a Triangle Equal to 180º?

Sum of the Measures of Angles Unit 9 Non-Euclidean Geometries When Is the Sum of the Measures of the Angles of a Triangle Equal to 180º? Overview: Objective: This activity illustrates the need for Euclid

### 1 CHAPTER 18 THE CATENARY. 18.1 Introduction

1 CHAPER 18 HE CAENARY 18.1 Introduction If a flexible chain or rope is loosely hung between two fixed points, it hangs in a curve that looks a little like a parabola, but in fact is not quite a parabola;

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### AB2.2: Curves. Gradient of a Scalar Field

AB2.2: Curves. Gradient of a Scalar Field Parametric representation of a curve A a curve C in space can be represented by a vector function r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k This is called

### Section 2.1 Rectangular Coordinate Systems

P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

### MA 105 D1 Lecture 20

MA 105 D1 Lecture 20 Ravi Raghunathan Department of Mathematics Autumn 2014, IIT Bombay, Mumbai Orientation and the Jordan curve theorem Green s Theorem Various examples Green s Theorem Other forms of

### 14.11. Geodesic Lines, Local Gauss-Bonnet Theorem

14.11. Geodesic Lines, Local Gauss-Bonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize

### MULTIPLE INTEGRALS. h 2 (y) are continuous functions on [c, d] and let f(x, y) be a function defined on R. Then

MULTIPLE INTEGALS 1. ouble Integrals Let be a simple region defined by a x b and g 1 (x) y g 2 (x), where g 1 (x) and g 2 (x) are continuous functions on [a, b] and let f(x, y) be a function defined on.

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =

Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Review Sheet for Test 1

Review Sheet for Test 1 Math 261-00 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And

### FINAL EXAM SOLUTIONS Math 21a, Spring 03

INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic

### AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

### Applications of Integration Day 1

Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

### Lecture L6 - Intrinsic Coordinates

S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6 - Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed

### Solutions - Homework sections 17.7-17.9

olutions - Homework sections 7.7-7.9 7.7 6. valuate xy d, where is the triangle with vertices (,, ), (,, ), and (,, ). The three points - and therefore the triangle between them - are on the plane x +

### A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

### Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

### Math 497C Sep 9, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say

### PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS

PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,

### If f is continuous on [a, b], then the function g defined by. f (t) dt. is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function g defined by g(x) = x a f (t) dt a x b is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

### 3.6 Cylinders and Quadric Surfaces

3.6 Cylinders and Quadric Surfaces Objectives I know the definition of a cylinder. I can name the 6 quadric surfaces, write their equation, and sketch their graph. Let s take stock in the types of equations

### Solutions to Vector Calculus Practice Problems

olutions to Vector alculus Practice Problems 1. Let be the region in determined by the inequalities x + y 4 and y x. Evaluate the following integral. sinx + y ) da Answer: The region looks like y y x x

### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

### DIFFERENTIAL GEOMETRY HW 2

DIFFERENTIAL GEOMETRY HW 2 CLAY SHONKWILER 2 Prove that the only orientation-reversing isometries of R 2 are glide reflections, that any two glide-reflections through the same distance are conjugate by

### Chapter 17. Review. 1. Vector Fields (Section 17.1)

hapter 17 Review 1. Vector Fields (Section 17.1) There isn t much I can say in this section. Most of the material has to do with sketching vector fields. Please provide some explanation to support your

### Extra Problems for Midterm 2

Extra Problems for Midterm Sudesh Kalyanswamy Exercise (Surfaces). Find the equation of, and classify, the surface S consisting of all points equidistant from (0,, 0) and (,, ). Solution. Let P (x, y,

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### Solutions to Practice Problems for Test 4

olutions to Practice Problems for Test 4 1. Let be the line segmentfrom the point (, 1, 1) to the point (,, 3). Evaluate the line integral y ds. Answer: First, we parametrize the line segment from (, 1,

### (x) = lim. x 0 x. (2.1)

Differentiation. Derivative of function Let us fi an arbitrarily chosen point in the domain of the function y = f(). Increasing this fied value by we obtain the value of independent variable +. The value

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### MATH 52 FINAL EXAM SOLUTIONS (AUTUMN 2003)

MATH 5 FINAL EXAM OLUTION (AUTUMN 3). Evaluate the integral by reversing the order of integration olution. 3 3y e x dxdy = = 3 3y 3 x/3 3 = ex 3 [ e x y e x dxdy. e x dxdy ] y=x/3 y= = e9 dx =. Rewrite

### Mathematics 433/533; Class Notes. Richard Koch

Mathematics 433/533; Class Notes Richard Koch March 4, 5 ii Contents Preface vii Curves. Introduction..................................... Arc Length.....................................3 Reparameterization................................

### SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

### Lines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line

Lines and planes in space (Sect. 2.5) Lines in space (Toda). Review: Lines on a plane. The equations of lines in space: Vector equation. arametric equation. Distance from a point to a line. lanes in space

### Solution: 2. Sketch the graph of 2 given the vectors and shown below.

7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit

### Fundamental Theorems of Vector Calculus

Fundamental Theorems of Vector Calculus We have studied the techniques for evaluating integrals over curves and surfaces. In the case of integrating over an interval on the real line, we were able to use

### Math 497C Sep 17, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 17, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 4 1.9 Curves of Constant Curvature Here we show that the only curves in the plane with constant curvature are lines and circles.

### 7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### HYPERBOLIC TRIGONOMETRY

HYPERBOLIC TRIGONOMETRY STANISLAV JABUKA Consider a hyperbolic triangle ABC with angle measures A) = α, B) = β and C) = γ. The purpose of this note is to develop formulae that allow for a computation of

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### Chapter 2. Parameterized Curves in R 3

Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,

### Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing.

Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing. The amplitude of the sun (at sunrise, say) is the direction you look to see the sun come up. If it s rising exactly

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### Green s Theorem Calculus III (MATH 2203)

Green s Theorem Calculus III (MATH 223) S. F. Ellermeyer November 2, 213 Green s Theorem gives an equality between the line integral of a vector field (either a flow integral or a flux integral) around

### Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve. cosh y = x + sin y + cos y

Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve at the point (0, 0) is cosh y = x + sin y + cos y Answer : y = x Justification: The equation of the

### Sample Problems. 3. Find the missing leg of the right triangle shown on the picture below.

Lecture Notes The Pythagorean Theorem page 1 Sample Problems 1. Could the three line segments given below be the three sides of a right triangle? Explain your answer. a) 6 cm; 10 cm; and 8 cm b) 7 ft,

### 2 Topics in 3D Geometry

2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and

### Section 8 Inverse Trigonometric Functions

Section 8 Inverse Trigonometric Functions Inverse Sine Function Recall that for every function y = f (x), one may de ne its INVERSE FUNCTION y = f 1 (x) as the unique solution of x = f (y). In other words,

### Chapter 1. Vector valued functions

Chapter 1. Vector valued functions This material is covered in Thomas (chapters 12 & 13 in the 11th edition, or chapter 10 in the 10th edition). The approach here will be only slightly different and we

### Homework 3 Model Solution Section

Homework 3 Model Solution Section 12.6 13.1. 12.6.3 Describe and sketch the surface + z 2 = 1. If we cut the surface by a plane y = k which is parallel to xz-plane, the intersection is + z 2 = 1 on a plane,

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Section 14.5 Directional derivatives and gradient vectors

Section 4.5 Directional derivatives and gradient vectors (3/3/08) Overview: The partial derivatives f ( 0, 0 ) and f ( 0, 0 ) are the rates of change of z = f(,) at ( 0, 0 ) in the positive - and -directions.

### Straight Line motion with rigid sets

Straight ine motion with rigid sets arxiv:40.4743v [math.mg] 9 Jan 04 Robert Connelly and uis Montejano January, 04 Abstract If one is given a rigid triangle in the plane or space, we show that the only

### APPLICATION OF DERIVATIVES

6. Overview 6.. Rate of change of quantities For the function y f (x), d (f (x)) represents the rate of change of y with respect to x. dx Thus if s represents the distance and t the time, then ds represents

### Sphere centered at the origin.

A Quadratic surfaces In this appendix we will study several families of so-called quadratic surfaces, namely surfaces z = f(x, y) which are defined by equations of the type A + By 2 + Cz 2 + Dxy + Exz

### Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011

Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often

7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 17 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### Related Rates Problems

These notes closely follow the presentation of the material given in James Stewart s textbook Calculus, Concepts and Contexts (2nd edition). These notes are intended primarily for in-class presentation

### alternate interior angles

alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate