Multivariable Calculus, at UC Berkeley, in the Summer of 2011
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Lecture 11 Line Integrals of Vector-Valued Functions (Cont'd)
Lecture 11 Line integrals of vector-valued functions (cont’d) In the previous lecture, we considered the following physical situation: A force, F(x), which is not necessarily constant in space, is acting on a mass m, as the mass moves along a curve C from point P to point Q as shown in the diagram below. z Q P m F(x(t)) x(t) C y x The goal is to compute the total amount of work W done by the force. Clearly the constant force/straight line displacement formula, W = F · d , (1) where F is force and d is the displacement vector, does not apply here. But the fundamental idea, in the “Spirit of Calculus,” is to break up the motion into tiny pieces over which we can use Eq. (1 as an approximation over small pieces of the curve. We then “sum up,” i.e., integrate, over all contributions to obtain W . The total work is the line integral of the vector field F over the curve C: W = F · dx . (2) ZC Here, we summarize the main steps involved in the computation of this line integral: 1. Step 1: Parametrize the curve C We assume that the curve C can be parametrized, i.e., x(t)= g(t) = (x(t),y(t), z(t)), t ∈ [a, b], (3) so that g(a) is point P and g(b) is point Q. From this parametrization we can compute the velocity vector, ′ ′ ′ ′ v(t)= g (t) = (x (t),y (t), z (t)) . (4) 1 2. Step 2: Compute field vector F(g(t)) over curve C F(g(t)) = F(x(t),y(t), z(t)) (5) = (F1(x(t),y(t), z(t), F2(x(t),y(t), z(t), F3(x(t),y(t), z(t))) , t ∈ [a, b] . -
The Mathematics of Fels Sculptures
THE MATHEMATICS OF FELS SCULPTURES DAVID FELS AND ANGELO B. MINGARELLI Abstract. We give a purely mathematical interpretation and construction of sculptures rendered by one of the authors, known herein as Fels sculptures. We also show that the mathematical framework underlying Ferguson's sculpture, The Ariadne Torus, may be considered a special case of the more general constructions presented here. More general discussions are also presented about the creation of such sculptures whether they be virtual or in higher dimensional space. Introduction Sculptors manifest ideas as material objects. We use a system wherein an idea is symbolized as a solid, governed by a set of rules, such that the sculpture is the expressed material result of applying rules to symbols. The underlying set of rules, being mathematical in nature, may thus lead to enormous abstraction and although sculptures are generally thought of as three dimensional objects, they can be created in four and higher dimensional (unseen) spaces with various projections leading to new and pleasant three dimensional sculptures. The interplay of mathematics and the arts has, of course, a very long and old history and we cannot begin to elaborate on this matter here. Recently however, the problem of creating mathematical programs for the construction of ribbed sculptures by Charles Perry was considered in [3]. For an insightful paper on topological tori leading to abstract art see also [5]. Spiral and twirling sculptures were analysed and constructed in [1]. This paper deals with a technique for sculpting works mostly based on wood (but not necessarily restricted to it) using abstract ideas based on twirls and tori, though again, not limited to them. -
Quadratic Approximation at a Stationary Point Let F(X, Y) Be a Given
Multivariable Calculus Grinshpan Quadratic approximation at a stationary point Let f(x; y) be a given function and let (x0; y0) be a point in its domain. Under proper differentiability conditions one has f(x; y) = f(x0; y0) + fx(x0; y0)(x − x0) + fy(x0; y0)(y − y0) 1 2 1 2 + 2 fxx(x0; y0)(x − x0) + fxy(x0; y0)(x − x0)(y − y0) + 2 fyy(x0; y0)(y − y0) + higher−order terms: Let (x0; y0) be a stationary (critical) point of f: fx(x0; y0) = fy(x0; y0) = 0. Then 2 2 f(x; y) = f(x0; y0) + A(x − x0) + 2B(x − x0)(y − y0) + C(y − y0) + higher−order terms, 1 1 1 1 where A = 2 fxx(x0; y0);B = 2 fxy(x0; y0) = 2 fyx(x0; y0), and C = 2 fyy(x0; y0). Assume for simplicity that (x0; y0) = (0; 0) and f(0; 0) = 0. [ This can always be achieved by translation: f~(x; y) = f(x0 + x; y0 + y) − f(x0; y0). ] Then f(x; y) = Ax2 + 2Bxy + Cy2 + higher−order terms. Thus, provided A, B, C are not all zero, the graph of f near (0; 0) resembles the quadric surface z = Ax2 + 2Bxy + Cy2: Generically, this quadric surface is either an elliptic or a hyperbolic paraboloid. We distinguish three scenarios: * Elliptic paraboloid opening up, (0; 0) is a point of local minimum. * Elliptic paraboloid opening down, (0; 0) is a point of local maximum. * Hyperbolic paraboloid, (0; 0) is a saddle point. It should certainly be possible to tell which case we are dealing with by looking at the coefficients A, B, and C, and this is the idea behind the Second Partials Test. -
Chapter 11. Three Dimensional Analytic Geometry and Vectors
Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1. -
Pencils of Quadrics
Europ. J. Combinatorics (1988) 9, 255-270 Intersections in Projective Space II: Pencils of Quadrics A. A. BRUEN AND J. w. P. HIRSCHFELD A complete classification is given of pencils of quadrics in projective space of three dimensions over a finite field, where each pencil contains at least one non-singular quadric and where the base curve is not absolutely irreducible. This leads to interesting configurations in the space such as partitions by elliptic quadrics and by lines. 1. INTRODUCTION A pencil of quadrics in PG(3, q) is non-singular if it contains at least one non-singular quadric. The base of a pencil is a quartic curve and is reducible if over some extension of GF(q) it splits into more than one component: four lines, two lines and a conic, two conics, or a line and a twisted cubic. In order to contain no points or to contain some line over GF(q), the base is necessarily reducible. In the main part of this paper a classification is given of non-singular pencils ofquadrics in L = PG(3, q) with a reducible base. This leads to geometrically interesting configur ations obtained from elliptic and hyperbolic quadrics, such as spreads and partitions of L. The remaining part of the paper deals with the case when the base is not reducible and is therefore a rational or elliptic quartic. The number of points in the elliptic case is then governed by the Hasse formula. However, we are able to derive an interesting combi natorial relationship, valid also in higher dimensions, between the number of points in the base curve and the number in the base curve of a dual pencil. -
Swimming in Spacetime: Motion by Cyclic Changes in Body Shape
RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing by ⌬ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding fixed, then increasing by (1). -
Art Concepts for Kids: SHAPE & FORM!
Art Concepts for Kids: SHAPE & FORM! Our online art class explores “shape” and “form” . Shapes are spaces that are created when a line reconnects with itself. Forms are three dimensional and they have length, width and depth. This sweet intro song teaches kids all about shape! Scratch Garden Value Song (all ages) https://www.youtube.com/watch?v=coZfbTIzS5I Another great shape jam! https://www.youtube.com/watch?v=6hFTUk8XqEc Cookie Monster eats his shapes! https://www.youtube.com/watch?v=gfNalVIrdOw&list=PLWrCWNvzT_lpGOvVQCdt7CXL ggL9-xpZj&index=10&t=0s Now for the activities!! Click on the links in the descriptions below for the step by step details. Ages 2-4 Shape Match https://busytoddler.com/2017/01/giant-shape-match-activity/ Shapes are an easy art concept for the littles. Lots of their environment gives play and art opportunities to learn about shape. This Matching activity involves tracing the outside shape of blocks and having littles match the block to the outline. It can be enhanced by letting your little ones color the shapes emphasizing inside and outside the shape lines. Block Print Shapes https://thepinterestedparent.com/2017/08/paul-klee-inspired-block-printing/ Printing is a great technique for young kids to learn. The motion is like stamping so you teach little ones to press and pull rather than rub or paint. This project requires washable paint and paper and block that are not natural wood (which would stain) you want blocks that are painted or sealed. Kids can look at Paul Klee art for inspiration in stacking their shapes into buildings, or they can chose their own design Ages 4-6 Lois Ehlert Shape Animals http://www.momto2poshlildivas.com/2012/09/exploring-shapes-and-colors-with-color.html This great project allows kids to play with shapes to make animals in the style of the artist Lois Ehlert. -
Automatic Estimation of Sphere Centers from Images of Calibrated Cameras
Automatic Estimation of Sphere Centers from Images of Calibrated Cameras Levente Hajder 1 , Tekla Toth´ 1 and Zoltan´ Pusztai 1 1Department of Algorithms and Their Applications, Eotv¨ os¨ Lorand´ University, Pazm´ any´ Peter´ stny. 1/C, Budapest, Hungary, H-1117 fhajder,tekla.toth,[email protected] Keywords: Ellipse Detection, Spatial Estimation, Calibrated Camera, 3D Computer Vision Abstract: Calibration of devices with different modalities is a key problem in robotic vision. Regular spatial objects, such as planes, are frequently used for this task. This paper deals with the automatic detection of ellipses in camera images, as well as to estimate the 3D position of the spheres corresponding to the detected 2D ellipses. We propose two novel methods to (i) detect an ellipse in camera images and (ii) estimate the spatial location of the corresponding sphere if its size is known. The algorithms are tested both quantitatively and qualitatively. They are applied for calibrating the sensor system of autonomous cars equipped with digital cameras, depth sensors and LiDAR devices. 1 INTRODUCTION putation and memory costs. Probabilistic Hough Transform (PHT) is a variant of the classical HT: it Ellipse fitting in images has been a long researched randomly selects a small subset of the edge points problem in computer vision for many decades (Prof- which is used as input for HT (Kiryati et al., 1991). fitt, 1982). Ellipses can be used for camera calibra- The 5D parameter space can be divided into two tion (Ji and Hu, 2001; Heikkila, 2000), estimating the pieces. First, the ellipse center is estimated, then the position of parts in an assembly system (Shin et al., remaining three parameters are found in the second 2011) or for defect detection in printed circuit boards stage (Yuen et al., 1989; Tsuji and Matsumoto, 1978). -
Exploding the Ellipse Arnold Good
Exploding the Ellipse Arnold Good Mathematics Teacher, March 1999, Volume 92, Number 3, pp. 186–188 Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). More than 200 books, videos, software, posters, and research reports are available through NCTM’S publication program. Individual members receive a 20% reduction off the list price. For more information on membership in the NCTM, please call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright May 1991 by the National Council of Teachers of Mathematics. All rights reserved. Arnold Good, Framingham State College, Framingham, MA 01701, is experimenting with a new approach to teaching second-year calculus that stresses sequences and series over integration techniques. eaders are advised to proceed with caution. Those with a weak heart may wish to consult a physician first. What we are about to do is explode an ellipse. This Rrisky business is not often undertaken by the professional mathematician, whose polytechnic endeavors are usually limited to encounters with administrators. Ellipses of the standard form of x2 y2 1 5 1, a2 b2 where a > b, are not suitable for exploding because they just move out of view as they explode. Hence, before the ellipse explodes, we must secure it in the neighborhood of the origin by translating the left vertex to the origin and anchoring the left focus to a point on the x-axis. -
Sculptures in S3
Sculptures in S3 Saul Schleimer and Henry Segerman∗ Abstract We describe the construction of a number of sculptures. Each is based on a geometric design native to the three- sphere: the unit sphere in four-dimensional space. Via stereographic projection, we transfer the design to three- dimensional space. All of the sculptures are then fabricated by the 3D printing service Shapeways. 1 Introduction The three-sphere, denoted S3, is a three-dimensional analog of the ordinary two-dimensional sphere, S2. In n+ general, the n–dimensional sphere is a subset of R 1 as follows: n n+1 2 2 2 S = f(x0;x1;:::;xn) 2 R j x0 + x1 + ··· + xn = 1g: Thus S2 can be seen as the usual unit sphere in R3. Visualising objects in dimensions higher than three is non-trivial. However for S3 we can use stereographic projection to reduce the dimension from four to three. n n n Let N = (0;:::;0;1) be the north pole of S . We define stereographic projection r : S − fNg ! R by x0 x1 xn−1 r(x0;x1;:::;xn) = ; ;:::; : 1 − xn 1 − xn 1 − xn See [1, page 27]. Figure 1a displays the one-dimensional case; this is also a cross-section of the n– dimensional case. For any point (x;y) 2 S1 − fNg draw the straight line L between N and (x;y). Then L meets R1 at a single point; this is r(x;y). N x 1−y (x;y) (a) Stereographic projection from S1 − (b) Two-dimensional stereographic projection applied to the Earth. -
The Divergence As the Rate of Change in Area Or Volume
The divergence as the rate of change in area or volume Here we give a very brief sketch of the following fact, which should be familiar to you from multivariable calculus: If a region D of the plane moves with velocity given by the vector field V (x; y) = (f(x; y); g(x; y)); then the instantaneous rate of change in the area of D is the double integral of the divergence of V over D: ZZ ZZ rate of change in area = r · V dA = fx + gy dA: D D (An analogous result is true in three dimensions, with volume replacing area.) To see why this should be so, imagine that we have cut D up into a lot of little pieces Pi, each of which is rectangular with width Wi and height Hi, so that the area of Pi is Ai ≡ HiWi. If we follow the vector field for a short time ∆t, each piece Pi should be “roughly rectangular”. Its width would be changed by the the amount of horizontal stretching that is induced by the vector field, which is difference in the horizontal displacements of its two sides. Thisin turn is approximated by the product of three terms: the rate of change in the horizontal @f component of velocity per unit of displacement ( @x ), the horizontal distance across the box (Wi), and the time step ∆t. Thus ! @f ∆W ≈ W ∆t: i @x i See the figure below; the partial derivative measures how quickly f changes as you move horizontally, and Wi is the horizontal distance across along Pi, so the product fxWi measures the difference in the horizontal components of V at the two ends of Wi. -
A Brief Tour of Vector Calculus
A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0.