1. Three Dimensional Coordinate System. Intro to Surfaces
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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. -
Calculus & Analytic Geometry
TQS 126 Spring 2008 Quinn Calculus & Analytic Geometry III Quadratic Equations in 3-D Match each function to its graph 1. 9x2 + 36y2 +4z2 = 36 2. 4x2 +9y2 4z2 =0 − 3. 36x2 +9y2 4z2 = 36 − 4. 4x2 9y2 4z2 = 36 − − 5. 9x2 +4y2 6z =0 − 6. 9x2 4y2 6z =0 − − 7. 4x2 + y2 +4z2 4y 4z +36=0 − − 8. 4x2 + y2 +4z2 4y 4z 36=0 − − − cone • ellipsoid • elliptic paraboloid • hyperbolic paraboloid • hyperboloid of one sheet • hyperboloid of two sheets 24 TQS 126 Spring 2008 Quinn Calculus & Analytic Geometry III Parametric Equations (§10.1) and Vector Functions (§13.1) Definition. If x and y are given as continuous function x = f(t) y = g(t) over an interval of t-values, then the set of points (x, y)=(f(t),g(t)) defined by these equation is a parametric curve (sometimes called aplane curve). The equations are parametric equations for the curve. Often we think of parametric curves as describing the movement of a particle in a plane over time. Examples. x = 2cos t x = et 0 t π 1 t e y = 3sin t ≤ ≤ y = ln t ≤ ≤ Can we find parameterizations of known curves? the line segment circle x2 + y2 =1 from (1, 3) to (5, 1) Why restrict ourselves to only moving through planes? Why not space? And why not use our nifty vector notation? 25 Definition. If x, y, and z are given as continuous functions x = f(t) y = g(t) z = h(t) over an interval of t-values, then the set of points (x,y,z)= (f(t),g(t), h(t)) defined by these equation is a parametric curve (sometimes called a space curve). -
Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vectors For our purposes we will think of a vector as a mathematical representation of a physical entity which has both magnitude and direction in a 3D space. Examples of physical vectors are forces, moments, and velocities. Geometrically, a vector can be represented as arrows. The length of the arrow represents its magnitude. Unless indicated otherwise, we shall assume that parallel translation does not change a vector, and we shall call the vectors satisfying this property, free vectors. Thus, two vectors are equal if and only if they are parallel, point in the same direction, and have equal length. Vectors are usually typed in boldface and scalar quantities appear in lightface italic type, e.g. the vector quantity A has magnitude, or modulus, A = |A|. In handwritten text, vectors are often expressed using the −→ arrow, or underbar notation, e.g. A , A. Vector Algebra Here, we introduce a few useful operations which are defined for free vectors. Multiplication by a scalar If we multiply a vector A by a scalar α, the result is a vector B = αA, which has magnitude B = |α|A. The vector B, is parallel to A and points in the same direction if α > 0. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
Solving the Geodesic Equation
Solving the Geodesic Equation Jeremy Atkins December 12, 2018 Abstract We find the general form of the geodesic equation and discuss the closed form relation to find Christoffel symbols. We then show how to use metric independence to find Killing vector fields, which allow us to solve the geodesic equation when there are helpful symmetries. We also discuss a more general way to find Killing vector fields, and some of their properties as a Lie algebra. 1 The Variational Method We will exploit the following variational principle to characterize motion in general relativity: The world line of a free test particle between two timelike separated points extremizes the proper time between them. where a test particle is one that is not a significant source of spacetime cur- vature, and a free particles is one that is only under the influence of curved spacetime. Similarly to classical Lagrangian mechanics, we can use this to de- duce the equations of motion for a metric. The proper time along a timeline worldline between point A and point B for the metric gµν is given by Z B Z B µ ν 1=2 τAB = dτ = (−gµν (x)dx dx ) (1) A A using the Einstein summation notation, and µ, ν = 0; 1; 2; 3. We can parame- terize the four coordinates with the parameter σ where σ = 0 at A and σ = 1 at B. This gives us the following equation for the proper time: Z 1 dxµ dxν 1=2 τAB = dσ −gµν (x) (2) 0 dσ dσ We can treat the integrand as a Lagrangian, dxµ dxν 1=2 L = −gµν (x) (3) dσ dσ and it's clear that the world lines extremizing proper time are those that satisfy the Euler-Lagrange equation: @L d @L − = 0 (4) @xµ dσ @(dxµ/dσ) 1 These four equations together give the equation for the worldline extremizing the proper time. -
Geodetic Position Computations
GEODETIC POSITION COMPUTATIONS E. J. KRAKIWSKY D. B. THOMSON February 1974 TECHNICALLECTURE NOTES REPORT NO.NO. 21739 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. GEODETIC POSITION COMPUTATIONS E.J. Krakiwsky D.B. Thomson Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton. N .B. Canada E3B5A3 February 197 4 Latest Reprinting December 1995 PREFACE The purpose of these notes is to give the theory and use of some methods of computing the geodetic positions of points on a reference ellipsoid and on the terrain. Justification for the first three sections o{ these lecture notes, which are concerned with the classical problem of "cCDputation of geodetic positions on the surface of an ellipsoid" is not easy to come by. It can onl.y be stated that the attempt has been to produce a self contained package , cont8.i.ning the complete development of same representative methods that exist in the literature. The last section is an introduction to three dimensional computation methods , and is offered as an alternative to the classical approach. Several problems, and their respective solutions, are presented. The approach t~en herein is to perform complete derivations, thus stqing awrq f'rcm the practice of giving a list of for11111lae to use in the solution of' a problem. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
Coordinate Transformation
Coordinate Transformation Coordinate Transformations In this chapter, we explore mappings – where a mapping is a function that "maps" one set to another, usually in a way that preserves at least some of the underlyign geometry of the sets. For example, a 2-dimensional coordinate transformation is a mapping of the form T (u; v) = x (u; v) ; y (u; v) h i The functions x (u; v) and y (u; v) are called the components of the transforma- tion. Moreover, the transformation T maps a set S in the uv-plane to a set T (S) in the xy-plane: If S is a region, then we use the components x = f (u; v) and y = g (u; v) to …nd the image of S under T (u; v) : EXAMPLE 1 Find T (S) when T (u; v) = uv; u2 v2 and S is the unit square in the uv-plane (i.e., S = [0; 1] [0; 1]). Solution: To do so, let’s determine the boundary of T (S) in the xy-plane. We use x = uv and y = u2 v2 to …nd the image of the lines bounding the unit square: Side of Square Result of T (u; v) Image in xy-plane v = 0; u in [0; 1] x = 0; y = u2; u in [0; 1] y-axis for 0 y 1 u = 1; v in [0; 1] x = v; y = 1 v2; v in [0; 1] y = 1 x2; x in[0; 1] v = 1; u in [0; 1] x = u; y = u2 1; u in [0; 1] y = x2 1; x in [0; 1] u = 0; u in [0; 1] x = 0; y = v2; v in [0; 1] y-axis for 1 y 0 1 As a result, T (S) is the region in the xy-plane bounded by x = 0; y = x2 1; and y = 1 x2: Linear transformations are coordinate transformations of the form T (u; v) = au + bv; cu + dv h i where a; b; c; and d are constants. -
Unit 5: Change of Coordinates
LINEAR ALGEBRA AND VECTOR ANALYSIS MATH 22B Unit 5: Change of Coordinates Lecture 5.1. Given a basis B in a linear space X, we can write an element v in X in a unique 3 way as a sum of basis elements. For example, if v = is a vector in X = 2 and 4 R 1 1 2 B = fv = ; v = g, then v = 2v + v . We say that are the B 1 −1 2 6 1 2 1 B 3 coordinates of v. The standard coordinates are v = are assumed if no other 4 basis is specified. This means v = 3e1 + 4e2. n 5.2. If B = fv1; v2; ··· ; vng is a basis of R , then the matrix S which contains the vectors vk as column vectors is called the coordinate change matrix. Theorem: If S is the matrix of B, then S−1v are the B coordinates of v. 1 1 6 −1 5.3. In the above example, S = has the inverse S−1 = =7. We −1 6 1 1 compute S−1[3; 4]T = [2; 1]T . Proof. If [v]B = [a1; : : : ; an] are the new coordinates of v, this means v = a1v1 + ··· + −1 anvn. But that means v = S[v]B. Since B is a basis, S is invertible and [v]B = S v. Theorem: If T (x) = Ax is a linear map and S is the matrix from a basis change, then B = S−1AS is the matrix of T in the new basis B. Proof. Let y = Ax. The statement [y]B = B[x]B can be written using the last theorem as S−1y = BS−1x so that y = SBS−1x. -
COORDINATE TRANSFORMATIONS Members of a Structural System Are Typically Oriented in Differing Directions, E.G., Fig
COORDINATE TRANSFORMATIONS Members of a structural system are typically oriented in differing directions, e.g., Fig. 17.1. In order to perform an analysis, the ele- ment stiffness equations need to be expressed in a common coor- dinate system – typically the global coordinate system. Once the element equations are expressed in a common coordinate system, the equations for each element comprising the structure can be Figure 17.1 – Frame Structure 1 (Kassimali, 1999) 2 assembled. Coordinate Transformations: Frame Elements Consider frame element m of Fig. 17.7. Y e x y m b X (a) Frame Q , u 6 6 Q , u m 4 4 Q3, u3 Q , u Q5, u5 1 1 Figure 17.7: Local – Global Q2, u2 Coordinate Relationships (b) Local Coordinate End Forces and Displacements 3 4 1 Figure 17.7 shows that the local – QcosFsinFxXY global coordinate transformations QsinFcosFyXY (17.9) can be expressed as QFzZ x = cos X + sin Y y = -sin X + cos Y where x, X = 1 or 4; y, Y = 2 or 5; and z, Z = 3 or 6. and since z and Z are parallel, this coordinate transformation is Utilizing (17.9) for all six member expressed as force components and expressing z = Z the resulting transformations in Using the above coordinate matrix form gives transformations, the end force and QFb [t] [0] b displacement transformations can QFee[0] [t] be expressed as or 5 {Q} = [T] {F} (17.11) T {Q} Q Q Q T where b 123 = member; {Q} = <<Q>b <Q>e> = beginning node local coordinate element local coordinate force T T force vector; {Q}e = <Q4 Q5 Q6> = vector; {F} = <<F>b <F>e> = end node local coordinate force element global coordinate force T vector; {F}b = <F1 F2 F3> = [t] [0] beginning node global coordinate vector; and [T] = = T [0] [t] force vector; {F}e = <F4 F5 F6> = end node global coordinate force element local to global coordinate cos sin 0 transformation matrix. -
1 Vectors & Tensors
1 Vectors & Tensors The mathematical modeling of the physical world requires knowledge of quite a few different mathematics subjects, such as Calculus, Differential Equations and Linear Algebra. These topics are usually encountered in fundamental mathematics courses. However, in a more thorough and in-depth treatment of mechanics, it is essential to describe the physical world using the concept of the tensor, and so we begin this book with a comprehensive chapter on the tensor. The chapter is divided into three parts. The first part covers vectors (§1.1-1.7). The second part is concerned with second, and higher-order, tensors (§1.8-1.15). The second part covers much of the same ground as done in the first part, mainly generalizing the vector concepts and expressions to tensors. The final part (§1.16-1.19) (not required in the vast majority of applications) is concerned with generalizing the earlier work to curvilinear coordinate systems. The first part comprises basic vector algebra, such as the dot product and the cross product; the mathematics of how the components of a vector transform between different coordinate systems; the symbolic, index and matrix notations for vectors; the differentiation of vectors, including the gradient, the divergence and the curl; the integration of vectors, including line, double, surface and volume integrals, and the integral theorems. The second part comprises the definition of the tensor (and a re-definition of the vector); dyads and dyadics; the manipulation of tensors; properties of tensors, such as the trace, transpose, norm, determinant and principal values; special tensors, such as the spherical, identity and orthogonal tensors; the transformation of tensor components between different coordinate systems; the calculus of tensors, including the gradient of vectors and higher order tensors and the divergence of higher order tensors and special fourth order tensors.