scholarly journals Student understanding of unit vectors and coordinate systems beyond cartesian coordinates in upper division physics courses

2016 ◽  
Author(s):  
Marlene Vega ◽  
Warren M. Christensen ◽  
Brian Farlow ◽  
Gina Passante ◽  
Michael E. Loverude
Keyword(s):  
Student Understanding ◽  
Coordinate Systems ◽  
Cartesian Coordinates ◽  
Physics Courses ◽  
Unit Vectors

Appendixes

1997 ◽  
Author(s):  
David Jon Furbish
Keyword(s):  
Fluid Mechanics ◽  
Coordinate System ◽  
Vector Analysis ◽  
College Level ◽  
Coordinate Systems ◽  
Cartesian Coordinates ◽  
Orthogonal Coordinate System ◽  
Unit Vectors ◽  
Orthogonal Coordinates

Definitions and formulae used at various points in the text to manipulate vectors are listed below. Additional useful formulae, including geometrical and physical interpretations complementary to those provided in this text, can be found in standard texts on vector analysis and in mathematical handbooks. The Standard Mathematical Tables published by CRC Press (Boca Raton, Florida) is a particularly handy resource, and most college-level calculus texts cover introductory vector analysis as part of the material intended for a third-semester course. Appendix A in Bird, Stewart, and Lightfoot (1960) is a very good summary of vector and tensor notation presented in the context of fluid mechanics. Section 17.1.1 begins with several basic definitions of vector quantities that generally apply to any orthogonal coordinate system. The notation for unit vectors in Cartesian coordinates, i, j, and k, are used in this section, but it is understood that this notation may be directly replaced with symbols for unit vectors associated with other orthogonal coordinates. Section 17.1.2 then covers differential operations for Cartesian coordinates. Although the notation used for these differential operations in Cartesian coordinates is the same as that for other coordinate systems, the actual operations connoted by the notation are different, and must be defined separately (Appendix 17.2). Let S and T denote scalar functions, and let U, V, and W denote vectors. If U = 〈U1, U2, U3〉, then . . . U = U1i + U2j + U3k . . . . . . (17.1) . . .

scholarly journals Tractography in Curvilinear Coordinates

2021 ◽  
Vol 15 ◽  
Author(s):  
Uzair Hussain ◽  
Corey A. Baron ◽  
Ali R. Khan
Keyword(s):  
Medical Imaging ◽  
Sensitivity And Specificity ◽  
Curvilinear Coordinates ◽  
Spherical Coordinates ◽  
Coordinate Systems ◽  
Cartesian Coordinates ◽  
Diffusion Tractography ◽  
Specificity And Sensitivity ◽  
Physical Laws ◽  
Coordinate Invariance

Coordinate invariance of physical laws is central in physics, it grants us the freedom to express observations in coordinate systems that provide computational convenience. In the context of medical imaging there are numerous examples where departing from Cartesian to curvilinear coordinates leads to ease of visualization and simplicity, such as spherical coordinates in the brain's cortex, or universal ventricular coordinates in the heart. In this work we introduce tools that enhance the use of existing diffusion tractography approaches to utilize arbitrary coordinates. To test our method we perform simulations that gauge tractography performance by calculating the specificity and sensitivity of tracts generated from curvilinear coordinates in comparison with those generated from Cartesian coordinates, and we find that curvilinear coordinates generally show improved sensitivity and specificity compared to Cartesian. Also, as an application of our method, we show how harmonic coordinates can be used to enhance tractography for the hippocampus.

Representation of Reservoir Geometry for Numerical Simulation

1970 ◽  
Vol 10 (04) ◽  
pp. 393-404 ◽  
Author(s):  
G.J. Hirasaki ◽  
P.M. O'Dell
Keyword(s):  
Coordinate System ◽  
Curvilinear Coordinates ◽  
Reference Plane ◽  
Cartesian Coordinate System ◽  
Curvilinear Coordinate System ◽  
Coordinate Systems ◽  
Cartesian Coordinates ◽  
Curvilinear Coordinate ◽  
Reservoir Geometry ◽  
Cartesian Coordinate

Abstract For most reservoirs the reservoir thickness and dip vary with position. For such reservoirs, the use of a Cartesian coordinate system is awkward as the coordinate surfaces are planes and the finite-difference grid elements are rectangular parallepipeds. However, these reservoirs may be efficiently parallepipeds. However, these reservoirs may be efficiently modeled with a curvilinear coordinate system that has coordinate surfaces that coincide with the reservoir surfaces. A procedure is presented that may be used to determine a curvilinear coordinate system that will conform with the geometry of the reservoir. The reservoir geometry is described by the depth of the top of the reservoir and the thickness. The mass conservation equations are presented in curvilinear coordinates. The finite-difference equations differ from the usual Cartesian coordinate formulation by a factor multiplying the pore volume and transmissibilities. A numerical example is presented to illustrate the magnitude of the error that may occur in the computed oil recovery if the Cartesian coordinate system is simply modified to yield the correct depth and pore volumes. Introduction Many reservoirs have a shape that is inconvenient and possibly inaccurate to model with Cartesian coordinates. The use of a curvilinear coordinate system that follows the shape of the reservoir can be advantageous for such reservoirs. The formulation discussed here will have the greatest advantage in modeling thin reservoirs but will have little advantage in modeling a reservoir whose thickness is greater than its radius of curvature, such as a pinnacle reef. pinnacle reef. In this paper the reader is introduced to various grid systems used to model reservoirs. A brief discussion of some concepts of differential geometry contrasts differences between Cartesian coordinates and curvilinear coordinates. A curvilinear coordinate system for modeling reservoir geometry is presented. Formulation of the conservation equations in curvilinear coordinates and the necessary modifications to pore volume and transmissibility are discussed. A numerical example illustrates the magnitude of the error that may result from some coordinate systems. COORDINATE SYSTEMS AND RESERVOIR GRID NETWORKS A reservoir is usually described with the depth, thickness, boundaries, etc., shown on a structure map with sea level as a reference plane. For example, the subsea depth may be shown as a contour map on the reference plane with a Cartesian coordinate grid superimposed on the reference plane as shown on Fig. 1. The Cartesian coordinates, plane as shown on Fig. 1. The Cartesian coordinates, (y1, y2), have been defined as the coordinates for the reference plane. If the reservoir surfaces are parallel planes, Cartesian coordinates may be used. The Cartesian coordinate may be rotated such that the coordinate surfaces coincide with the reservoir surfaces. SPEJ P. 393

Probing Student Understanding of Basic Concepts and Principles in Introductory Engineering Thermodynamics

2007 ◽  
Author(s):  
Christian H. Kautz ◽  
Gerhard Schmitz
Keyword(s):  
Student Understanding ◽  
State Function ◽  
Instructional Tool ◽  
Ongoing Research ◽  
Physics Courses ◽  
University Of Technology ◽  
Traditional Lectures ◽  
Upper Level ◽  
Curricular Materials ◽  
Interactive Lecture

We report on an ongoing research study on student understanding of thermodynamic concepts and principles in the context of an introductory engineering thermodynamics course at Hamburg University of Technology (TUHH). Through analysis of student responses to mostly qualitative questions, we have identified prevalent and persistent difficulties. In this paper, we describe the research methods, present some preliminary results, and discuss the implications of our work for instruction and the development of curricular materials. We also illustrate the use of interactive lecture questions as an instructional tool. In recent decades, research on student understanding in science and engineering has revealed that traditional quantitative problems often are not a suitable tool for the assessment of conceptual understanding. On the basis of results from prior investigations in the context of thermal physics we have therefore begun to administer “conceptual” questions to students of engineering thermodynamics. These questions are delivered through ungraded quizzes, course examinations, and as interactive lecture questions (ILQs or “clicker questions”) via a classroom communication system. While only the two written formats require students to explain the reasoning supporting their answers, we have found that there is good agreement between the results obtained through different methods. Our work so far has concentrated on probing student understanding of (1) work and the application of the first law to closed systems and flow processes, (2) the distinction between state and process quantities, in particular student understanding of entropy as a state function, and (3) the application of the second law, especially to refrigeration cycles. Conceptual difficulties that we have observed include, for example, the students’ tendency to associate an increase in entropy of the system with any irreversible process even if the state function property of the entropy leads to a different result. Similar difficulties have been documented in the context of introductory and upper-level physics courses. While ILQs serve as a research instrument, we also recognize their potential as an effective instructional tool. Data from post-tests suggest that the use of such questions can enhance student learning in traditional lectures. In addition, we discuss how results from this study contributed to the writing of a textbook on engineering thermodynamics.

Analysis of Spatial Open-Loop System by Means of Direction Cosine Transformation Matrices

1989 ◽  
Vol 111 (4) ◽  
pp. 508-512 ◽  
Author(s):  
T. C. Yih ◽  
Y. Youm
Keyword(s):  
Industrial Robot ◽  
Direction Cosine ◽  
Open Loop ◽  
Coordinate Systems ◽  
Open Loop System ◽  
Transformation Matrices ◽  
Joint Axis ◽  
Cosine Transformation ◽  
Direction Cosine Matrix ◽  
Unit Vectors

In this paper, an analytical approach for the displacement analysis of spatial openloop systems by means of direction cosine transformation matrices is presented. Two local coordinate systems at each joint are designated to formulate the direction cosine matrices, in recursive form, of the joint axis and link vector. Elements of the 3×3 direction cosine transformation matrices are computed based on the geometry of successive link elements, the unit vectors of preceding joint axis and link vector, and the cofactors of direction cosine matrix. The analysis using direction cosine matrix method will provide the “exact” joint positions in space. A computer algorithm is developed to investigate the workspaces of spatial n-R open-loop systems that projected onto the X-Y, Y-Z, and Z-X coordinate planes, respectively. Numerical examples for the workspaces of an industrial robot and the human upper extremity are illustrated.

Computable simulated transformations of the Cartesian coordinates for random vectors

2021 ◽  
pp. 62-84
Author(s):  
Анжела Романовна Абдразакова ◽  
Антон Вацлавович Войтишек
Keyword(s):  
Scientific Literature ◽  
Computer Modelling ◽  
Three Dimensional ◽  
Random Variables ◽  
Computer Algorithms ◽  
Coordinate Systems ◽  
Cartesian Coordinates ◽  
Practical Applications ◽  
Isotropic Vectors ◽  
Modelling Simulation

Рассмотрены специальные преобразования декартовых координат, позволяющие строить эффективные (экономичные) алгоритмы численного моделирования многомерных случайных величин. В качестве иллюстративных и практически значимых примеров таких преобразований рассмотрены переходы к полярным, сферическим, параболическим и цилиндрическим координатам. The purpose of the paper was to expand the range of efficient (economical) computer algorithms for simulation of multi-dimensional random variables. The authors noticed that in a number of applied problems (for example, when modelling twoor three-dimensional isotropic vectors), transitions from Cartesian to other coordinate systems (for example, to polar or spherical) are effective. In this regard, the new generalizing notation of the computable simulated transformation of Cartesian coordinates is introduced in the paper. Such transformations allow constructing effective (economical) algorithms for the numerical modelling (simulation) of multi-dimensional random variables. The problem of finding examples of constructive and practical applications for computer simulated transformations of Cartesian coordinates is formulated. Transitions to polar, spherical, parabolic and cylindrical coordinates are considered as illustrative examples of such transformations. The practical applications of computable simulated transformations of Cartesian coordinates found in the scientific literature are described in detail by the authors. These applications are associated with both computer modelling (simulation) of isotropic vectors and Gaussian distribution along with the numerical solution of boundary value problems and problems of radiation transfer. Thus, the introduced notion of the computable simulated transformations of Cartesian coordinates is quite constructive. It opens up prospects for the scientific search for new transformations of this type with the aim of using them in stochastic computer models for important processes and phenomena.

scholarly journals Student Understanding of Period in Introductory and Quantum Physics Courses

2016 ◽  
Author(s):  
Tong Wan ◽  
Paul J. Emigh ◽  
Gina Passante ◽  
Peter S. Shaffer
Keyword(s):  
Quantum Physics ◽  
Student Understanding ◽  
Physics Courses

Numerical Simulation of Single Drops in a Vertical Pipe Using Various Coordinate Systems

2007 ◽  
Author(s):  
Keishun Hirokawa ◽  
Kosuke Hayashi ◽  
Akira Sou ◽  
Akio Tomiyama ◽  
Naoki Takada
Keyword(s):  
Spatial Resolution ◽  
Velocity Ratio ◽  
Three Dimensional ◽  
Terminal Velocity ◽  
Continuous Phase ◽  
Curvilinear Coordinates ◽  
Coordinate Systems ◽  
Vertical Pipe ◽  
Cylindrical Coordinates ◽  
Cartesian Coordinates

The effects of the diameter ratio λ (= d/D, where d and D are the diameters of a drop and a pipe, respectively), the Morton number M and the viscosity ratio κ (= μd/μc, where μ is the viscosity and the subscripts d and c are the dispersed fluid particle and the continuous phase, respectively) on terminal velocities and shapes of single drops rising through stagnant liquids in a vertical pipe are investigated experimentally. Then, the drops in the pipe are simulated using a volume tracking method with various coordinate systems, i.e., three-dimensional (3D) cylindrical coordinates, 3D general curvilinear coordinates and 3D Cartesian coordinates. Predicted velocities and shapes of the drops using three coordinate systems are compared with the measured data to examine the effects of coordinate systems on the accuracy of prediction. As a result, (1) The velocity ratio VT/VT0 (VT and VT0 are the terminal velocity in a pipe and infinite liquid, respectively) decreases as λ increases, and it depends not only on λ but also on M and κ, (2) Good predictions for the terminal velocities and shapes of drops are obtained not only with cylindrical coordinates and curvilinear coordinates but also with Cartesian coordinates, provided that the spatial resolution is high, (3) When the spatial resolution is low, effects of coordinate systems on a drop shape are larger for Cartesian coordinate systems than for cylindrical coordinate and general curvilinear coordinate systems, and (4) Errors in predicted drop velocities are not so large even with very low spatial resolution.

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