Observability and Vector Algebra

To simulate echoes from the earth’s surface in the low flight mode, it is necessary to reproduce reliably the delayed reflected sounding signal of the radar in real time. For this, it is necessary to be able to calculate accurately and quickly the dependence of the distance to the object being measured from the angular position of the line of sight of the radar station. Obviously, the simplest expressions for calculating the range can be obtained for a segment or a plane. In the text of the article, analytical expressions for the calculation of range for two-dimensional and three-dimensional cases are obtained. Methods of statistical physics, vector algebra, and the theory of the radar of extended objects were used. Since the calculation of the dependence of the range of the object to the target from the angular position of the line of sight is carried out on the analytical expressions found in the paper, the result obtained is accurate, and due to the relative simplicity of the expressions obtained, the calculation does not require much time.

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Stress and Deformation

10.1093/oso/9780195095036.001.0001 ◽

1996 ◽

Cited By ~ 1

Author(s):

Gerhard Oertel

Keyword(s):

Differential Equations ◽

Mathematical Description ◽

Finite Strain ◽

Stress Strain ◽

Complex Problems ◽

Vector Algebra ◽

Stress And Deformation

Students of geology who may have only a modest background in mathematics need to become familiar with the theories of stress, strain, and other tensor quantities, so that they can follow, and apply to their own research, developments in modern, quantitative geology. This book, based on a course taught by the author at UCLA, can provide the proper introduction. Included throughout the eight chapters are 136 complex problems, advancing from vector algebra in standard and subscript notations, to the mathematical description of finite strain and its compounding and decomposition. Fully worked solutions to the problems make up the largest part of the book. With their help, students can monitor their progress, and geologists will be able to utilize subscript and matrix notations and formulate and solve tensor problems on their own. The book can be successfully used by anyone with some training in calculus and the rudiments of differential equations.

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Kinematics of Meshing Surfaces Using Geometric Algebra

Volume 4: 9th International Power Transmission and Gearing Conference, Parts A and B ◽

10.1115/detc2003/ptg-48086 ◽

2003 ◽

Author(s):

Scott M. Miller

Keyword(s):

Geometric Algebra ◽

Screw Theory ◽

Three Dimensional ◽

Normal Vector ◽

The Past ◽

Geometric Concepts ◽

Vector Algebra ◽

The Common ◽

Two Bodies ◽

Theoretical Developments

As is well known, analysis of two surfaces in mesh plays a fundamental role in gear theory. In the past, special coordinate systems, vector algebra, or screw theory was used to analyze the kinematics of meshing. The approach here instead relies on geometric algebra, an extension of conventional vector algebra. The elegance of geometric algebra for theoretical developments is demonstrated by examining the so-called “equation of meshing,” which requires that the relative velocity of two bodies at a point of contact be perpendicular to the common surface normal vector. With surprisingly little effort, several alternative forms of the equation of meshing are generated and, subsequently, interpreted geometrically. Via straightforward algebraic manipulations, the results of screw theory and vector algebra are unified. Due to the simplicity with which complex geometric concepts are expressed and manipulated, the effort required to grasp the general three-dimensional meshing of surfaces is minimized.

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