Vector Analysis in Cartesian Coordinates

2020 ◽  
pp. 1-15
Author(s):  
Victor Ilisie
Keyword(s):  
Vector Analysis ◽  
Cartesian Coordinates

scholarly journals A general vector analysis, with applications to electrodynamical theory

1925 ◽  
Vol 108 (747) ◽  
pp. 418-455 ◽  
Keyword(s):  
Three Dimensions ◽  
Vector Analysis ◽  
Cartesian Coordinates ◽  
Straight Line ◽  
Straight Lines ◽  
General Vector

The vector analyses in use up to the present, as a rule, are concerned with quantities which are represented by straight lines, and the space to which they are applicable is Euclidean in its properties. The straight line, AB, in space of three dimensions, is represented by a vector a, and if B has Cartesian coordinates ( x, y, z ) with respect to A, we write: a = i x + j y + k z , where i, j, k, are fundamental vectors. An account will be given of a vector analysis in which a vector is represented by δa' = Σ n i n δx n . The vector is of infinitesimal length and represents a component measured in any system of co-ordinates.

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) . . .

DESCRIPTIVE RELIABILITY OF ACTIVITY VECTOR ANALYSIS

1958 ◽  
Vol 4 ◽  
pp. 435 ◽  
Author(s):  
HAROLD R. MUSIKER
Keyword(s):  
Vector Analysis ◽  
Activity Vector

Correlation between quality of Pyrola decorata and its ecological factors based on hierarchy- vector analysis

2013 ◽  
Vol 36 (9) ◽  
pp. 992-1003
Author(s):  
Zhen-Jiang LÜ ◽  
Dong-Mei WANG ◽  
Deng-Wu LI
Keyword(s):  
Ecological Factors ◽  
Vector Analysis

Vector geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Vector Space ◽  
Velocity Vector ◽  
Cartesian Coordinates

This chapter defines the mathematical spaces to which the geometrical quantities discussed in the previous chapter—scalars, vectors, and the metric—belong. Its goal is to go from the concept of a vector as an object whose components transform as Tⁱ → 𝓡ⱼ ⁱTj under a change of frame to the ‘intrinsic’ concept of a vector, T. These concepts are also generalized to ‘tensors’. The chapter also briefly remarks on how to deal with non-Cartesian coordinates. The velocity vector v is defined as a ‘free’ vector belonging to the vector space ε‎3 which subtends ε‎3. As such, it is not bound to the point P at which it is evaluated. It is, however, possible to attach it to that point and to interpret it as the tangent to the trajectory at P.

Vector Analysis

1933 ◽  
Vol 17 (225) ◽  
pp. 281
Author(s):  
E. H. Neville ◽  
H. B. Phillips
Keyword(s):  
Vector Analysis

scholarly journals Evaluating the impacts of major cyclonic catastrophes in coastal Bangladesh using geospatial techniques

2021 ◽  
Vol 3 (8) ◽  
Author(s):  
Mashoukur Rahaman ◽  
Md. Esraz-Ul-Zannat
Keyword(s):  
Deep Water ◽  
Water Body ◽  
Crop Production ◽  
Vector Analysis ◽  
Agricultural Crop ◽  
Geospatial Technologies ◽  
Catastrophic Events ◽  
Change Vector Analysis ◽  
Change Vector ◽  
Post Disaster

AbstractCyclonic catastrophes frequently devastate coastal regions of Bangladesh that host around 35 million people which represents two-thirds of the total population. They have caused many problems like agricultural crop loss, forest degradation, damage to built-up areas, river and shoreline changes that are linked to people’s livelihood and ecological biodiversity. There is an absence of a comprehensive assessment of the major cyclonic disasters of Bangladesh that integrates geospatial technologies in a single study. This study aims to integrate geospatial technologies with major disasters and compares them, which has not been tried before. This paper tried to identify impacts that occurred in the coastal region by major catastrophic events at a vast level using different geospatial technologies. It focuses to identify the impacts of major catastrophic events on livelihood and food production as well as compare the impacts and intensity of different disasters. Furthermore, it compared the losses among several districts and for that previous and post-satellite images of disasters that occurred in 1988, 1991, 2007, 2009, 2019 were used. Classification technique like machine learning algorithm was done in pre- to post-disaster images. For quantifying change in the indication of different factors, indices including NDVI, NDWI, NDBI were developed. “Change vector analysis” equation was performed in bands of the images of pre- and post-disaster to identify the magnitude of change. Also, crop production variance was analyzed to detect impacts on crop production. Furthermore, the changes in shallow to deep water were analyzed. There is a notable change in shallow to deep water bodies after each disaster in Satkhira and Bhola district but subtle changes in Khulna and Bagerhat districts. Change vector analysis revealed greater intensity in Bhola in 1988 and Satkhira in 1991. Furthermore, over the years 2007 and 2009 it showed medium and deep intense areas all over the region. A sharp decrease in Aus rice production is witnessed in Barishal in 2007 when cyclone “Sidr” was stricken. The declination of potato production is seen in Khulna district after the 1988 cyclone. A huge change in the land-use classes from classified images like water body, Pasture land in 1988 and water body, forest in 1991 is marked out. Besides, a clear variation in the settlement was observed from the classified images. This study explores the necessity of using more geospatial technologies in disastrous impacts assessment around the world in the context of Bangladesh and, also, emphasizes taking effective, proper and sustainable disaster management and mitigation measures to counter future disastrous impacts.

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