Vector Fields

9 Key excerpts on "Vector Fields"

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  eBook - ePub

  Vectors in Physics and Engineering

  • Alan Durrant(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  Fig 4.1b ). These fields are obviously time-dependent when observed over say a 24 hour period but are approximately independent of time when observed over a short enough period, say an hour or so in the case of room temperature. We shall consider only fields that are independent of time.

  Fig 4.1 (a) A scalar field. The temperature inside the room is shown at a selection of points P, Q etc. (b) A vector field. The wind velocity is shown at a selection of points by drawing arrows.

  Other examples of scalar fields are the barometric pressure in the atmosphere, the distribution of mass in a material body, and the electrostatic potential in the region near an electrically charged object. Other examples of Vector Fields are the gravitational field of a planet and the magnetic field produced by a magnet.

  Most fields of interest are three-dimensional, i.e. the scalar or vector quantity is distributed throughout a region of three-dimensional space. Some fields however are restricted to a surface. An example is the static electric charge produced on the surface of a dry glass plate when it is rubbed with a silk cloth; this is a two-dimensional scalar field since the scalar (electric charge) is distributed over the surface of the glass. The water velocity on the surface of a river is an example of a two-dimensional vector field.

  The concept of a field involves two sets of objects: the set of points in the region of space where the field exists; and the set of field values of the physical quantity, scalar or vector, that exists in the region. The field consists of an association of field values to points in a region of space. This leads to a mathematical description of fields by scalar or vector functions of position. Scalar and vector field functions are introduced and described in Sections 4.2 to 4.4 of this chapter. Section 4.5

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  Mathematical Methods for Physics

  Using MATLAB and Maple

  • J. R. Claycomb(Author)
  • 2018(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  Chapter 165 4 Chapter V ECTOR C ALCULUS Chapter Outline 4.1 Vector and Scalar Fields 4.2 Gradient of Scalar Fields 4.3 Divergence of Vector Fields 4.4 Curl of Vector Fields 4.5 Laplacian of Scalar and Vector Fields 4.6 Vector Identities 4.7 Integral Theorems 4.1 VECTOR AND SCALAR FIELDS Vector Fields have both magnitude and direction while scalar fields are directionless. Both scalar and Vector Fields can have physical units and be spatially varying. A vector field can be described by the spatial variation of a scalar field. In this section, we seek a mathematical description of the spatial distribution of scalar and Vector Fields. 166 MATHEMATICAL METHODS FOR PHYSICS USING MATLAB AND MAPLE 4.1.1 Scalar Fields Examples of scalar fields include temperature T(r) and pressure P(r) with values that depend on coordinates r. A two-dimensional scalar field such as the temperature ( ) ( ) 2 2 0 , x y T x y T e − + = (4.1.1) may describe the temperature in a plate with a hot spot at the center. A scalar field describes the electrical potential of a point charge located at r  0 in spherical coordinates ( ) 2 0 1 4 q V r r  = (4.1.2) Scalar fields may be represented graphically using contour plots, surface plots or density plots. Contour plots show lines of constant field values. Surface plots represent scalar field values by different heights. Density plots may use variations in grayscale or color to illustrate regions with different field values. 4.1.2 Vector Fields Examples of Vector Fields include the electric field E(r), magnetic field B(r) and the velocity field v(r) of a fluid. A two-dimensional vector field has x and y components that can vary over space ( ) ( ) ( ) ˆ ˆ , , , x y x y F x y F x y = + F i j (4.1.3) The magnitude of a vector field will produce a scalar field ( ) ( ) ( ) 2 2 , , , x y x y F x y F x y = + F (4.1.4) We may plot a vector field by drawing arrows with lengths proportional to the magnitude of the field at equally spaced locations on a grid.

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  What Goes Up... Gravity and Scientific Method

  • Peter Kosso(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  reference point, this thermal system requires a physical phenomenon, water freez- ing, as a reference. Scalar quantities like mass are written into equations in a nor- mal, not bold, print, as in the mass m. There is no arrow associated with a scalar. These are the tools we need for a precise description of motion, the science of kinematics. If we had these numbers, units, and arrows on the blackboard, anyone looking in would be able to tell it’s a science class. But we’re not really doing science yet, since we haven’t really described anything in nature yet. We have the words but we haven’t constructed any sentences. The next step, when we’ll start doing science, is to make connections to and among the terms, connecting them individually to things in nature and connecting them to each other. The distance formula, the Pythagorean theorem applied to Figure 2.1, was a start. It said that d 2 = x 2 + y 2 . This relation between distance and coordinate positions is true for any two points in any Cartesian coordinate system. This sort of generalization is exactly what to expect from science. It’s not just a precise description of a particular situation; it’s a universal formula that applies to all such situations. It’s a law. Now that we have the vocabulary of kinematics, the concepts of vectors and scalars, and the will to generalize, we can turn to the concept of a field. The most general characterization is that a field is a physical parameter that depends on pos- ition. It is a smoothly varying, continuous array of parameter values. This is perhaps too general and abstract to help, so consider some examples of fields. The distribution of air temperatures at all points across the country is a field, a temperature field. A weather map in the newspaper shows a few selected values of the field at points of interest, but the field itself is continuous. There is a temperature at every single point in the country, measured or not, reported or not.

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  eBook - ePub

  Applied Engineering Analysis

  • Tai-Ran Hsu(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  direction. Examples of vectorial quantities include the velocity of an object traveling either in a (2D) plane or in a (3D) space. As well as the example of the velocity of a moving vehicle, the concept includes related quantities such as acceleration. The force acting on an object is another vectorial quantity that has to be defined in terms of its magnitude and the direction in which the force is acting. Other vectorial quantities include the electric field, current flow, and heat transmission in solids and fluids.

  In contrast to vectorial quantities, common physical quantities in engineering analysis such as the temperature, speed, mass of an object, heat and energy, and electric potential are scalar quantities.

  A vector may be represented as a directed line segment as shown in Figure 3.1 . We will use boldfaced letters and notation to designate the vector quantities throughout this book.

  Figure 3.1

  Graphical representation of a vector.

  In Figure 3.1 the vector A is characterized by its magnitude |A| and its direction indicated with an arrowhead. Graphical representation of this vector includes an “initial point” shown as a solid circle at one end and a “terminal point” indicated by the arrowhead. A vector with the same magnitude of vector A but acting in the opposite direction carries a negative sign (i.e., ) as also shown in Figure 3.1 .

  Vector quantities are usually defined by a coordinate system in engineering analyses, for example in an x–y coordinate system as shown in Figure 3.2 .

  Figure 3.2

  Rectangular x–y coordinate system for a vector. (a) Vector A in x–y coordinates. (b) Decomposition of vector A

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  eBook - PDF

  Introduction to Engineering Electromagnetic Fields

  • K Umashankar(Author)
  • 1989(Publication Date)
  • WSPC

    (Publisher)

  Chapter: 1 UECTOR CALCULUS 1.0 INTRODUCTION The basic knowledge of scalars, vectors and their mathematical representation is important for understanding the phenomena associated with electric and magnetic fields. By definition, a scalar quantity represents just a magnitude, while a vector quantity represents both magnitude and direction. For example, mass of an object, temperature or electric charge distribution in a given region, time elapsed for a certain object to move, can be mathematically represented in terms of a scalar quantity. Similarly, force acting on an object, electric field or magnetic field distribution in a given region, electric current flowing along a conducting wire, can be mathematically represented in terms of a vector quantity. In general, the scalar and the vector quantities can be functions of one or more dependent physical parameters including spatial coordinate variables. The spatial dependence of a given problem under consideration, such as, the one dimensional problem, the two dimensional problem or the three dimensional problem associated with the electric field and the magnetic field, can complicate the scalar quantity or the vector quantity representation. For higher dimensionality problems, the vector quantities are relatively difficult to manage. Sometimes, the graphical representation of the scalar quantity or the vector quantity helps to understand their physical significance. In fact, the scalar quantity can be graphically represented by a straight line of certain length with a specified scale factor assumed. The magnitude of the vector quantity can be graphically represented in the same manner, in addition, a straight bar or an arrow is included to depict the actual direction of the vector. For example, in figure 1.1, two arbitrary vectors Rand B are represented graphically.

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  An Introduction to the Theory of Microwave Circuits

  • K. Kurokawa(Author)
  • 1969(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R 2 ELECTROMAGNETIC FIELD VECTORS In order to discuss microwave circuits, Maxwell's equations must be studied since these describe the relations between electric and magnetic fields. If the concept of a vector is introduced, the relations between the fields become simpler to describe and easier to understand. Therefore, we shall make extensive use of vectors in this book. This chapter reviews some of the important theorems on vector analysis which will facilitate our later study. In Section 2.1, elements of vectors and vector analysis are presented, and in Section 2.2, Maxwell's equations are explained in terms of the field vectors in order to refresh the reader's understanding of electromagnetic theory. Section 2.3 gives an analysis of plane waves, first in terms of scalar quantities resolving the field vectors into their components, and then in terms of vectors after which the results are compared. 2.1 Vectors A vector is a quantity with magnitude and direction. To visualize it, we usually consider an arrow whose length and direction expresses the magnitude and direction of the vector, respectively, as shown in Fig. 2.1. The space of A Fig. 2.1. Arrow representing vector A. 40 2.1. Vectors 41 this figure is not the actual space in which we live, but rather it is an abstract space where the magnitude and direction of the vector is represented by the length and direction of the corresponding arrow. Vectors are generally functions of time and position existing in actual space, therefore, depending on the position and time, the lengths and the directions of the vector arrows vary in abstract space. Sometimes however, the tail of the arrow is located at the point in the actual space where the vector is considered. An example will be shown in Fig. 2.10. In such a case, the illustrated space has two meanings; one is the actual space, and the other is the abstract space in which we consider the arrows representing vectors.

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  Differential Geometry and Topology

  With a View to Dynamical Systems

  • Keith Burns, Marian Gidea(Authors)
  • 2005(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 2 Vector Fields and Dynamical Systems 2.1 Introduction Many physical systems are described by differential equations. As an example, we consider the two-body problem (the Kepler prob-lem). It concerns the motion of two bodies under mutual gravitation. By placing a reference frame at one of the bodies, the problem can be reduced to the motion of a single body in a central gravitational force field. The force is expressed by the universal law of gravitation F = -μ x k x k 3 , where the gravitational constant and the total mass of the system are normalized to 1. At every position x in the phase space R 3 { (0 , 0 , 0) } there is a force vector that points towards the origin and has its magni-tude inverse proportional to the square of the distance to the origin. The motion of the body is governed by the second order differential equation d 2 x dt 2 = -x k x k 3 . This translates into a first order system dx dt = v, (2.1.1) dv dt = -x k x k 3 , 71 72 2. Vector Fields AND DYNAMICAL SYSTEMS with the variables x and v representing the position and the velocity of the body, respectively. The theory of differential equations asserts that the future motion of the body is completely determined by the initial position x 0 and initial velocity v 0 of the body. We should note that many differential systems do not have closed-form solutions. This is the case, for example, of the n -body problem with n ≥ 3, which studies the general motion of n bodies interacting by mutual gravitation forces (we will discuss this problem in Section 9.2). In such instances, one usually attempts to perform a qualitative study of differential equations. In the sequel, we will find explicit solutions to the central force prob-lem, and, implicitly, to the two-body problem. We denote by · the dot product and by × the vector product in R 3 .

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  eBook - ePub

  Matrix Vector Analysis

  • Richard L. Eisenman(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  chapter III      VECTOR AND SCALAR FIELDS

  This chapter exploits the vector calculus of three variables. You are urged to be constantly aware of the natural generalizations of ideas from scalar calculus and of the beneficial interplay among algebraic, geometric, and physical ideas.

  3.1 | POINT FUNCTIONS

  The terminology “vector point function” and “scalar point function” is essentially self-explanatory. By a scalar point function we mean a machine (function) into which you feed a point and out of which comes a scalar; i.e., the domain consists of points, and the range consists of scalars. By a vector point function or vector field we mean a machine which generates a vector for each point fed to it.

  Definitions: A vector point function or vector field or vector function of position is a relation which associates one vector with each point of a given region of space.

       A scalar point function or scalar field or scalar function of position is a relation which associates one scalar with each point of a given region of space.

       Point functions have physical, algebraic, and geometric significance. Temperature, length of radius vector, electrostatic potential, and gravitational potential are physical representations of scalar point functions. Heat flow, the radius vector, velocity of fluid particles, and the force of gravity are physical examples of vector point functions.

       Algebraically, the most basic scalar point functions are the coordinates which one chooses to describe a point. In a rectangular coordinate system in three dimensions, the three displacements of a point are the scalar point functions commonly designated by the three independent variables x, y, z. These three scalar point functions are so fundamental that the algebraic representation of every other point function is commonly given in terms of x, y, and z. For example, the scalar point function representing distance from the origin is written as (x2 + y2 + z2 )½

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  eBook - ePub

  Tensor and Vector Analysis

  With Applications to Differential Geometry

  • C. E. Springer(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  10

  Scalar and Vector Fields

  10–1. Fields. A tensor has been defined as an object with a set of components which transform in a specified manner when the coordinates are changed. These components are, in general, functions of position and therefore vary from point to point in space. The totality of these sets of functions at all points of a region of space is called a tensor field. Because the covariant derivative of a tensor is a tensor, it is seen that the covariant derivative of a tensor field is a tensor field of one higher order in the co-variant indices.

  It has been pointed out that a tensor field of zero order is represented by a scalar point function. As a physical model for this, one may consider the scalar function as representing the temperature at any point x i of space. In general, the temperature varies from point to point in space.

  A tensor field of the first order is a vector field. With each coordinate system there are associated three functions of position, which may be denoted by λ i (or by λ i ). As a physical model for this, one may consider the motion of a fluid in three-space. At each point a vector with components λ i specifies the velocity of a particle of the fluid at the point. A particular kind of vector field is obtained by taking the covariant derivative of a scalar field, which is the same as the partial derivative. For instance, the vector field represented by is the gradient vector field of the scalar field . In the temperature model, are the three rates of change of the temperature function with respect to the respective x i coordinates, and are the components of the gradient vector , which will now be shown to point in the direction of the greatest rate of change of the scalar field. If a i are the components of a unit vector A, the scalar projection of on the direction of A is given by . On the surface , the function clearly does not change. The total differential of the equation of the surface gives , which means that the a i are proportional to dx i , the direction of any tangent vector to the surface, so that the scalar projection of on the surface is zero. This shows that the direction of is normal to the surface

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