Scalar and Vector Fields

12 Key excerpts on "Scalar and 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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  Mathematical Methods for Physicists

  A Concise Introduction

  • Tai L. Chow(Author)
  • 2000(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 Vector and tensor analysis Vectors and scalars Vector methods have become standard tools for the physicists. In this chapter we discuss the properties of the vectors and vector fields that occur in classical physics. We will do so in a way, and in a notation, that leads to the formation of abstract linear vector spaces in Chapter 5. A physical quantity that is completely specified, in appropriate units, by a single number (called its magnitude) such as volume, mass, and temperature is called a scalar. Scalar quantities are treated as ordinary real numbers. They obey all the regular rules of algebraic addition, subtraction, multiplication, division, and so on. There are also physical quantities which require a magnitude and a direction for their complete specification. These are called vectors if their combination with each other is commutative (that is the order of addition may be changed without a€ecting the result). Thus not all quantities possessing magnitude and direction are vectors. Angular displacement, for example, may be characterised by magni-tude and direction but is not a vector, for the addition of two or more angular displacements is not, in general, commutative (Fig. 1.1). In print, we shall denote vectors by boldface letters (such as A ) and use ordin-ary italic letters (such as A ) for their magnitudes; in writing, vectors are usually represented by a letter with an arrow above it such as ~ A . A given vector A (or ~ A ) can be written as A ˆ A ^ A ; 1 : 1 † where A is the magnitude of vector A and so it has unit and dimension, and ^ A is a dimensionless unit vector with a unity magnitude having the direction of A . Thus ^ A ˆ A = A . 1 A vector quantity may be represented graphically by an arrow-tipped line seg-ment. The length of the arrow represents the magnitude of the vector, and the direction of the arrow is that of the vector, as shown in Fig.

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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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  Elements of Theoretical Mechanics for Electronic Engineers

  International Series of Monographs in Electronics and Instrumentation

  • Franz Bultot(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  6 Scalar and Vector Fields 6.1. Scalar and Vector Fields-DEFINITIONS A scalar field is a space, all the points of which are characterised by numerical values (a temperature field, for example). A vector field is a space, all the points of which are origins of vectors (a field of force, for example). The function characterising the field is called the field function. 6.2. GRADIENT OF A SCALAR FIELD FUNCTION Let a scalar field be referred to a set of axes Oxyz and let U(x, y, z) be the field function. All the points for which the field function takes the same value C are situated on a surface of equation U(x, y, z) = C. If different values are given to C, a family of surfaces is obtained called level surfaces of the scalar field (the isothermal surfaces of a temperature field for example). The increase d U of the field function U (x, y, z) when we pass from point P (r), where r = c1 x + y1 & -}-z1 2» to another infinitely near arbitrary point Q (r -{- dr), with r ± dr = (c + dx) I +(y+dy)1 u +(z+dz)1,, FIG. 6.1 is expressed (Fig. 6.1) d U= ax dx -f- ay dy -~~ dz dz. (6.1) 199 200 ELEMENTS OF THEORETICAL MECHANICS The right-hand side of (6.1) can be written in the form of a scalar product, namely dU=( az 1 x-{ au au az 1 z ).(dx1 x -}-dy I y -{-dz1.) (6.2) y where dx 1 x + dy 1, dz 1, = dr. The first factor depends only on point P (not on Q); it is called the gradient of the field at point P. Thus we have -E + u = grad (6.3) au 1 x ~y 1, 1 z U. The relation (6.2) can therefore be written dU= grad U .dr. (6.4) If it is agreed to indicate by V (nabla) the vector operator V i x ± ó y l y + a ax (6.5) relation (6.4) takes the condensed form dU =VU.dr =dr.VU. (6.6) As there is a gradient vector V U at each point of the scalar field these gradients constitute a vector field. If the infinitesimal displacement dr is taken on a level surface U = C, we have d U = 0 and, consequently, (from (6.6)), V U.

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  Mathematical Methods and Physical Insights

  An Integrated Approach

  • Alec J. Schramm(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  Part II The Calculus of Vector Fields Mathematics is the music of reason. James Joseph Sylvester 11 Prelude: Visualizing Vector Fields One of the most fundamental concepts of mathematical physics is the vector. Whether in the perhaps mundane but powerful second law of Newton; as the abstract fields of gravity, electricity, and magnetism; or the complex and often perplexing wavefunction of quantum mechanics, vectors appear innumerable times in mathematics, physics, and engineering. Over the course of this book, we will discuss different mathematical manifestations and representations of vectors, including vector-valued functions of space and time, column vectors obeying the rules of matrix manipulation, and even functions such as Gaussians, which can be regarded as vectors in an infinite-dimensional space. In this Part, we explore the calculus of vector fields in the familiar confines of R 3 (good physics references include Griffiths, 2017 and Jackson, 1998). Unlike a scalar field f ( r), which assigns to every point a number (say, the mass or temperature at  r), a vector-valued field  F( r) assigns both a number and a direction. These numbers and directions can be easily depicted by drawing arrows from representative points. Example 11.1  F(x, y , z) = ˆ ı x + ˆ j y + ˆ kz Allowing the relative lengths of the arrows to repre- sent the field strength at each point, we easily appre- ciate that this vector field explodes out from the ori- gin. Though  F may appear to be algebraically three- dimensional, because of its clear spherical symmetry it is essentially a one-dimensional field — as can be demonstrated by changing to spherical coordinates,  F(x, y , z) = ˆ ı x + ˆ j y + ˆ kz =  r = r ˆ r, (11.1) where ˆ r ≡  r/| r|. So this field has only a radial com- ponent. Moreover, in this form it’s obvious that the magnitude of  F depends not on separate values of x, y, and z, but only on the distance r from the origin.

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  eBook - ePub

  Mathematics for Engineers and Scientists

  • Alan Jeffrey(Author)
  • 2004(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  ϕ of position will define a scalar field within its domain of definition. A typical physical example of a scalar field is provided by the temperature at each point of a body.

  Similarly, if F is a vector function of position, we say that F defines a vector field throughout its domain of definition in the sense that it assigns a specific vector to each point. Thus the vector function F = (sinx )i + xy j +y ez k defines a vector field throughout all space.

  As heat flows in the direction of decreasing temperature, it follows that associated with the scalar temperature field within a body there must also be a vector field which assigns to each point a vector describing the direction and maximum rate of flow of heat. Other physical examples of vector fields are provided by the velocity field v throughout a fluid, and the magnetic field H throughout a region.

  To examine more closely the nature of a scalar field, and to see one way in which a special type of vector field arises, we must now define what is called the gradient of a scalar function. This is a vector differentiation operation that associates a vector field with every continuously differentiable scalar function.

  Definition 11.5 (gradient of scalar function)

  If the scalar function ϕ (x ,y , z ) is a continuously differentiable function with respect to the independent variables x , y and z , then the gradient of ϕ , written erad ϕ , is defined to be the vector

  grad   ϕ =

  ∂ ϕ

  ∂ x

  i +

  ∂ ϕ

  ∂ y

  j +

  ∂ ϕ

  ∂ z

  k .

  For the moment let it be understood that r = x i +y j +z k is a specific point, and consider a displacement from it, dr = dx i +dy j +dz k . Then it follows from the definition of grad ϕ that

  d r . grad   ϕ =

  ∂ ϕ

  ∂ x

  d x +

  ∂ ϕ

  ∂ y

  d y +

  ∂ ϕ

  ∂ z

  d z ,

  in which it is supposed that grad ϕ is evaluated at r = x i +y j +z k . Theorem 5.19 then asserts that the right-hand side of this expression is simply the total differential dϕ of the scalar function ϕ

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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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  eBook - ePub

  Vector Analysis and Cartesian Tensors, Third edition

  • P C Kendall, D.E. Bourne(Authors)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  x, y, z ) relative to the origin, the notation

  may also be used to signify that Ω , F are functions of position.

  Simple examples of scalar fields are

  and

  Examples of vector fields are

  and

  The scalar field (4.5 ) and the vector field (4.7 ) are defined over the whole of space. The scalar field (4.6 ) is defined at all points except those lying on the plane x = 0, and the vector field (4.8 ) is defined only at points lying inside or on the sphere

  Scalar and Vector Fields arise naturally in a variety of physical situations. For example, when a gas flows along a pipe there are associated with any point in the pipe the gas pressure p , the density ρ , and the velocity v at that point. Thus p and ρ are scalar fields and visa vector field associated with the motion.

  4.4 Gradient of a Scalar Field

  If the scalar field Ω (x, y, z ) is defined and continuously difierentiable in some open region , then the gradient of Ω is defined as

  The gradient of Ω is a vector field on

  Proof

  At every point of , grad Ω clearly satisfies condition 1. of the definition of a vector given in section 2.2 . To complete the proof we must establish (a) that the components are invariant under a translation of the coordinate axes and (b) that the components transform according to the vector law (see equations (2.1 )) under a rotation of the axes.

  (a) Consider a translation to new coordinate axes O′XYZ such that the coordinates (X, Y, Z ) are related to the original coordinates (x, y, z ) by the equations

  where a, b, c are constants. By the chain rule (equations (4.3

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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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  Gravitation

  Foundations and Frontiers

  • T. Padmanabhan(Author)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Scalar and electromagnetic fields in special relativity 2.1 Introduction This chapter develops the ideas of classical field theory in the context of spe-cial relativity. We use a scalar field and the electromagnetic field as examples of classical fields. The discussion of scalar field theory will allow us to under-stand concepts that are unique to field theory in a somewhat simpler context than electromagnetism; it will also be useful later on in the study of topics such as infla-tion, quantum field theory in curved spacetime, etc. As regards electromagnetism, we concentrate on those topics that will have direct relevance in the development of similar ideas in gravity (gauge invariance, Hamilton–Jacobi theory for particle motion, radiation and radiation reaction, etc.). The ideas developed here will be used in the next chapter to understand why a field theory of gravity – developed along similar lines – runs into difficulties. The concept of an action principle for a field will be extensively used in Chapter 6 in the context of gravity. Other topics will prove to be valuable in studying the effect of gravity on different physical systems. 1 2.2 External fields of force In non-relativistic mechanics, the effect of an external force field on a particle can be incorporated by adding to the Lagrangian the term − V ( t, x ) , thereby adding to the action the integral of − V dt . Such a modification is, however, not Lorentz invariant and hence cannot be used in a relativistic theory. Our first task is to determine the form of interactions which are permitted by the Lorentz invariance. The action for the free particle was the integral of dτ (see Eq. (1.72) ), which is Lorentz invariant. We can modify this expression to the form A = − L ( x a , u a ) dτ, (2.1) 54 2.3 Classical scalar field 55 where L ( x a , u a ) is a Lorentz invariant scalar dependent on the position and veloc-ity of the particle, and still maintain Lorentz invariance.

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