Vector Calculus

6 Key excerpts on "Vector Calculus"

• 

  eBook - ePub

  Mathematics for Physicists

  • Brian R. Martin, Graham Shaw(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  12 Vector Calculus

  In Chapter 8 we introduced the idea of a vector as a quantity with both magnitude and direction and we discussed vector algebra, particularly as applied to analytical geometry, and the differentiation and integration of vectors with respect to a scalar parameter. In this chapter we extend our discussion to include directional derivatives and integration over variables that are themselves vectors. This topic is called Vector Calculus or vector analysis. It plays a central role in many areas of physics, including fluid mechanics, electromagnetism and potential theory.

  12.1 Scalar and vector fields

  If scalars and vectors can be defined as continuous functions of position throughout a region of space, they are referred to as fields and the region of space in which they are defined is called a domain. An example of a scalar field would be the distribution of temperature T within a fluid. At each point the temperature is represented by a scalar field T(r) whose value depends on the position r at which it is measured. A useful concept when discussing scalar fields is that of an equipotential surface, that is, a surface joining points of equal value. This is somewhat analogous to the contour lines on a two-dimensional map, which join points of equal height. An example of a vector field is the distribution of velocity v(r) in a fluid. At every point r, the velocity is represented by a vector of definite magnitude and direction, both of which can change continuously throughout the domain. In this case, we can define flow lines such that the tangent to a flow line at any point gives the direction of the vector at that point. Flow lines cannot intersect. This is illustrated in Figure 12.1 .

  Figure 12.1

  The motion of a fluid around a smooth solid. The coloured lines are the flow lines and the arrows show the direction of the vector field, in this case the velocity v(r

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  eBook - PDF

  Calculus

  Concepts and Contexts, Enhanced Edition

  • James Stewart(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  905 Vector Calculus In this chapter we study the calculus of vector fields. (These are functions that assign vectors to points in space.) In particular we define line integrals (which can be used to find the work done by a force field in moving an object along a curve). Then we define surface integrals (which can be used to find the rate of fluid flow across a surface). The connections between these new types of integrals and the single, double, and triple integrals that we have already met are given by the higher-dimensional versions of the Fundamental Theorem of Calculus: Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. 13 Jorg Hackemann/Shutterstock.com Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 906 CHAPTER 13 Vector Calculus The vectors in Figure 1 are air velocity vectors that indicate the wind speed and direction at points 10 m above the surface elevation in the San Francisco Bay area. (Notice that the wind patterns on consecutive days are quite different.) Associated with every point in the air we can imagine a wind velocity vector. This is an example of a velocity vector field. Other examples of velocity vector fields are illustrated in Figure 2: ocean currents and flow past an airfoil. Another type of vector field, called a force field , associates a force vector with each point in a region. An example is the gravitational force field that we will look at in Example 4.

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  Available until 16 Feb |Learn more

  Advanced Engineering Mathematics with MATLAB

  • Dean G. Duffy(Author)
  • 2021(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Vector Calculus

  Physicists invented vectors and vector operations to facilitate their mathematical expression of such diverse topics as mechanics and electromagnetism. In this chapter we focus on multivariable differentiations and integrations of vector fields, such as the velocity of a fluid, where the vector field is solely a function of its position.

  4.1 REVIEW

  The physical sciences and engineering abound with vectors and scalars. Scalars are physical quantities that only possess magnitude. Examples include mass, temperature, density, and pressure. Vectors are physical quantities that possess both magnitude and direction. Examples include velocity, acceleration, and force. We shall denote vectors by boldfaced letters.

  Two vectors are equal if they have the same magnitude and direction. From the limitless number of possible vectors, two special cases are the zero vector 0, which has no magnitude and unspecified direction, and the unit vector, which has unit magnitude.

  The most convenient method for expressing a vector analytically is in terms of its components. A vector a in three-dimensional real space is any order triplet of real numbers (components) al , a2 , and a3 such that a = a1 i a2 j + a3 k, where a1 i, a2 j, and a3 k are vectors that lie along the coordinate axes and have their origin at a common initial point. The magnitude, length, or norm of a vector a, |a|, equals

  a 1 2

  +

  a 2 2

  +

  a 3 2

  .

  A particularly important vector is the position vector, defined by r = xi + yj+zk.

  As in the case of scalars, certain arithmetic rules hold. Addition and subtraction are very similar to their scalar counterparts:

  a + b =

  (

  a 1

  +

  b 1

  )

  i +

  (

  a 2

  +

  b 2

  )

  j +

  (

  a 3

  +

  b 3

  )

  k ,

  (4.1.1)

  and

  a − b =

  (

  a 1

  −

  b 1

  )

  i +

  (

  a 2

  −

  b 2

  )

  j +

  (

  a 3

  −

  b 3

  )

  k .

  (4.1.2)

  In contrast to its scalar counterpart, there are two types of multiplication. The dot product

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  eBook - ePub

  Applied Engineering Analysis

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

    (Publisher)

  Chapter 3 Vectors and Vector Calculus

  Chapter Learning Objectives

  • Recap the distinction between scalar and vector quantities in engineering analysis.
  • Learn Vector Calculus and its applications in engineering analysis.
  • Learn to manipulate expressions of vectors and vector functions.
  • Refresh vector algebra.
  • Learn the dot and cross products of vectors and their physical meanings.
  • Learn about derivatives, gradient, divergence, and curl in Vector Calculus.
  • Learn to apply Vector Calculus in engineering analysis.
  • Learn to apply Vector Calculus in rigid body dynamics in rectilinear and plane curvilinear motion along paths and in both rectangular and cylindrical polar coordinate systems.

  3.1 Vector and Scalar Quantities

  In Section 2.2.3 , we introduced functions that represent physical quantities in engineering analyses, and whose values vary with the values of the associated independent variables in space (x, y, z) in a rectangular coordinate system) and time (t). These quantities are called as scalar quantities.

  There is another group of physical quantities for which not only the magnitude but also the position and the direction are significant and must be represented.. These are called vector quantities. Thus, a speed of 80 km/h of a moving car is a scalar quantity, but a velocity of 80 km/h implies that the car is traveling in specific direction on the road at this speed, so that it is a vector quantity. Engineering analyses involving vectorial quantities will require the use of Vector Calculus, which will be described later in Section 3.5 .

  A vector, as stated, is characterized by both its magnitude and its direction

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  eBook - PDF

  Calculus

  Late Transcendental

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  971 15 Results in this chapter provide tools for analyzing and understanding the behavior of hurricanes and other fluid flows. TOPICS IN Vector Calculus We begin this chapter by introducing the concept of a vector field, an important tool for the study of gravitational and electrostatic force fields, the flow of fluids, and conservation of energy. Next, we will introduce the “line integral,” a new type of integral with a variety of applications to the analysis of vector fields. Finally, we conclude with three major theorems, Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem. These theorems provide deep insight into the nature of vector fields and are the basis for many of the most important principles in physics and engineering. 15.1 VECTOR FIELDS In this section we will consider functions that associate vectors with points in 2-space or 3-space. We will see that such functions play an important role in the study of fluid flow, gravitational force fields, electromagnetic force fields, and a wide range of other applied problems. VECTOR FIELDS Consider a unit point-mass located at any point in the Universe. According to Newton’s Law of Universal Gravitation, the Earth exerts an attractive force on the mass that is di- rected toward the center of the Earth and has a magnitude that is inversely proportional to the square of the distance from the mass to the Earth’s center (Figure 15.1.1). This associa- tion of force vectors with points in space is called the Earth’s gravitational field. A similar Figure 15.1.1 association occurs in fluid flow. Imagine a stream in which the water flows horizontally at every level, and consider the layer of water at a specific depth. At each point of the layer, the water has a certain velocity, which we can represent by a vector at that point (Figure 15.1.2). This association of velocity vectors with points in the two-dimensional layer is called the velocity field at that layer.

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  eBook - PDF

  Classical Dynamics of Particles and Systems

  • Jerry B. Marion(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R 2 Vector Calculus 2.1 Introduction The application of vector methods to physical problems most frequently takes the form of differential operations. The rate of change of a vector function with respect to the spatial coordinates or with respect to the time are of particular importance. Such operations allow us, for example, to define the velocity vector of the motion of a particle or to describe the flow properties of a fluid. In this chapter we begin by defining the elemen-tary differential operations which immediately allow us to calculate the velocity and acceleration vectors in the commonly used coordinate systems. Angular velocity is considered next and this leads to a discussion of infinitesimal rotations. Treated next is the important differential operator, the gradient. The fact that the gradient operator may act on vector functions in different ways, leads finally to the divergence and the curl. The chapter concludes with a brief discussion of the simple integral concepts that are necessary in mechanics. 32 2.2 DIFFERENTIATION OF A VECTOR WITH RESPECT TO A SCALAR 33 2.2 Differentiation of a Vector with Respect to a Scalar If a scalar function φ = cp(s) is differentiated with respect to the scalar variable s, then since neither part of the derivative can change under a coordinate transformation, the derivative itself cannot change and must therefore be a scalar. That is, in the x f and x coordinate systems, φ = φ' and s — s', so that άφ = άφ' and ds = ds'. Hence, dcp/ds = d(p'/ds' = (άφ/ds)' (2.1) In a similar manner, we may formally define the differentiation of a vector A with respect to a scalar s.

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